Solution Manual for Data Science For All, 1st edition by Brennan Davis Hunter Glanz

INSTRUCTOR SOLUTIONS MANUAL ADAM POLIAK Bryn Mawr College

D ATA S CIENCE FOR A LL Brennan Davis California Polytechnic State University

Hunter Glanz California Polytechnic State University

Data Science for All

Table of Contents Chapter 1: What Is Data Science? .............................................................................................. xii Chapter Review Questions ...................................................................................................... xii Section 1. Introduction to Data Science ................................................................................... xii Section 2: Data in Tables ....................................................................................................... xii Section 3: Data Preparation..................................................................................................... 2 Section 4: Data Analysis and Storytelling ................................................................................. 2 Section 5: Data Science in Society and Industry ........................................................................ 3 Explaining the Concepts .......................................................................................................... 4 Try It Yourself ........................................................................................................................ 4 Try It Yourself: Identify Variable Uses and Types (1.2) ............................................................. 4 Try It Yourself: Tidy Messy Data (1.2) .................................................................................... 5 Try It Yourself: Tidy Data for Better Analysis (1.2) ................................................................... 5 Try It Yourself: Investigate Metadata (1.2) ............................................................................... 6 Try It Yourself: Illustrate Unstructured Data Using Structured Tables (1.3) .................................. 6 Try It Yourself: Identify Analysis Types (1.4) ........................................................................... 6 Try It Yourself: Optimize the Narrative (1.4) ............................................................................ 7 Applying the Concepts............................................................................................................. 7 Activity: Identify Variable Uses and Types ............................................................................... 7 Activity: Tidy Messy Data ...................................................................................................... 7 Activity: Tidy Data for Better Analysis .................................................................................... 8 Activity: Investigate Metadata................................................................................................. 9 Activity: Illustrate Unstructured Data Using Structured Tables ................................................... 9 Activity: Optimize the Narrative ............................................................................................. 9 Activity: Identify Analysis Types ............................................................................................ 9 Chapter 2: Data Wrangling: Preprocessing .................................................................................10 Chapter Review Questions ......................................................................................................10 Section 1: What Is Data Wrangling? .......................................................................................10 Section 2: Cleaning Missing Data ...........................................................................................10 Section 3: Cleaning Anomalous Values ...................................................................................11 Section 4: Transforming Quantitative Variables .......................................................................11 Section 5: Transforming Categorical Variables ........................................................................16 iii © 2026 Pearson Education, Inc.

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Section 6: Reshaping a Dataset...............................................................................................19 Section 7: Combining Datasets...............................................................................................19 Explaining the Concepts .........................................................................................................22 Try It Yourself .......................................................................................................................22 Try It Yourself: Identifying Missing Data (2.2) ........................................................................22 Try It Yourself: Impute Missing Values from Internal Data (2.2)................................................23 Try It Yourself: Identify Extreme Data Values (2.3) .................................................................23 Try It Yourself: Clean Incorrect Formats (2.3) .........................................................................24 Try It Yourself: Create a Categorical Variable from a Quantitative Variable (2.4) ........................24 Try It Yourself: Create a Boolean Variable that Represents a Categorical Variable (2.5) ...............25 Try It Yourself: Splitting a Variable into Multiple Variables (2.5) ..............................................25 Applying the Concepts............................................................................................................26 Activity: Identify Missing Data ..............................................................................................26 Activity: Impute Missing Values from Internal Data .................................................................26 Activity: Identify Extreme Data Values ...................................................................................26 Activity: Clean Incorrect Formats ...........................................................................................27 Activity: Create a Categorical Variable from a Quantitative Variable ..........................................27 Activity: Create a Boolean Variable that Represents a Categorical Variable.................................27 Activity: Splitting a Variable into Multiple Variables................................................................28 Chapter 3: Making Sense of Data Through Visualization ............................................................28 Chapter Review Questions ......................................................................................................28 Section 1: The Grammar of Graphics ......................................................................................28 Section 2: Visualizations with One Quantitative Variable ..........................................................29 Section 3: Visualizations with One Categorical Variable ...........................................................30 Section 4: Visualizations with Two Variables ..........................................................................30 Section 5: Visualizations with Three or More Variables ............................................................32 Section 6: The Dangers of Visual Misrepresentation .................................................................33 Section 7: Data Visualization Guidelines .................................................................................33 Explaining the Concepts .........................................................................................................34 Try It Yourself .......................................................................................................................35 Try It Yourself: Interpret a Histogram (3.2) .............................................................................35 Try It Yourself: Describe a Density Plot (3.2) ..........................................................................35 Try It Yourself: Work with a Box Plot (3.2).............................................................................36 iv © 2026 Pearson Education, Inc.

Data Science for All

Try It Yourself: Interpret a Violin Plot (3.2).............................................................................36 Try It Yourself: Ponder a Word Cloud (3.3) .............................................................................37 Try It Yourself: Interpret a Column Chart (3.4) ........................................................................37 Try It Yourself: Navigate a Treemap (3.4) ...............................................................................37 Try It Yourself: Contemplate a Box Plot with a Categorical Variable (3.4) ..................................38 Try It Yourself: Interpret a Violin Plot with a Categorical Variable (3.4) .....................................38 Try It Yourself: Interpret a Scatterplot (3.4) .............................................................................38 Try It Yourself: Explore a Line Chart (3.4) ..............................................................................39 Try It Yourself: Describe a Bubble Chart (3.5) ........................................................................39 Try It Yourself: Interpret a Heatmap (3.5) ...............................................................................40 Try It Yourself: Interpret a Geographic Map (3.5) ....................................................................40 Try It Yourself: Navigate the Dangers of Cherry-Picking (3.6)...................................................40 Applying the Concepts............................................................................................................41 Activity: Interpret a Histogram ...............................................................................................41 Activity: Describe a Density Plot............................................................................................42 Activity: Work with a Box Plot ..............................................................................................43 Activity: Interpret a Violin Plot ..............................................................................................44 Activity: Ponder a Word Cloud ..............................................................................................45 Activity: Interpret a Column Chart .........................................................................................45 Activity: Navigate a Treemap ................................................................................................46 Activity: Contemplate a Box Plot with a Categorical Variable ...................................................47 Activity: Interpret a Violin Plot with a Categorical Variable ......................................................48 Activity: Interpret a Scatterplot ..............................................................................................49 Activity: Explore a Line Chart ...............................................................................................50 Activity: Describe a Bubble Chart ..........................................................................................51 Activity: Interpret a Heatmap .................................................................................................52 Activity: Interpret a Geographic Map ......................................................................................53 Activity: Navigate the Dangers of Cherry-Picking ....................................................................53 Chapter 4: Exploratory Data Analysis ........................................................................................54 Chapter Review Questions ......................................................................................................54 Section 1: Central Tendency ..................................................................................................54 Section 2: Variability ............................................................................................................54 Section 3: Shape ...................................................................................................................55 v © 2026 Pearson Education, Inc.

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Section 4. Resistant Central Tendency and Variability ..............................................................56 Section 5. Data Associations ..................................................................................................57 Section 6. Identifying Outliers................................................................................................58 Explaining the Concepts .........................................................................................................60 Try It Yourself .......................................................................................................................64 Try It Yourself: Calculate the Mean, Mode, Median, and 90% Trimmed Average (4.1) .................64 Try It Yourself: Calculate the Range, Variance, and Standard Deviation (4.2) ..............................65 Try It Yourself: Describe the Shape of the Histogram (4.3)........................................................65 Try It Yourself: Investigate Resistance (4.4) ............................................................................65 Try It Yourself: Plot Correlations between Two Quantitative Variables (4.5) ...............................65 Try It Yourself: Match Box Plots with Histograms (4.5)............................................................65 Try It Yourself: Associate Two Categorical Variables (4.5) .......................................................66 Try It Yourself: Determine Outliers Using the Quartile and Mean/SD Methods (4.6) ....................66 Applying the Concepts............................................................................................................67 Activity: Calculate the Mean, Mode, Median, and 90% Trimmed Average ..................................67 Activity: Calculate the Range, Variance, and Standard Deviation ...............................................67 Activity: Describe the Shape of the Histogram .........................................................................67 Activity: Investigate Resistance..............................................................................................68 Activity: Plot Correlations between Two Quantitative Variables ................................................69 Activity: Match Box Plots with Histograms .............................................................................70 Activity: Associate Two Categorical Variables ........................................................................71 Activity: Determine Outliers Using the Quartile and Mean/SD Methods .....................................71 Chapter 5: Data Management ....................................................................................................72 Chapter Review Questions ......................................................................................................72 Section 1: Asking Questions of Data .......................................................................................72 Section 2: Selecting Variables ................................................................................................73 Section 3: Filtering and Ordering Observations ........................................................................73 Section 4: Summarizing and Structuring Data ..........................................................................74 Section 5. Merging Tables .....................................................................................................75 Explaining the Concepts .........................................................................................................76 Try It Yourself .......................................................................................................................77 Try It Yourself: Select Variables (5.2).....................................................................................77 Try It Yourself: Select Distinct Values of Variables (5.2) ..........................................................77 vi © 2026 Pearson Education, Inc.

Data Science for All

Try It Yourself: Filter Text Variables (5.3) ..............................................................................77 Try It Yourself: Filter with the Not-Equal Variable (5.3) ...........................................................77 Try It Yourself: Filter Using a Wildcard (5.3) ..........................................................................77 Try It Yourself: Filter a Quantitative Variable (5.3) ..................................................................78 Try It Yourself: Filter Missing Values Variable (5.3) ................................................................78 Try It Yourself: Filter Multiple Conditions Simultaneously (5.3) ................................................78 Try It Yourself: Order by a Variable (5.3) ...............................................................................78 Try It Yourself: Summarize a Quantitative Variable (5.4) ..........................................................78 Try It Yourself: Summarize a Quantitative Variable by a Categorical Variable (5.4) .....................78 Try It Yourself: Create a New Variable (5.4) ...........................................................................78 Try It Yourself: Merge Two Tables (5.5) .................................................................................78 Applying the Concepts............................................................................................................79 Activity: Select Variables ......................................................................................................79 Activity: Select Distinct Values of Variables ...........................................................................79 Activity: Filter Text Variables................................................................................................79 Activity: Filter with the Not-Equal Operator ............................................................................79 Activity: Filter Using a Wildcard............................................................................................79 Activity: Filter a Quantitative Variable....................................................................................79 Activity: Filter Missing Values ..............................................................................................79 Activity: Filter Multiple Conditions Simultaneously .................................................................79 Activity: Order by a Variable .................................................................................................79 Activity: Summarize a Quantitative Variable ...........................................................................80 Activity: Summarize a Quantitative Variable by a Categorical Variable ......................................80 Activity: Create a New Variable .............................................................................................80 Activity: Merge Two Tables ..................................................................................................80 Chapter 6: Understanding Uncertainty, Probability, and Variability ...........................................80 Chapter Review Questions ......................................................................................................80 Section 1: Variability and Uncertainty.....................................................................................80 Section 2: Probability............................................................................................................81 Section 3: Sampling Methods.................................................................................................82 Section 4: Simulation ............................................................................................................83 Section 5: Working with Probabilities and Common Fallacies....................................................84 Explaining the Concepts .........................................................................................................85 vii © 2026 Pearson Education, Inc.

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Try It Yourself .......................................................................................................................85 Try It Yourself: Assess Variability in Histograms (6.1) .............................................................85 Try It Yourself: Relate Variability to Uncertainty and Predictability (6.1) ...................................86 Try It Yourself: Estimating Likelihood of Values in Data (6.1) ..................................................86 Try It Yourself: Predictability from Histograms (6.1) ................................................................87 Try It Yourself: Predictability with Two Variables (6.1) ............................................................87 Try It Yourself: Describe Outcomes in a Sample Space (6.2) .....................................................87 Try It Yourself: Values of Variables as Samples Spaces (6.2) ....................................................87 Try It Yourself: Estimation from Different Sampling Methods (6.3) ...........................................87 Try It Yourself: Simulate Flipping a Loaded Coin (6.4).............................................................88 Try It Yourself: Simulate Samples (6.4) ..................................................................................88 Applying the Concepts............................................................................................................88 Activity: Assess Variability in Histograms ..............................................................................88 Activity: Relate Variability to Uncertainty and Predictability .....................................................89 Activity: Estimating Likelihood of Values in Data ....................................................................90 Activity: Predictability from Histograms .................................................................................91 Activity: Predictability with Two Variables .............................................................................91 Activity: Describe Outcomes in a Sample Space ......................................................................93 Activity: Values of Variables as Sample Spaces .......................................................................93 Activity: Estimation from Different Sampling Methods .............................................................93 Activity: Simulate Flipping a Loaded Coin (Roulette Wheel) .....................................................94 Activity: Simulate Samples ....................................................................................................96 Chapter 7: Drawing Conclusions from Data................................................................................98 Chapter Review Questions ......................................................................................................98 Section 1: Introduction to Statistical Inference .........................................................................98 Section 2: Data Collection and Study Design ...........................................................................98 Section 3: The Language of Statistical Inference ......................................................................99 Section 4: Exploratory Data Analysis to Begin Inference .........................................................100 Section 5: Drawing Conclusions in an Observational Study .....................................................100 Section 6: A/B Testing as a Case of Experiments ...................................................................100 Explaining the Concepts .......................................................................................................102 Try it Yourself .....................................................................................................................102 Try It Yourself: Estimate the Probability of an Event (7.3) ......................................................102 viii © 2026 Pearson Education, Inc.

Data Science for All

Try It Yourself: Make a Statistical Inference (7.3) ..................................................................103 Try It Yourself: Evaluate a Hypothesis (7.5) ..........................................................................103 Try It Yourself: Perform a t-Test on Observational Data (7.5) ..................................................103 Try It Yourself: Perform a t-Test Using a Different Alternative Hypothesis (7.5) .......................103 Try It Yourself: Draw Conclusions from a Correlation (7.5) ....................................................104 Try It Yourself: Identify Type I and II Errors (7.5) .................................................................104 Try It Yourself: Identify an A/B Tests’ Null and Alternative Hypotheses (7.6) ...........................105 Try It Yourself: Perform t-Tests on A/B Testing Data (7.6) .....................................................105 Try It Yourself: Identify Type I and Type II errors in A/B Testing Results (7.6) .........................105 Applying the Concepts..........................................................................................................105 Activity: Estimate the Probability of an Event ........................................................................105 Activity: Make a Statistical Inference....................................................................................106 Activity: Evaluate a Hypothesis ...........................................................................................107 Activity: Perform a t-Test on Observational Data ...................................................................107 Activity: Draw Conclusions from a Correlation ......................................................................107 Activity: Identify Type I and Type II Errors ...........................................................................108 Activity: Identify an A/B Test’s Null and Alternative Hypotheses ............................................108 Activity: Perform t-Tests on A/B Testing Data .......................................................................108 Activity: Identify Type I and Type II Errors in A/B Testing Results..........................................109 Chapter 8: Machine Learning ..................................................................................................109 Chapter Review Questions ....................................................................................................109 Section 1: Artificial Intelligence ...........................................................................................109 Section 2: Three Steps in the Machine Learning Process .........................................................109 Section 3: Characteristics of Machine Learning Method ..........................................................110 Section 4: Machine Learning Method Evaluation ...................................................................111 Section 5: Deep Learning ....................................................................................................113 Section 6: Use High-Quality Data in Machine Learning ..........................................................114 Explaining the Concepts .......................................................................................................114 Try it Yourself .....................................................................................................................114 Try It Yourself: Identify Artificial Intelligence (8.1) ...............................................................114 Try It Yourself: Identify Data Ingestion, Data Analysis, and Results (8.2) .................................114 Try It Yourself: Reading a Decision Tree (8.3) .......................................................................115 Try It Yourself: Regression versus Classification (8.3) ............................................................115 ix © 2026 Pearson Education, Inc.

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Try It Yourself: Inference versus Prediction (8.3) ...................................................................115 Try It Yourself: Interpretability versus Complexity (8.3) .........................................................115 Try It Yourself: Calculate Accuracy, Precision, Recall, and Specificity Using a Confusion Matrix (8.4)..................................................................................................................................116 Try It Yourself: Overfitted, Underfitted, and Well-Fitted Models (8.4)......................................116 Applying the Concepts..........................................................................................................117 Activity: Identify Artificial Intelligence ................................................................................117 Activity: Identify Data Ingestion, Data Analysis and Results ...................................................117 Activity: Reading a Decision Tree ........................................................................................118 Activity: Regression versus Classification .............................................................................119 Activity: Inference versus Prediction ....................................................................................119 Activity: Interpretability versus Complexity ..........................................................................119 Activity: Calculate Accuracy, Precision, Recall, and Specificity Using a Confusion Matrix .........122 Activity: Overfitted, Underfitted, and Well-Fitted Models .......................................................122 Chapter 9: Supervised Learning ...............................................................................................123 Chapter Review Questions ....................................................................................................123 Section 1: Linear Regression with No Explanatory Variables ...................................................123 Section 2: Linear Regression with a Categorical Explanatory Variable ......................................123 Section 3: Linear Regression with a Quantitative Explanatory Variable.....................................124 Section 4: Multiple Linear Regression...................................................................................124 Section 5: Nonparametric Regression Models ........................................................................125 Section 6: Classification Models ..........................................................................................126 Explaining the Concepts .......................................................................................................127 Try it Yourself .....................................................................................................................130 Try It Yourself: Perform a Two-Sample t-Test (9.2) ...............................................................130 Try It Yourself: Connecting Linear Regression and t-Test (9.2)................................................130 Try It Yourself: Compute a Correlation and Fit a Regression Line (9.3) ....................................130 Try It Yourself: Extrapolation (9.3) ......................................................................................130 Try It Yourself: Exploring R-Squared in Multiple Linear Regression (9.4) ................................130 Try It Yourself: Fit, Evaluate, and Interpret Decision Trees (9.5)..............................................130 Try It Yourself: Fit, Evaluate, and Interpret Random Forests (9.5) ............................................131 Try It Yourself: Fit and Evaluate k-Nearest Neighbors (9.5) ....................................................132 Try It Yourself: Compare Regression Models (9.5) .................................................................132

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Try It Yourself: Fit, Evaluate, and Interpret Logistic Regression (9.6) .......................................132 Try It Yourself: Compare Classification Models (9.6) .............................................................133 Applying the Concepts..........................................................................................................133 Activity: Perform a Two-Sample t-Test .................................................................................133 Activity: Connecting Linear Regression and t-Tests................................................................133 Activity: Compute a Correlation and Fit a Regression Line......................................................134 Activity: Extrapolation ........................................................................................................134 Activity: Exploring R-Squared in Multiple Linear Regression ..................................................135 Activity: Fit, Evaluate, and Interpret Decision Trees ...............................................................135 Activity: Fit, Evaluate, and Interpret Random Forests .............................................................137 Activity: Fit and Evaluate k-Nearest Neighbors ......................................................................138 Activity: Compare Regression Models ..................................................................................138 Activity: Fit, Evaluate, and Interpret Logistic Regression ........................................................138 Activity: Compare Classification Models ..............................................................................139 Chapter 10: Unsupervised Learning .........................................................................................139 Chapter Review Questions ....................................................................................................139 Section 1: What Is Unsupervised Learning? ...........................................................................139 Section 2: Getting to Know Cluster Analysis .........................................................................140 Section 3: Introduction to k-Means Clustering .......................................................................141 Section 4: Introduction to Hierarchical Clustering ..................................................................142 Section 5: Assessing the Quality of Clusters ..........................................................................142 Explaining the Concepts .......................................................................................................143 Try it Yourself .....................................................................................................................143 Try It Yourself: Detect Anomalies (10.1) ..............................................................................143 Try It Yourself: Recognize Unsupervised Learning (10.1) .......................................................144 Try It Yourself: Calculate Similarity Using Distance (10.2) .....................................................144 Try It Yourself: Choose Variables for Clustering (10.3) ..........................................................145 Try It Yourself: Compare Cluster Analysis Results (10.4) .......................................................145 Try It Yourself: Examine the Elbow of a Cluster Analysis (10.5) .............................................145 Applying the Concepts..........................................................................................................145 Activity: Detect Anomalies ..................................................................................................145 Activity: Recognize Unsupervised Learning ..........................................................................146 Activity: Calculate Similarity Using Distance ........................................................................146 xi © 2026 Pearson Education, Inc.

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Activity: Choose Variables for Clustering .............................................................................146 Activity: Compare Cluster Analysis Results ..........................................................................147 Activity: Examine the Elbow of Cluster Analysis ...................................................................154

Chapter 1: What Is Data Science?

Chapter Review Questions Section 1. Introduction to Data Science 1.

What is the interdisciplinary field that uses scientific methods to extract knowledge from data?

Data science.

2.

What is the primary purpose of data analysis in data science?

To generate insights and inform decision-making.

3.

Define a population in the context of data science.

A population is the set of every and all items or individuals that are of interest for a question.

4.

What other field, often confused with data science, applies data science principles to domain-specific problems?

Data analytics.

5.

What are some of the modern challenges that have shaped data science?

Increased complexity of data and analysis methods, as well as rapidly increasing amounts of the available data.

6.

What constitutes a sample in data science?

A subset of the population.

7.

In the context of data science, what is the significance of understanding samples and populations?

To inform better decision-making in society.

8.

What combination of fields does data science encompass?

Statistics, computer science, and domain knowledge.

9.

When was the term “data science” first proposed as an alternative for “computer science”?

1974

10. What is the role of a data scientist in the context of the data science lifecycle?

To extract actionable insights from data.

11. What is the most significant reason for the importance of data and their analysis, according to the text?

To describe and draw conclusions about a population based on the data collected for a sample.

Section 2: Data in Tables 12. In data science, what are the entities about which data are recorded called? xii © 2026 Pearson Education, Inc.

Table of Contents

Activity: Choose Variables for Clustering .............................................................................146 Activity: Compare Cluster Analysis Results ..........................................................................147 Activity: Examine the Elbow of Cluster Analysis ...................................................................154

Chapter 1: What Is Data Science?

Chapter Review Questions Section 1. Introduction to Data Science 1.

What is the interdisciplinary field that uses scientific methods to extract knowledge from data?

Data science.

2.

What is the primary purpose of data analysis in data science?

To generate insights and inform decision-making.

3.

Define a population in the context of data science.

A population is the set of every and all items or individuals that are of interest for a question.

4.

What other field, often confused with data science, applies data science principles to domain-specific problems?

Data analytics.

5.

What are some of the modern challenges that have shaped data science?

Increased complexity of data and analysis methods, as well as rapidly increasing amounts of the available data.

6.

What constitutes a sample in data science?

A subset of the population.

7.

In the context of data science, what is the significance of understanding samples and populations?

To inform better decision-making in society.

8.

What combination of fields does data science encompass?

Statistics, computer science, and domain knowledge.

9.

When was the term “data science” first proposed as an alternative for “computer science”?

1974

10. What is the role of a data scientist in the context of the data science lifecycle?

To extract actionable insights from data.

11. What is the most significant reason for the importance of data and their analysis, according to the text?

To describe and draw conclusions about a population based on the data collected for a sample.

Section 2: Data in Tables 12. In data science, what are the entities about which data are recorded called? xii © 2026 Pearson Education, Inc.

Data Science for All

Observational units 13. What are the recorded characteristics of observational units called?

Variables of interest 14. What term is used to refer to the data for a single observational unit?

An observation 15. From the following set of numbers: 0.0, 2.71828, -3, 5, which one is most likely to be stored as a float variable?

2.71828 16. Which type of variable is used to designate categories?

Categorical 17. In the context of data science, what does it mean for data to be tidy?

When data is tidy, it is structured so that each column represents a distinct variable, each row corresponds to a unique observation, and each cell holds only one value.

18. What do we call the form when data are represented as a table with rows and columns?

Tabular form 19. Integers and floats are examples of what kind of variables?

Quantitative 20. What kind of variables only take on values of TRUE and FALSE, sharing aspects of both categorical and quantitative variables?

Boolean 21. What form of quantitative variables consists only of umbers with no decimal points?

Integer 22. Given the numbers 5000, -2.01, 1.3, and 1/2, which one is most likely to be stored as an integer variable?

5000 23. What form of quantitative variables consists of numbers that may have decimal points?

Float 24. Given the set of numbers 0, 2, 5, and 1.5, which is most likely to be converted to a Boolean value in a data analysis context?

0 25. What form of categorical variables contains words or characters without order or numerical utility?

Text 26. Given the numbers -1.0, 0.0, 3, and 4, which one could be stored as a float or converted to a Boolean variable?

0.0 27. Given the numbers 2.71828, -5, 3.14, and 2.01, which one is most likely to be stored as an integer in a data structure?

-5 28. What is metadata?

Data about data.

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Chapter 1: What Is Data Science?

29. When is data considered messy?

When it is not yet organized in a well-defined structure.

30. Which of the following numbers is likely to be stored as a float variable in a data set: 1, 4.44, 144, or 0?

4.44 31. When data are in tabular form, what is stored in the columns? What is stored in the rows?

The variables are stored in the columns, and the observations are stored in the rows.

32. When the rows and columns of tabular data are fully populated, and each cell of the table corresponds to a single value, we consider the dataset clean and organized. What is the term we use to describe this?

Tidy 33. Given the numbers 1243424, 1.2, -1.01, and 2.03, which is most likely to be stored as an integer variable?

1243424 34. What is the primary role of a data key or codebook?

To store metadata 35. In a tidy dataset, what does each column represent?

A variable

Section 3: Data Preparation 36. In the context of data science, what does it mean to prepare data?

Accessing, cleaning, transforming, and integrating data.

37. What is the first step in data wrangling?

Clean the data to ensure its quality.

38. For what purpose is data transformation essential?

Organizing and formatting data for efficient analysis.

39. What is the goal of integrating datasets?

To combine two or more sources for richer insights.

40. In the context of data preparation, what does data wrangling refer to?

Converting raw data into formats necessary for analysis.

41. Describe the process of data collection.

Gathering data systematically from various sources.

42. What is the significance of data quality and relevance in data collection?

To form the foundation of analysis and insights.

43. What does the term “unstructured data” mean?

Data that are not organized in predefined formats.

44. What is the process of organizing data (often into tables) called?

Data structuring.

Section 4: Data Analysis and Storytelling 45. What is descriptive analysis in data science is primarily concerned with?

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Data Science for All

Reporting what has happened.

46. What is the goal of diagnostic analytics in data science?

To explain why something might have happened.

47. What is the goal of predictive analysis in data science?

To estimate future behaviors based on past data.

48. What is the component of data storytelling that allows us to communicate statistics and analysis results in form of graphs and charts?

Visualization 49. Describe prescriptive analysis in the context of data science.

The recommendation of actions based on predictive analysis.

50. What is the purpose of communication, in the context of data science?

To make information more accessible and guide decision-making.

51. How is data visualization useful in data science?

Answers may vary. Example: Data visualization is useful because it allows the data scientist to show information concisely in a graphical format, which facilitates storytelling.

52. What is the ultimate goal of data science storytelling?

To effectively communicate analysis results to a wider audience and optimize decision-making across various fields.

53. What does effective storytelling in data science require?

The ability to replicate results based on the process description.

Section 5: Data Science in Society and Industry 54. What is the primary driver for the growth of data science in society and industry?

B. The increasing amounts of available data.

55. How does data science help businesses engage with customers?

By showing which engagement strategies are most cost-effective.

56. In the context of data science, what does the term “data deluge” refer to?

The unprecedented amount of data available.

57. According to the text, for whom are data science skills essential?

All professional roles, even those not traditionally quantitative.

58. What is the benefit of integrating data science into traditionally quantitative jobs?

Enhanced understanding of data interactions across various fields.

59. What does the message of “data science for all” emphasize?

Data science affects virtually every aspect of modern life.

60. According to the text, how can an organization’s management use data science for decision-making?

Data science can be a primary tool for addressing top technology opportunities.

61. According to the chapter, what benefits does data science offer to consumers, researchers, and companies?

Data science can enable informed consumption and production choices.

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Chapter 1: What Is Data Science?

Explaining the Concepts 1.

Answers may vary. Sample answer: Data are information that can be used to answer a question or motivate a decision.

This idea of data is different from tidy data, since tidy data is a cleaned version of data. Specifically, “Tidy data is a data structure where each column represents a distinct variable, each row corresponds to a unique observation, and each cell contains only one value” (Section 1.2).

Excel sheets are a common way to work with data on a day-to-day basis. Excel sheets are similar to tidy data in that they are often in a tabular form. But Excel sheets are not guaranteed to be tidy data since tidy data is typically cleaned data.

2.

Answers may vary. Sample answer: Almost everyone has done data science in their life. Many college student jobs use data science. For example, when working at a campus store, one might track sales, and test to see if inventory changes or how products are placed in the store lead to an increase in sales.

3.

Answers may vary. Sample answer: When analyzing a large collection of TikTok videos, characteristics, or variables, might include: the length of the video, number of views, author of the video, and additional users who are tagged in the video. Each video would be an observation. Each column would represent a specific characteristic. Each row would represent a video. A given cell in the table would record the value of the given video’s corresponding variable.

When analyzing Amazon reviews, characteristics, or variables, might include: the name of the product, an ID for the product (each product should have a unique ID), the text of the review, the rating (an integer from 1 through 5), the name of the author writing the review, a link to the product being reviewed, etc.

Each review would be an observation. Each column would represent a specific characteristic. Each row would represent a review. A given cell in the table would record the value of the given review’s variable.

Identifying the observations and variables makes the unit of data being studied clear. It will help us perform meaningful analysis, draw conclusions, and create useful visualizations of the data.

4.

Answers may vary. Sample answer: Data science enables informed consumption and production choices.

Consuming news and social media is a way to gather information. Data science also encompasses the tools needed to question the information we gather.

5.

Answers may vary. Sample answer: Some examples of how one provides data to people, companies, or organizations include: posting to social media, liking and sharing social media posts, website activity, etc.

Be more mindful about sharing and reacting to content on social media and disabling cookies on websites that track users.

Some arguments for why this is a good thing include: by providing more information/data we can see better ads. Some arguments why this is a bad thing include: arguments that oversharing our personal information can lead to theft.

Try It Yourself Try It Yourself: Identify Variable Uses and Types (1.2) • • • • • •

height: Float, quantitative age: Integer, quantitative name: Text, categorical gpa: Float, quantitative graduated: Boolean, categorical pets: Integer, quantitative 4 © 2026 Pearson Education, Inc.

Data Science for All

• • • •

birth_month: Text, categorical siblings: Integer, quantitative fulltime_student: Boolean, categorical shoe_size: Integer, quantitative

Try It Yourself: Tidy Messy Data (1.2) student_id

subject

score

1

Math

85

1

Chemistry

90

2

Physics

88

2

Math

88

3

Math

88

3

Chemistry

85

Table 1.16. Tidy table

Try It Yourself: Tidy Data for Better Analysis (1.2) student_jersey

sport

rating

endurance

1

Volleyball

87

7.11

1

Basketball

90

7.09

1

Water Polo

88

7.10

2

Volleyball

97

6.75

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Chapter 1: What Is Data Science?

2

Basketball

88

6.57

2

Water Polo

90

6.66

3

Volleyball

81

8.11

3

Basketball

89

7.75

3

Water Polo

85

8.01

4

Volleyball

97

5.98

4

Basketball

85

6.03

4

Water Polo

90

6.12

Table 1.17. The data are now tidy for Coach John’s purpose. [AS1]

Try It Yourself: Investigate Metadata (1.2) Metadata help us understand what the data mean. For example, we understand that “Survived” coded as “1” means the passenger this record represents survived, which is crucial for analyzing survival patterns in the Titanic dataset.

Try It Yourself: Illustrate Unstructured Data Using Structured Tables (1.3) 1.

2.

3.

4.

product_name, products table. review_title, reviews product_id customer_id

Try It Yourself: Identify Analysis Types (1.4) • • • •

Prescriptive: A targeted marketing campaign in regions with a higher appreciation for bakery books should increase Cake Pops book ratings by 4%.

Diagnostic: Customers who bought at least two other bakery books rated the Cake Pops book 4.71 on average, which is higher than the 4.55 average rating from customers who only bought the Cake Pops book. This indicates that a general interest in bakery played a role in its ratings.

Predictive: Based on current bakery book review trends, the average rating of the Cake Pops book is projected to decrease by 0.5 stars (half a star) in the next year.

Descriptive: The Cake Pops book has an average rating of 4.68, whereas the overall average rating for general bakery books is 4.79.

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Data Science for All

Try It Yourself: Optimize the Narrative (1.4) 1.

2.

3.

B D STAR Method: a. Scope the Data's Context: The line chart displays the average monthly Amazon ratings for the Cake Pops book over the past three years, plotted monthly on the x-axis against the rating value on the y-axis, which ranges from about 4.4 to 4.9. b. Track the Question Behind the Exploration: This chart aims to explore how the average ratings for the “Cake Pops” book fluctuate over the course of a year. c. Articulate the Results: The trend shows that the ratings generally increase throughout the year, with a noticeable uptick starting in August and reaching the peak in November. d. Respond with Next Steps: Further investigation could involve examining factors influencing the increase in ratings toward the end of the year, such as seasonal purchasing habits, book content, or promotional activities.

Applying the Concepts Activity: Identify Variable Uses and Types Part I 1.

2.

3.

The observational units are students because the measurements were taken on them.

The variables that your professor recorded about each student were the student’s estimated sleep the prior night and their final exam scores.

Presuming that the final exam score is a numbered score (not a letter score), then both variables are quantitative. The estimated sleep is a float variable because it can take on decimal values. The final exam score is an integer.

Part II Tabular format: 1.

2.

3.

4.

The observational units stored in the rows are the patient visits.

The doctor would use the measurements record during each visit to populate the rows.

The variables are stored in the columns.

These variables that should then be placed in the columns are date, blood sugar, weight, eaten in the last hour, and symptom description.

Matching: 1.

2.

3.

4.

Blood sugar: Float Weight: Integer Eaten in the last hour: Boolean Symptom description: Text

Activity: Tidy Messy Data date

blood_sugar

weight

eaten_in_the_last_hour

symptom_description

2022/08/02

75.3 mg/dL

154.3 lbs

TRUE

Dizziness, migraines

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Chapter 1: What Is Data Science?

2022/08/16

77.2 mg/dL

155.2 lbs

FALSE

Dizziness, excessive hunger

2022/08/30

74.5 mg/dL

155.1 lbs

TRUE

Low energy, lethargic

2022/09/13

73.2 mg/dL

153.2 lbs

FALSE

Migraines, lethargic

Activity: Tidy Data for Better Analysis Tidy Table Format

student_id

dish

skill_rating

preparation_time (min)

1

Pizza

78

40

1

Pasta

82

35

1

Cake

75

60

2

Pizza

85

37

2

Pasta

88

30

2

Cake

90

55

3

Pizza

65

45

3

Pasta

70

40

3

Cake

80

50

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Data Science for All

4

Pizza

90

35

4

Pasta

95

25

4

Cake

85

45

Activity: Investigate Metadata 1.

2.

3.

4.

5.

It simplifies analysis by representing survival status in a numeric format.

To track individual passenger data throughout the dataset.

Ticket class of passengers.

It dictates the appropriate methods for data handling and analysis. p_class and survived

Activity: Illustrate Unstructured Data Using Structured Tables 1.

2.

3.

4.

user_id user_name; users profile_picture_url; users user_id; posts

Activity: Optimize the Narrative 1.

2.

3.

The average number of Cake Pops book reviews is larger during the beginning and end of the year.

Answers may vary. Sample answer: More people buy Cake Pops books during this time, maybe as gifts during the holidays.

STAR: a. Scope the Data's Context: This line chart represents the average number of monthly reviews for the Cake Pops book over the past three years, with the months on the x-axis and the average review count on the y-axis, ranging from 0 to 16. b. Track the Question behind the Exploration: The visualization seeks to understand the relationship between the time of year and consumer engagement with the Cake Pops book on Amazon, potentially to gauge the impact of seasonal trends or events on its popularity. c. Articulate the Results: The data shows a significant drop in reviews after January, with a steady count throughout the year and a sharp increase in December. d. Respond with Next Steps: To comprehend the variations in review counts, it would be prudent to examine any marketing strategies or seasonal occurrences that may align with the heightened review activity observed in January and December. The uptick may be linked to the winter holidays when the book could be a popular gift or more frequently used.

Activity: Identify Analysis Types Part I 1.

2.

3.

4.

Descriptive Prescriptive Diagnostic Predictive

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Chapter 1: What Is Data Science?

Part II 1.

2.

3.

4.

5.

6.

7.

8.

9.

Data Collection Data Preparation Data Analysis Descriptive Diagnostic Storytelling Predictive Prescriptive Data science lifecycle

Chapter 2: Data Wrangling: Preprocessing Chapter Review Questions Section 1: What Is Data Wrangling?

1.

What term refers to the process of converting raw data into a tidy, organized format suitable for analysis?

Data wrangling

2.

What are the three key components of data wrangling?

Cleaning, transforming, integrating

3.

Explain the data wrangling process.

Answers may vary. Example: In data wrangling, the data scientist takes collected data and prepares it for analysis or storytelling, or both. It involves 3 stages: cleaning the data, transforming the data, and integrating the data.

4.

What is the first step in data wrangling?

Cleaning the data

5.

In the context of data wrangling, what does “integrating the data” refer to?

Combining multiple data sets

6.

What is the purpose of cleaning data in data wrangling?

To enhance the quality of the data for accurate analysis

7.

What does the term “dataframe” mean in the context of data wrangling?

A tabular data structure with named columns

Section 2: Cleaning Missing Data 8.

What are three ways of dealing with missing data values?

Removing observations with missing values; imputation from internal data; imputation from external data.

9.

What is an advantage of imputing missing values over leaving them missing?

Bigger sample size, because you do not need to remove missing values for analyses

10. What issue might arise when dropping rows with any missing values from a dataset?

Answers may vary. Example: Biased sample 11. When cleaning data, under which condition would it make most sense to remove an entire row of data?

10 © 2026 Pearson Education, Inc.

Chapter 1: What Is Data Science?

Part II 1.

2.

3.

4.

5.

6.

7.

8.

9.

Data Collection Data Preparation Data Analysis Descriptive Diagnostic Storytelling Predictive Prescriptive Data science lifecycle

Chapter 2: Data Wrangling: Preprocessing Chapter Review Questions Section 1: What Is Data Wrangling?

1.

What term refers to the process of converting raw data into a tidy, organized format suitable for analysis?

Data wrangling

2.

What are the three key components of data wrangling?

Cleaning, transforming, integrating

3.

Explain the data wrangling process.

Answers may vary. Example: In data wrangling, the data scientist takes collected data and prepares it for analysis or storytelling, or both. It involves 3 stages: cleaning the data, transforming the data, and integrating the data.

4.

What is the first step in data wrangling?

Cleaning the data

5.

In the context of data wrangling, what does “integrating the data” refer to?

Combining multiple data sets

6.

What is the purpose of cleaning data in data wrangling?

To enhance the quality of the data for accurate analysis

7.

What does the term “dataframe” mean in the context of data wrangling?

A tabular data structure with named columns

Section 2: Cleaning Missing Data 8.

What are three ways of dealing with missing data values?

Removing observations with missing values; imputation from internal data; imputation from external data.

9.

What is an advantage of imputing missing values over leaving them missing?

Bigger sample size, because you do not need to remove missing values for analyses

10. What issue might arise when dropping rows with any missing values from a dataset?

Answers may vary. Example: Biased sample 11. When cleaning data, under which condition would it make most sense to remove an entire row of data?

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Data Science for All

Duplicate observations 12. What term refers to estimating missing values based on other available information?

Imputing 13. Describe a potential issue that arises from imputing missing data values.

Answers may vary. Example: Influencing analyses with artificial data 14. What is the mean GPA, ignoring only the missing GPA value?

3.22 15. What is the mean GPA, ignoring any row with a missing value?

3.25 16. Impute the CA student's missing age by taking the mean age of other CA students. What is their imputed age?

20 (mean of other CA students) 17. Impute the missing favorite color by the most frequent favorite color of students with the same GPA. What is the imputed color?

Blue (mode for students with 3.3 GPA)

Section 3: Cleaning Anomalous Values 18. What term refers to data values that are conspicuously different from other values of the same variable?

Outliers 19. If a data set contains a person's age as 2995, how might this be addressed in data cleaning?

Answers may vary. Sample answer: Average all ages and replace it with the mean 20. In data cleaning, what term refers to values that are clearly invalid or impossible?

Implausible values 21. Describe an example of incorrect formatting in data.

Answers may vary. Example: Age entered as text 22. What is the best way to handle an extreme data value that is verified to be correct?

Answers may vary. Example: Keep it

Section 4: Transforming Quantitative Variables 23. What term refers to creating a new categorical variable by assigning categories to intervals of an existing quantitative variable?

Categorization 24. How might a quantitative variable, such as annual income measured in dollars, be transformed for easier visualization?

Convert to a scale such as income measured in thousands of dollars 25. Describe an example of creating a new quantitative variable by combining two existing quantitative variables.

Answers may vary. Example: Calculating BMI from weight and height 26. How does scaling a quantitative variable by dividing its values help create better visualizations?

It reduces the number of digits.

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Chapter 2: Data Wrangling: Preprocessing

27. In general, what is the best practice when transforming variables?

To create new variables leaving originals intact 28. What can be done to make a quantitative variable with a wide range easier to visualize and interpret without distorting the data?

Answers may vary. Example: Rescale the variable by a constant.

29. How does creating a quantitative variable from a categorical one enable more types of analysis?

It allows for different numerical summary statistics.

30. student

data_usage (MB)

major

data_usage (GB)

Jackson

10587

Art

10.34

Oneal

15789

CS

15.42

Ortega

12678

Bio

12.38

White

14567

Math

14.23

Martinez

16987

Eng

16.59

Dixon

12356

Psych

12.07

Garcia

13489

Hist

13.17

Harris

15782

Econ

15.41

Brown

14256

Socio

13.92

Baker

13567

Music

13.25

Mean data usage in Gigabytes: 13.68 GB Median data usage in Gigabytes: 13.59 GB 31. student

height (in)

major

12 © 2026 Pearson Education, Inc.

height (cm)

Data Science for All

Washington

71

Football

180.34

Jefferson

74

Basketball

187.96

Patel

68

Soccer

172.72

Gupta

69

Baseball

175.26

Li

74

Tennis

187.96

Kim

75

Swimming

190.50

Rodriguez

73

Track

185.42

Garcia

72

Hockey

182.88

Jackson

66

Wrestling

167.64

Sharma

70

Golf

177.80

Mean height in centimeters: 180.85 cm 32. In data cleaning, what might be done with text values in a quantitative variable like height?

Convert to numeric equivalent 33. Create a new variable, BMI, by filling in the values of the final column’s cells using the formula (rounded to one decimal place): bmi = weight (lb) / [height (in)]2 x 703 student

weight (lb)

height (in)

major

bmi

A

120

62

Math

21.9

B

135

64

Bio

23.2

C

150

66

CS

24.2

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Chapter 2: Data Wrangling: Preprocessing

D

165

68

Eng

25.1

E

180

70

Psych

25.8

F

155

64

Hist

26.6

G

160

66

Art

25.8

H

175

68

Econ

26.6

I

190

70

Socio

27.3

J

205

72

Music

27.8

34. What is the mean BMI for this data set (rounded to one decimal place)?

Mean BMI = 25.4 35. Calculate the cost per mile for each trip using the formula: Cost per Mile ($) = (Fuel Consumption (gallons) x Cost per Gallon ($)) / Distance (miles) student_id

distance (miles)

fuel_consumption (gallons)

cost_per_g allon ($)

major

cost_per_mile ($)

345634

50

2

3

Art

0.12

534534

100

4

2.5

CS

0.10

832344

150

5

3.5

Bio

0.12

242345

200

6

3

Math

0.09

859285

250

7

2.5

Eng

0.07

894993

100

3

3

Psych

0.09

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Data Science for All

388539

150

4

2.5

Hist

0.07

393559

200

5

3.5

Econ

0.09

395222

250

6

3

Socio

0.07

825234

300

7

2.5

Music

0.06

36. What is the mean cost per mile? Use two decimal places for all answers.

Mean cost per mile: $ 0.09 37. Calculate the force (in newtons) applied to each object of the final column using the formula (to one decimal place): force = mass x acceleration object

mass (kg)

acceleration (m/s^2)

force (N)

1

5

0.5

2.5

2

10

1

10.0

3

15

1.5

22.5

4

20

2

40.0

5

25

2.5

62.5

6

30

3

90.0

7

35

3.5

122.5

8

40

4

160.0

9

45

4.5

202.5

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Chapter 2: Data Wrangling: Preprocessing

10

50

5

250.0

38. What is the mean force applied to the objects (one decimal place)?

Mean Force: 96.3 N 39. Create a categorical age_group variable using the following categories: 18-35: Young Adult 36-55: Middle Aged 56+: Senior participant age height weight age_group 1 32 5’11 175 Young Adult 2 55 5’5 132 Middle Aged 3 62 5’9 195 Senior 4 18 5’3 120 Young Adult 5 23 6’1 180 Young Adult 6 43 5’7 150 Middle Aged 7 51 5’8 170 Middle Aged 8 28 5’4 135 Young Adult 9 47 6’0 200 Middle Aged 10 61 5’2 142 Senior 40. How many participants would be in the Young Adult category?

4 41. How many would be Middle Aged?

4 42. How many would be Senior?

2

Section 5: Transforming Categorical Variables 43. How might the uncommon (low frequency) categories of a categorical variable be simplified, given there are no concerns about othering?

Merging uncommon categories into one new, slightly larger category 44. What term refers to extracting multiple variables from a single categorical variable?

Splitting 45. Why is it important to assign every possible value to a category when creating categories from a quantitative variable?

It ensures that every value is properly assigned and prevents missing values.

46. What is a Boolean variable? When is it used?

It is a variable with two values, usually true or false. It is used to encode categorical data, converting it to a binary form.

47. What are some other names for Boolean variables?

Indicator variables, binary variables, dummy variables, logical variables, dichotomist variables.

48. When encoding a categorical variable numerically, why is the choice of values important?

The differences between the quantitative values should mirror differences between categories.

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Data Science for All

49. What term refers to reducing the number of distinct values a categorical variable can take?

Condensing 50. Describe the process of condensing a categorical variable.

Answers may vary. Example: the process involves merging small categories into broader ones, which can simplify the variable for analysis while reducing its informational detail.

51. When would splitting a categorical variable into multiple new variables be useful?

When a single value encompasses multiple distinct attributes 52. Provide three examples of categorical variables.

Answers may vary. Examples: college major, type of donation; animal species; country 53. What term is used to refer to the creation of quantitative variables from a categorical variable?

Encoding 54. What is an example of creating a categorical variable from a quantitative one?

Answers may vary. Example: Converting test scores to letter grades.

55. What is a potential mistake that can arise when encoding categorical data numerically?

Answers may vary. Example: Numerical codes not aligning with the meaning of categories.

56. How many people have January as their birth month?

1 (January) 57. How many were born in the last 15 days of a month?

5 (Last 15 days) 58. What is the mean birth year, rounded to the nearest year?

1989 (Mean year) 59. Recode the responses as follows: Strongly Disagree = 1 | Disagree = 2 | Neutral = 3 | Agree = 4 | Strongly Agree = 5 participant

response

age

1

1

35

2

4

29

3

3

42

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Chapter 2: Data Wrangling: Preprocessing

4

5

27

5

4

38

6

2

31

7

3

44

8

4

24

9

5

36

10

1

39

60. What is the mean of the recorded responses rounded to one decimal place?

Mean: 3.3 61. What is the median?

Median: 3.5 62. Split the major into Department and Major columns: student

major

department

major

1

Computer Science - Software Engineering

Computer Science

Software Engineering

2

Physics - Applied Physics

Physics

Applied Physics

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Data Science for All

3

Business - Accounting

Business

Accounting

4

History - American History

History

American History

5

Engineering - Electrical Engineering

Engineering

Electrical Engineering

6

English - Creative Writing

English

Creative Writing

7

Chemistry - Biochemistry

Chemistry

Biochemistry

8

Education - Elementary Education

Education

Elementary Education

9

Math - Applied Math

Math

Applied Math

10

Nursing - Nursing

Nursing

Nursing

63. How many students have Business majors?

1

Section 6: Reshaping a Dataset 64. Describe the process of converting a data set from wide format to long format.

Answers may vary. Example: It involves increasing the number of rows and reducing columns in the data set.

65. Why might a data scientist prefer a long-format dataset?

It is better for computing systems to perform analysis.

Section 7: Combining Datasets 66. What is an advantage of stacking two datasets with the same variables but different observations?

It increases the number of observations.

67. Which data integration technique combines datasets with all the same variables but additional observations?

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Chapter 2: Data Wrangling: Preprocessing

Stacking 68. When would it be appropriate to augment two datasets by binding them horizontally?

Answers may vary. Example: When the data sets contain the same observations but different variables 69. What does “merging datasets” typically involve?

Answers may vary. Example: When two datasets are merged, they will have at least one common variable.

Observations between the two datasets are matched on at least one of their common variables.

70. For which type of data integration must datasets have at least one common variable?

Merging 71. When merging two datasets, what must they have in common?

At least one variable 72. What term refers to binding two datasets together horizontally when they have the same observations but completely different variables?

Augmenting 73. What is the primary purpose of combining datasets?

To facilitate more comprehensive analysis by integrating diverse data 74. If these tables were augmented by binding them horizontally, how many rows and how many columns would the resulting table have? (Don’t count the column headings as rows.)

3 rows, and 6 columns.

75. If the graduate student stacks these data tables, how many rows would the new stacked table contain?

6 rows: one each for Amanda, Diego, Sara, Tia, Anna, and Sarah (which is distinct from Sara) 76. How many columns would the new stacked table have?

4 columns: name, math_test_1, math_test_2, section 77. If the professor stacks these two data tables, how many rows would the stacked table have?

4 rows: one each for Diego’s Semester 1, Sam’s Semester 2, Alex’s Semester 1, and Alex’s Semester 2.

78. How many columns would the stacked table have?

5 columns: name, exam_1, exam_2, final_exam, semester 79. If the teacher merges these tables using Name, how many rows would the new table have?

4 rows: one each for Anna, Diego, Sara, and Tia 80. How many columns would it have?

6 columns: name, reading_test_1, reading_test_2, reading_test_3, math_test_1, math_test_2 81. If they merge the datasets, how many unique Customer ID values would exist in the merged dataset?

3 Customer IDs 82. How many rows would the merged dataset contain?

3 rows (corresponding to the 3 Customer IDs) 83. Merge the categories of the categorical variable, major, into 3 categories: ● STEM: Biology, Chemistry, Physics, Math, Computer Science, Engineering ● Humanities: English, History, Arts ● Other: Any other majors student

major

gpa

clubs 20

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Data Science for All

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

STEM Humanities STEM Humanities STEM STEM Humanities STEM Humanities STEM STEM STEM Humanities STEM Humanities STEM Humanities STEM STEM Humanities

3.2 3.0 3.5 3.2 3.8 3.4 3.1 3.7 3.0 3.5 3.6 3.2 3.9 3.3 2.9 3.8 3.0 3.5 3.7 3.4

1 2 0 1 0 2 3 1 0 2 1 0 2 1 0 2 3 1 0 2

84. How many students have a STEM major?

12 (STEM) 85. How many have a Humanities major?

8 (Humanities) 86. How many have an Other major?

0 (Other) 87. Merge the categories of the categorical variable, Crop, into 2 categories: ● Row crops: Corn, Soybeans, Cotton ● Small grains: Wheat farm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

crop Row crops Row crops Small grains Row crops Row crops Row crops Small grains Row crops Row crops Row crops Small grains Row crops Row crops Row crops Small grains

acres 80 160 240 100 200 150 300 90 140 180 220 160 110 170 260

yield 8500 9200 11,000 9000 10,000 8000 12,000 8000 7500 9500 10,500 8500 9500 9000 11,500

88. How many farms grow row crops?

11 (row crops)

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Chapter 2: Data Wrangling: Preprocessing

89. How many farms grow small grains?

4 (small grains)

Explaining the Concepts 1.

Pieces of information provided on some of these forms might include name, address, age, gender, or ID number.

Different people might fill out their names differently. For example, one person might put their first name followed by their last name. Another person might put their last name first followed by their first name.

Similarly, different people might fill out their age differently. One person might use integers while another might write out the number, 20 vs twenty.

Changing prompts on a form can help make the data more consistent. Some changes can include providing examples or detailed instructions.

These issues should be handled at each stage of the data pipeline, including collection and cleaning.

2.

For categorical data, standardize the labels/names of the categories. For example, if there is no difference in the meaning between Dog, dog, and DOG, then they should be standardized when cleaning the data.

Lowercasing (dog) or title casing (Dog) are commonly used standards. For quantitative data with decimal points, e.g. 10.5 or 3.45, keep the number of digits consistent. For example, we might want to replace 10.5 with 10.50.

3.

Retaining variables is a good practice that helps ensure the work is reproducible. Additionally, since data science is an iterative process, replacing variables might remove some information that we will only realize later is important. We might not be able to recover the original data if we replace the original variables.

When replacing variables by condensing variables, we might remove information. For example, merging pet_preference into just “dog”, “cat”, “none”, or “other” and removing pet_preference would prevent us from keeping track of the other responses (Section 2.5). Categorization is another example of a data transformation method that can result in a loss of information if we replace a variable. For example, replacing the ages with “child”, “adult”, or “senior” and removing the original age will prevent us from computing quantitative statistics, e.g. the mean or median age.

4.

Cleaning and transforming the data of each dataset separately first can help integrate data correctly by making the datasets match each other better. For example, imagine that the seasons in Table 2.10 are in lowercase, e.g. “fall”. When augmenting Tables 2.10 and 2.11 together, we would not be able to identify that average_rainfall and average_temperature are variables for the same observation.

5.

Detailed instructions and explanations of Mirabel’s data-wrangling steps would help Miles reproduce Mirabel's work. Also, if Miles wants to compare a new group of students with the students in Mirabel’s group, Miles will need to repeat the exact same process that Mirabel used. This process would include the exact data wrangling steps.

Ensuring that work can be replicated on her data makes Mirabel confident in the results of her analysis.

Ensuring that the work can be applied accurately to new data enables researchers to continue Mirabel’s work or use the same approach to study another population.

Try It Yourself Try It Yourself: Identifying Missing Data (2.2) 1.

The number of missing values in the GPA column is 596.

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Data Science for All

2.

3.

4.

The total number of rows in the dataset is 5187.

The percentage of rows with missing GPA values is approximately 11.49%, a high amount that should elicit some concern.

STAR Framework: • Scope the Data’s Context: The dataset under consideration includes GPA data alongside other variables, such as voting behavior for students. A substantial number of missing GPA values were identified, which is critical for understanding the educational background and its possible association with voting behavior.

• Track the Question behind the Exploration: The primary objective is to assess the extent of missing GPA records to evaluate the dataset’s reliability, and to inform potential data cleaning or imputation strategies. This assessment is crucial to ensure the accuracy and relevance of any subsequent analysis involving GPA data.

• Articulate the Results: In the dataset comprising 5187 rows, there are 596 missing values in the GPA column. This constitutes approximately 11.49% of the dataset, a noticeable proportion that could impact the validity of any analysis that relies on GPA data.

• Respond with Next Steps: Given the high percentage of missing GPA data, it is essential to consider methods for handling these gaps, such as data imputation, or to adjust the analysis approach to account for this limitation. Further exploration into the reasons behind these missing values might also provide insights into data collection processes and help improve data quality for future studies.

Try It Yourself: Impute Missing Values from Internal Data (2.2) 1.

2.

3.

4.

5.

The mean GPA is approximately 2.98.

The median GPA is 3.00.

The mean GPA remained approximately 2.98.

The median GPA changed to approximately 2.98.

STAR Framework: • Scope the Data’s Context: The voter survey dataset, which includes GPA data, has missing values in the GPA column. The mean imputation method is used to address these missing values, aiming to maintain the integrity of the dataset for reliable analysis.

• Track the Question Behind the Exploration: The exploration focuses on understanding the impact of replacing missing GPA values with the mean GPA. The key questions involve assessing the mean and median GPA before and after the imputation process to gauge how this method affects these central measures.

• Articulate the Results: Initially, the mean GPA in the dataset was approximately 2.98, and the median GPA was 3.0. After employing mean imputation to replace missing values, the mean GPA remained around 2.98. However, the median GPA shifted to approximately 2.98, showing a slight change due to the imputation process.

• Respond with Next Steps: The results suggest that while mean imputation preserves the mean GPA, it slightly alters the median GPA. This finding should be considered when analyzing the dataset postimputation, particularly in studies where the median GPA is a critical measure. Further, exploring other imputation methods or considering the implications of missing data on the overall analysis would be beneficial for comprehensive understanding and interpretation.

Try It Yourself: Identify Extreme Data Values (2.3) 1.

2.

3.

4.

5.

The oldest 5% of respondents are between 22 and 115 years old.

The youngest 5% of respondents are between 3 and 18 years old.

There are three respondents who are above 100 years old.

There are nine respondents who are below 18 years old.

STAR Framework: • Scope the Data’s Context: In evaluating the vote dataset, primarily consisting of university campus respondents, our aim is to scrutinize the age data for potential inaccuracies. Identifying extreme age

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•

•

•

values is crucial for maintaining data quality and ensuring reliable analysis, especially considering university policies regarding minor participation.

Track the Question Behind the Exploration: Our exploration is centered on determining the ages of the oldest and youngest 5% of respondents, identifying individuals above 100 years old, and those below 18. This analysis helps in discerning whether these age values are plausible or constitute outliers.

Articulate the Results: The analysis reveals that the ages of the oldest 5% of respondents range from 22 to 115 years, and the youngest 5% range from 3 to 18 years. There are three respondents over 100 years old and nine below 18. The presence of respondents below 18 is an outlier, as the university prohibits their participation as minors. While respondents above 100 are unlikely on a university campus, it’s less clear-cut and may warrant a more inclusive approach.

Respond with Next Steps: Given these findings, the immediate step involves addressing the presence of minors in the dataset, as they represent clear data entry errors or policy violations. For the respondents above 100, a deeper investigation might be necessary to verify the authenticity of these entries. Adjusting the dataset by removing or further scrutinizing these extreme values will enhance its validity and reliability for subsequent analyses.

Try It Yourself: Clean Incorrect Formats (2.3) 1.

2.

2.

Before the basic data cleaning, in the dataset: a. The number of people who indicated their favorite pet as "dog" was 1646. b. The number of people who indicated their favorite pet as "cat" was 1808.

After the basic data cleaning, the numbers changed as follows: a. The number of people who indicated their favorite pet as "dog" increased to 1833. b. The number of people who indicated their favorite pet as "cat" increased to 2004.

STAR Framework: • Scope the Data’s Context: The focus is on the Favorite_pet[AS1] column in the vote dataset. The goal is to standardize the data by correcting variations in the entries for favorite pets, specifically targeting the categories "dog" and "cat."

• Track the Question Behind the Exploration: The exploration aims to quantify the impact of data cleaning on the representation of "dog" and "cat" in the dataset. This involves counting the occurrences of these pets before and after applying specific cleaning rules to unify similar entries under standard labels.

• Articulate the Results: Before cleaning, the count for "dog" was 1646 and for "cat" was 1808. After applying cleaning rules—standardizing variations of "dogs" to "dog," "cats" to "cat," and including words starting with "d" and ending in "g" under "dog," and those starting with "c" and ending in "t" under "cat" — the count for "dog" increased to 1833 and for "cat" to 2004.

• Respond with Next Steps: The results indicate that the cleaning process successfully identified and corrected inconsistent entries, leading to a more accurate representation of pet preferences in the dataset. Establishing guidelines or using tools for data entry can prevent such inconsistencies in future data collection.

Try It Yourself: Create a Categorical Variable from a Quantitative Variable (2.4) 1.

2.

There are 2349 students with a high GPA (with values from 3.00 to 4.00), and 2242 students with a low GPA (with values from 2.00 to 2.99) in the dataset.

STAR Framework: • Scope the Data’s Context: The analysis involves the GPA variable in the vote dataset, with an objective to categorize GPAs into two groups for a simplified analysis. The new variable, GPA_category, is created to classify GPAs as “low” for the values of 2.00–2.99 and “high” for values of 3.00–4.00.

• Track the Question Behind the Exploration: The exploration seeks to understand the distribution of GPAs among students in terms of these two categories. This categorization helps in analyzing the

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•

•

data more effectively by reducing the complexity of continuous GPA values into simpler, categorical terms.

Articulate the Results: The categorization resulted in identifying 2349 students with a “high” GPA (with values from 3.00 to 4.00), and 2242 students with a “low” GPA (with values from 2.00 to 2.99).

This distinction provides a clearer picture of the overall academic performance distribution within the dataset.

Respond with Next Steps: Given these results, further analysis could explore associations between GPA categories and other variables in the dataset, such as voting behaviors or other demographic factors. Additionally, considering different GPA categories could offer alternative perspectives on the data.

Try It Yourself: Create a Boolean Variable that Represents a Categorical Variable (2.5) 1. a. b.

2081 drinking_age values are 1.

3106 drinking_age values are 0.

2.

3.

a. Number of people of drinking age who voted: 1199 b. Number of people not of drinking age who voted: 1715 c. Percentage of drinking age people who voted: 57.6% d. Percentage of non-drinking age people who voted: 55.2% STAR Framework: • Scope the Data’s Context: A drinking_age Boolean variable was added to the dataset to differentiate between individuals above and below the legal drinking age, facilitating an analysis of voting behavior across different age groups.

• Track the Question Behind the Exploration: The analysis focused on determining how many individuals fell into each drinking age category and assessed their respective voting participation rates.

• Articulate the Results: In the dataset, 2081 individuals were classified as of legal drinking age (21 and over), with 1199 having voted, representing 57.6% of this group. In contrast, 3106 individuals were under the drinking age, with 1715 having voted, making up 55.2% of this younger group.

• Respond with Next Steps: These findings suggest a marginal difference in voting rates between the two age groups. Further investigation could involve exploring demographic and psychographic factors that might influence these voting behaviors, providing deeper insights for voter engagement strategies.

Try It Yourself: Splitting a Variable into Multiple Variables (2.5) 1.

• • • • • • • • • • • • 2.

January: 474 people February: 398 people March: 426 people April: 432 people May: 436 people June: 406 people July: 432 people August: 437 people September: 443 people October: 433 people November: 459 people December: 411 people

STAR Framework:

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•

• • •

Scope the Data’s Context: In the voter survey dataset, we isolate the birth month from the existing birthdate information, creating a new column for this purpose. This step enables an analysis focused on discerning potential associations between voters’ birth months and their voting behaviors.

Track the Question Behind the Exploration: The primary question centers on quantifying the distribution of voters across different birth months to explore if and how the birth month might be associated with voting patterns.

Articulate the Results: The breakdown by month revealed varying numbers of people born in each month, with January having the highest at 474 and February the lowest at 398. The distribution is relatively even across the year, with most months having between 400 and 450 people.

Respond with Next Steps: Given this distribution, the next phase of analysis can delve into comparing these birth month groupings with voting behaviors. This could uncover trends or patterns that are specific to certain times of the year, potentially offering insights into voter behavior associated with demographic factors such as age.

Applying the Concepts Activity: Identify Missing Data 1.

2.

3.

4.

5.

Overall, there were 891 passengers recorded in this dataset.

177 passengers were missing a value for the age variable.

As a percentage, this means that rounded to the nearest whole percentage point, 20% of all passengers did not have their age recorded.

If child passengers were less likely to have their age recorded in this dataset than adult passengers, then the average age we calculate would be higher than the true average age.

This strategy for handling missing data is called imputation, where the missing values are filled with an educated guess. This guess could be made using means, medians, or prediction models. Deciding on the most appropriate method requires careful consideration.

Activity: Impute Missing Values from Internal Data 1.

2.

The mean imputation method produces an average life expectancy of 63.18 years.

The median imputation method produces an average life expectancy of 63.23 years.

Activity: Identify Extreme Data Values 1.

2.

3.

4.

5.

6.

The youngest age of the oldest 5% is 101 The oldest age for the youngest 5% is 22 There are 1,333 birdwatchers who are above 100 years old.

There are 11 birdwatchers who are below 18 years old.

There are at least four impossible age values: -1, 0, 134, 723. Validity of values from 3 to 9 can also be questioned.

STAR Framework: • Scope the Data's Context: The birdwatcher dataset contains information about birdwatchers, including their ages. Our focus is on understanding the distribution of ages, especially identifying any extreme or impossible values.

• Track the Question behind the Exploration: We aim to uncover if there are any unusually high or low ages, as these could be errors affecting data integrity. Specifically, we want to know the percentile values of the oldest and youngest 5% of birdwatchers, the number of birdwatchers above 100 and below 18, and any impossible age values. Identifying and addressing extreme data

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•

•

values in the birdwatcher dataset ensures data accuracy and reliability, facilitating informed decision-making and robust statistical analyses.

Articulate the Results: The analysis reveals that the youngest age of the oldest 5% of birdwatchers is 101 years, and the oldest age of the youngest 5% is 22 years. There are 1,333 birdwatchers above 100 years and 11 below 18. Additionally, 4 impossible values were found: a negative age, 0, 134, and 723, were found. These results suggest significant data quality issues, particularly with impossibly high and low age values.

Respond with Next Steps: Further investigation is needed to validate the data source and correct these anomalies. Cleaning the data set by removing or correcting these extreme values is crucial for reliable analyses. Additionally, setting up data validation rules to prevent future data entry errors could be beneficial.

Activity: Clean Incorrect Formats 1.

2.

n/a a. b. c.

“sparrow” Answer: 7232 “woodpecker” Answer: 7142 “hummingbird” Answer: 7109

3.

4.

a. “sparrow” Answer: 8019 b. “woodpecker” Answer: 7950 c. “hummingbird” Answer: 7924 STAR Framework • Scope the Data’s Context: The birdwatcher dataset from an Airbnb birdwatching experience is being cleaned for analysis. The focus is on the Favorite_Bird column, standardizing entries like “sparrows” to “sparrow”, “woodpeckers” to “woodpecker”, and “hummingbirds” to “hummingbird”.

• Track the Question behind the Exploration: The objective is to determine how basic data cleaning affects the count of participants' favorite birds, specifically “sparrow”, “woodpecker”, and “hummingbird”, to understand better the relationship between bird preferences and interest in repeat experiences.

• Articulate the Results: Before cleaning, the counts were 7232 for “sparrow”, 7142 for “woodpecker”, and 7109 for “hummingbird”. Post-cleaning, the counts increased to 8019 for “sparrow”, 7950 for “woodpecker”, and 7924 for “hummingbird”. This indicates that cleaning helped rectify inconsistent entries and provided a more accurate representation of bird preferences.

• Respond with Next Steps: With the data cleaned, further analysis can be conducted to explore any correlations between participants' favorite birds and their willingness to sign up for another tour.

Understanding these preferences can aid the guide in tailoring future birdwatching experiences and improving participant satisfaction.

Activity: Create a Categorical Variable from a Quantitative Variable 1.

2.

3.

4.

5.

6.

Most houses have prices greater than $100,000 but less than $200,000. n/a The number of houses in the “Cheap Moderate” category is 1019.

The largest category is now “Cheap Moderate”, followed by Expensive and Expensive Moderate There are 857 “Expensive” houses.

There are 802 “Expensive Moderate” houses.

Activity: Create a Boolean Variable that Represents a Categorical Variable 1.

2.

There are 41 mammals in the dataset.

There are 20 birds and 5 reptiles in this dataset.

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Chapter 2: Data Wrangling: Preprocessing

Activity: Splitting a Variable into Multiple Variables 1.

2.

• January: 2016 entries • February: 1965 entries • March: 1996 entries • April: 1973 entries • May: 1947 entries • June: 1960 entries • July: 2085 entries • August: 1972 entries • September: 1959 entries • October: 1986 entries • November: 2006 entries • December: 2028 entries STAR Framework: • Scope the Data’s Context: The birdwatcher dataset was modified to include a new column representing the month, extracted from the existing Date variable. This change aims to investigate seasonal birdwatching patterns and the correlation between the months and types of birds observed.

• Track the Question behind the Exploration: The exploration sought to quantify the number of birdwatching entries each month, enabling an analysis of how birdwatching experiences and preferences vary seasonally.

• Articulate the Results: The data revealed a relatively even distribution of birdwatching entries throughout the year, with the highest in July (2,085 entries) and the lowest in May (1,947 entries).

This suggests that birdwatching activity is consistently popular across different months, with slight variations.

• Respond with Next Steps: These findings provide a foundation for further analysis to explore potential correlations between specific months and the types of birds observed. Such insights could be invaluable for understanding the seasonal preferences of various bird species and enhancing the birdwatching experience for participants.

Chapter 3: Making Sense of Data Through Visualization Chapter Review Questions Section 1: The Grammar of Graphics 1.

What does the term "grammar of graphics" refer to in data visualization?

A structured method to map data components to graphical attributes

2.

What elements are important to include in most data visualizations?

Answers may vary. Examples: a source dataset; a horizontal axis, a vertical axis; color; geometry/shape

3.

In the context of the grammar of graphics, what does geometry refer to?

The shape used to represent data (e.g., bars, lines, points)

4.

According to the grammar of graphics, what role does color typically play in data visualization?

It differentiates between data categories or represents values.

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Chapter 2: Data Wrangling: Preprocessing

Activity: Splitting a Variable into Multiple Variables 1.

2.

• January: 2016 entries • February: 1965 entries • March: 1996 entries • April: 1973 entries • May: 1947 entries • June: 1960 entries • July: 2085 entries • August: 1972 entries • September: 1959 entries • October: 1986 entries • November: 2006 entries • December: 2028 entries STAR Framework: • Scope the Data’s Context: The birdwatcher dataset was modified to include a new column representing the month, extracted from the existing Date variable. This change aims to investigate seasonal birdwatching patterns and the correlation between the months and types of birds observed.

• Track the Question behind the Exploration: The exploration sought to quantify the number of birdwatching entries each month, enabling an analysis of how birdwatching experiences and preferences vary seasonally.

• Articulate the Results: The data revealed a relatively even distribution of birdwatching entries throughout the year, with the highest in July (2,085 entries) and the lowest in May (1,947 entries).

This suggests that birdwatching activity is consistently popular across different months, with slight variations.

• Respond with Next Steps: These findings provide a foundation for further analysis to explore potential correlations between specific months and the types of birds observed. Such insights could be invaluable for understanding the seasonal preferences of various bird species and enhancing the birdwatching experience for participants.

Chapter 3: Making Sense of Data Through Visualization Chapter Review Questions Section 1: The Grammar of Graphics 1.

What does the term "grammar of graphics" refer to in data visualization?

A structured method to map data components to graphical attributes

2.

What elements are important to include in most data visualizations?

Answers may vary. Examples: a source dataset; a horizontal axis, a vertical axis; color; geometry/shape

3.

In the context of the grammar of graphics, what does geometry refer to?

The shape used to represent data (e.g., bars, lines, points)

4.

According to the grammar of graphics, what role does color typically play in data visualization?

It differentiates between data categories or represents values.

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Data Science for All

Section 2: Visualizations with One Quantitative Variable 5.

What can be inferred about a dataset with a histogram that is skewed to the right?

The mean is greater than the median.

6.

What statistic is calculated as the third quartile (Q3) minus the first quartile (Q1)?

Interquartile range

7.

Why would a violin plot be preferred over a box plot?

It provides a richer understanding of the data distribution.

8.

What is the main purpose of “binning” data in a histogram?

To group data into intervals for easier visualization.

9.

What visual technique is often used to show the distribution of a variable and identify potential outliers simultaneously?

Box plot

10. What is a histogram used to display?

A histogram is used to display the distribution of a set of continuous data.

11. In a box plot, what do the ends of the whiskers represent?

The highest and lowest values that are not outliers 12. What type of visualization displays one quantitative variable with a smooth continuous line to show the distribution?

A density plot 13. Which type of plot is specifically useful for identifying the median, quartiles, and outliers in a dataset?

Box plot 14. When data is left-skewed, what does it imply about the distribution?

The median is greater than the mean.

15. What does a density plot help visualize?

The distribution of a continuous variable 16. When constructing a histogram, what does the y-axis typically represent?

The count or frequency of observations 17. What can skewness in a histogram indicate?

The asymmetry in the distribution of data 18. How are outliers represented in a box plot?

With dots outside the whiskers 19. Which iris species tends to have the smallest petal width?

Setosa 20. Which species of irises have flowers with petals between 1.6 and 1.8 cm wide?

Virginica and versicolor 21. How many centimeters are the bins of this histogram incremented by?

0.2 22. Which species have the widest range of petal width?

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Virginica 23. In the histogram of Figure 3.61, which displays the distribution of grades in a class, approximately how many students scored 80?

20 24. In the histogram of Figure 3.62, which displays the distribution of student ages, approximately how many students are 22 years old?

15 25. What is the median value of Group 1 in the box plot of Figure 3.63?

8 26. In Figure 3.64, what is the interquartile range of Group 3?

4 27. Which group shows the highest density around the median in the violin plots of Figure 3.65?

2

Section 3: Visualizations with One Categorical Variable 28. What is a chart type that would be ineffective when trying to display a dataset with hundreds of unique categories?

Histogram. (Answers may vary.)

29. How is a categorical variable most effectively represented in a visualization?

Through discrete bars or segments 30. How are word clouds used in data visualization?

Word clouds are used to highlight critical components of textual data (such as a speech, blog, or article).

The size of the word in a word cloud reflects its frequency in the corresponding textual data.

31. When are bar charts most useful in data visualization?

When comparing different categories 32. What is the primary visual element used in a bar chart to represent data?

Bars 33. According to Figure 3.66, what is the usage percentage of Python in programming?

67% 34. According to the bar graph in Figure 3.67, how many students are enrolled in the Arts major?

800 35. What is the value for Category D in the bar chart of Figure 3.68?

20

Section 4: Visualizations with Two Variables 36. What does the x-axis typically represent in data visualization?

The independent variable or time 37. What kind of relationship does a scatterplot illustrate?

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38. In what context would a line chart be most appropriate?

To display changes in a quantitative variable over a period of time 39. What type of data does a treemap display?

Hierarchical data 40. In a scatterplot, which variable is typically placed on the x-axis?

Explanatory 41. For which type of data would a line chart not be appropriate?

Frequency of categorical outcomes. (Answers may vary.)

42. Describe a situation where a scatterplot would not effectively communicate your data.

Answers may vary. Example: Because a scatterplot is used to show the relationship between two quantitative variables, it would not be useful for showing the relationship between a quantitative variable and a categorical variable, such as a relationship between age and favorite color.

43. How does a line chart typically display data?

As a series of points connected by straight lines 44. What type of plot is this visualization?

Density plot 45. The filter in this visualization allows users to specify what?

Format of music.

46. In approximately which year does the peak in CD sales appear to have occurred?

2000 47. How would you describe the curve of the plot?

Symmetric 48. Approximately how many CDs were sold in 1990?

10,000 49. What kind of visualization is used in this graph?

Side-by-side column chart 50. Which continents had data points for countries classified as low income?

Africa and Asia 51. The highest mean life expectancy is among countries in which group?

Europe, high income 52. Approximately what is the lowest mean life expectancy represented among the groups?

60 53. Which variety of diamond cut has the greatest variation in depth?

Fair 54. Which variety of diamond cut has the least variation in depth?

Premium 55. What is the approximate median depth for diamonds with a premium cut?

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56. Approximately what is the largest depth among all the diamonds in the dataset?

79% 57. Which diamond cuts have clear depth outliers?

Ideal 58. Looking at Figure 3.71, what was the popularity of Data Science in the year 2021?

63% 59. From the scatterplot’s line of Figure 3.72, if a student studies for approximately 30 hours weekly, what could be the expected GPA (in the absence of outliers)?

3.0 60. From the scatterplot’s line in Figure 3.73, what GPA would we expect to be associated with studying 16 to 16 hours?

Acceptable answer is anything from 1.6 to 1.9.

61. From Figure 3.74, if a student studied approximately 50 hours, what could be the expected test score in the absence of outliers?

49 or 50 is acceptable 62. From Figure 3.75, what was the test score of the person who studied for the least number of hours?

40 (accept 40, 41) 63. From Figure 3.76’s scatterplot, one person in the dataset studied 15 - 16 hours. What was their GPA? Acceptable answer is anything from 2.0 to 2.2.

64. In the line graph of Figure 3.77 which shows student enrollment over the years, what was the number of students enrolled in 2020?

600

Section 5: Visualizations with Three or More Variables 65. Describe an effective way of visualizing a large dataset with multiple categories to quickly spot trends and patterns.

Designing a heatmap 66. What is a key feature of a heatmap in data visualization?

It uses color shades to represent data values.

67. What would be the ideal visualization to represent spatial relationships involving countries, states, and zip codes?

Geographic map 68. In a bubble chart, what does the size of a bubble typically represent?

A third quantitative variable 69. What additional information does a bubble chart provide compared to a traditional scatterplot?

Magnitude of a third variable 70. How do heatmaps differ from traditional bar or column charts?

They use color gradients to represent data values.

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71. What do the colors of the bubbles in this visualization tell us?

Continent 72. What do the sizes of the bubbles in this visualization tell us?

Population 73. Which continent appears to have the largest proportion of countries with low GNI per capita?

Africa 74. Which continent appears to have the largest proportion of countries with high GGEI? Europe 75. Which continent appears to hold the largest countries, by population?

Asia 76. Consider the Titanic visualization in Figure 3.79, which illustrates the relationship between age, fare, and passenger class in relation to the survival of passengers. How many third-class passengers aged between 20 and 30 years old, and paying a fare of $100-200, did not survive?

2

Section 6: The Dangers of Visual Misrepresentation 77. What characteristic of a visualization assists in avoiding the misinterpretation of data due to outliers?

Using a box plot to highlight outliers 78. When examining data for misrepresentation, what aspect should be carefully considered to maintain integrity?

The scale of the axes 79. What is a common result of cherry-picking in a visualization?

Biased or distorted representation of the data 80. Why is the use of misleading colors considered a danger in data visualization?

It can lead to misinterpretation of data significance.

81. What can misinterpreting association and causation in a visualization cause viewers to do?

Incorrectly infer a cause-effect relationship 82. What is a common issue with mislabeling in data visualizations?

It leads to confusion about what the data represent.

83. Explain the danger of using truncated axes in data visualizations.

It can lead to misleading interpretation of data magnitude.

Section 7: Data Visualization Guidelines 84. Why is data democratization important in data visualization?

It makes data accessible to non-technical users.

85. How do interactive data visualizations enhance user engagement?

By allowing users to explore and manipulate the data.

86. What is the benefit of incorporating storytelling in data visualization?

It helps convey a narrative that can make the data more relatable.

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87. When aiming to make data visualizations interactive, what technique can be employed to engage users?

Enabling filtering and detailed view on hover 88. What is a primary benefit of using artificial intelligence (AI) in data visualization?

AI helps discover new insights from the data.

89. Why is it important to consider mobile-friendliness in data visualization design?

Increasing consumption of information on mobile devices 90. What is a quality that data visualizations should have in order to communicate data effectively to a broader audience?

Data visualizations should be democratically designed for varied literacies.

91. How does the inclusion of interactive functions in data visualizations enhance the user experience?

By involving users as participants rather than passive viewers 92. What is the advantage of incorporating storytelling into data visualization?

It combines narrative and visuals to improve information digestion.

Explaining the Concepts 1.

We typically first look at a title that describes what is displayed in the visualization. We then next look at the axes titles and labels to inform us about the variables being displayed. Characteristics of good data visualization include integrity of the data, appropriate use of colors, descriptive titles, informative labels/legends, and labeled axes.

2.

A common understanding of how data are visualized and communicated makes visualizations an effective storytelling tool. A common understanding of which visualizations are appropriate for data with given characteristics—certain numbers of variables, certain variable types, etc.—makes it easy for someone viewing a visualization to understand what type of data are being displayed quickly.

3.

Text in an infographic can describe the sources of the data. The bottom of Figure 3.2 in the Introduction includes the source as a footer of the visualization. We can indicate the scope of the data in the title, e.g., results of a poll among students of a particular college. Another way might be to add a graph with demographic information for the individuals, e.g., histograms for age and income. We can use specific characteristics to describe limitations. For example, in Figure 3.2, a specific color indicates which countries are missing in their dataset, i.e., those that are not ranked.

4.

One common type of visual misrepresentation is misleading bar graphs with a y-axis that does not starting at 0. This is a common misrepresentation that exaggerates the differences between different categories.

This misrepresentation is often seen in advertisements when a company wants to show the difference between their product and another. In ads, this misrepresentation might be deliberate when a company tries to demonstrate that its product is less expensive than a competitor's.

Another common type of visual misrepresentation is presenting an association between variables as a causal relation between them. Some media publications can use a plot demonstrating correlation between two variables as a proof of a direct causation, while, in fact, correlation alone doesn’t explain to us how the variables are connected. Any statements claiming otherwise should be approached with a deal of reasonable criticism.

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5.

Using graphing techniques appropriate to the data we are visualizing helps impart the same general message to all viewers. Using the correct visualizations enables viewers to quickly identify the specific numbers and types of variables depicted in the figure.

Visualizing a coded categorical variable can be worse or misrepresentative compared to visualizing the original categorical variable since a viewer can inadvertently make the wrong assumptions and takeaways from the visualization. A positive is that it can make the axis labels cleaner. For example, if the name of a category is very long, it might be better to just use numbers in the axis labels and a legend to indicate the numerical values associated with each category.

Visualizing categorized quantitative variables can be worse or misrepresentative compared to visualizing the original quantitative variable by removing the quantitative difference between values. However, doing so is necessary to plot histograms where similar values are binned together into groups.

Try It Yourself Try It Yourself: Interpret a Histogram (3.2) 1.

2.

3.

4.

The range of GNI per capita values is less than $25,000 for most countries.

Yes. The distribution of GNI per capita appears skewed right, 50 countries.

STAR Framework: ● Scope the Data’s Context: This histogram visualizes the counts of Gross National Income (GNI) per capita (x-axis) among various countries around the globe, offering a snapshot of their relative economic prosperity. ● Track the Question Behind the Exploration: We examine disparities in economic prosperity among countries worldwide, as measured by GNI per capita. ● Articulate the Results: The distribution is right-skewed. Many countries are clustered at the lower end of the GNI per capita spectrum, while few reach the higher end. ● Respond with Next Steps: Further investigate global policies to see what might improve underlying factors leading to growth in GNI per capita, such as education, infrastructure, and social safety nets.

Try It Yourself: Describe a Density Plot (3.2) 1.

2.

The distribution is skewed left.

STAR Framework: • Scope the Data’s Context: This density plot illustrates the density of the mean years of schooling (x-axis) for countries worldwide (observations). This type of visualization helps us understand how educational attainment is represented globally.

• Track the Question Behind the Exploration: Some countries may have a higher or lower standard for years of schooling. How do countries differ worldwide?

• Articulate the Results: Peaks show where there is a higher proportion (or density) of values, and valleys or tails show where there is a lower proportion of values. The plot peaks at around 12 years, suggesting a commonality in schooling duration for many countries. The density decreases toward a second peak at 5.5 years but remains relatively high between those values. However, the breadth of the distribution indicates notable differences, with some countries achieving much less (5 years or fewer) or much more (12 years or more) education.

• Respond with Next Steps: Further investigation into why some countries have more or fewer years of education can help tailor educational initiatives to bridge gaps and promote equal educational opportunities worldwide.

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Try It Yourself: Work with a Box Plot (3.2) 1.

2.

3.

4.

The IQR of WHR scores has a narrow range of about 15; 50% of countries received WHR scores between about 50 and 65. Only one country has a WHR score deemed an outlier, and it has a score less than 30.

Values between 55 and 58 are valid answers.

79 STAR Framework: • Scope the Data’s Context: The box plot visualizes the distribution of the World Happiness Report scores (y-axis) across different countries. This visualization helps us see the range of happiness scores on a global scale.

• Track the Question Behind the Exploration: Considering the pursuit of happiness as a fundamental human goal, we observe variations in happiness scores across nations. Are there global happiness gaps?

• Articulate the Results: The box plot shows that the median happiness score is just below 60, with the interquartile range extending from about 50 to 65. Notably, there is an outlier indicating at least one country with a noticeably lower score, which suggests disparities in happiness levels.

• Respond with Next Steps: Further exploration is needed to understand the factors contributing to the lower happiness scores across various countries. This can lead to better insights into improving happiness.

Try It Yourself: Interpret a Violin Plot (3.2) 1.

2.

3.

4.

Looking at the curvature of the plot, the distribution of mortality rates is most dense between 0 and 15 deaths per 1000 births. So, low values for under-5 mortality rates are most common. We can also see that the median under-5 mortality rate is about 15 deaths per 1000 births. Increasingly fewer countries have higher than 30 deaths per 1000 births.

5 40 STAR Framework: • Scope the Data’s Context: The violin plot illustrates the distribution of the mortality rate for children under five years old (y-axis) across different countries.

• Track the Question Behind the Exploration: Child mortality is a critical indicator of a country’s health and development. The distribution of these rates raises questions about global child health disparities. What does the distribution of child mortality look like worldwide?

• Articulate the Results: The median mortality rate is just below 15 deaths per 1000 live births.

The interquartile range extends from approximately 5 to 40 deaths per 1000 live births, meaning that half of the countries fall within this bracket. The widest part of the violin, representing the highest density of data points, appears between 0 and 15 deaths, which indicates where the highest density of countries’ rates is concentrated. Notable outliers are present, some reaching rates over 100 deaths per 1000 live births, highlighting extreme cases of child mortality.

• Respond with Next Steps: A detailed examination of the factors causing higher mortality rates, particularly in the outlier regions with rates above 100, is crucial. Insights gained from this analysis can guide targeted health interventions and policies to improve child survival rates.

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Try It Yourself: Ponder a Word Cloud (3.3)

1.

“Earth” is mentioned more than the others. “Planet” and “world” are more abstract words, while “earth” more concretely refers to a physical place. “Earth” implies a place where we live, while “planet” refers to one of many celestial bodies. So, the term earth may be intentionally used more to help people think about the place that we all call home.

Try It Yourself: Interpret a Column Chart (3.4) 1.

2.

3.

4.

Among the continents, Oceania and Europe have the highest average percentages of people with internet access, while Africa and North America have the lowest. The means for the continents range between about 37.34% and 90.55%.

B. Continent Africa STAR Framework: ● Scope the Data’s Context: The column chart displays the mean percentage of people with internet access (y-axis) in countries by continent (x-axis), providing a clear comparison across different geographic regions. ● Track the Question Behind the Exploration: As the digital divide remains a global challenge, this visualization prompts us to consider the extent of internet access disparities between continents. What is the average proportion of people with internet access for each continent and how do they compare? ● Articulate the Results: The chart shows that Oceania and Europe have the highest mean internet access rates, at 90.55% and 85.71%, respectively. South America and Asia follow, with access rates of 74.49% and 70.28%. North America’s mean access rate is 66.13%, while Africa has the lowest at 37.34%, indicating a significant gap compared to other continents. ● Respond with Next Steps: Recognizing the gap in internet access, particularly in Africa, calls for a deeper look into the underlying factors and potential initiatives to increase connectivity and reduce the digital divide.

Try It Yourself: Navigate a Treemap (3.4) 1.

2.

3.

New Zealand and Australia Micronesia and Vanuatu STAR Framework: ● Scope the Data’s Context: The treemap visualizes the countries with the highest mean years of schooling in each continent (variable of interest). The size of each block corresponds to the relative mean years of schooling in that country.

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● ● ●

Track the Question Behind the Exploration: Education levels vary greatly around the world. This figure highlights which countries per continent lead, allowing us to ask: Which countries have higher mean years of schooling?

Articulate the Results: Canada, the United States, South Africa, Mauritius, Botswana, Japan, Israel, Australia, and New Zealand show large blocks, reflecting higher relative education levels within their continents.

Respond with Next Steps: Investigating the policies, economic conditions, and cultural factors that contribute to higher educational attainment in these countries could provide valuable insights.

Try It Yourself: Contemplate a Box Plot with a Categorical Variable (3.4) 1.

2.

3.

4.

85 Africa Oceania STAR Framework ● Scope the Data’s Context: The box plot provides a comparison of mean life expectancies (y-axis) by continent (x-axis and color), displaying a summary of the central tendency and dispersion for Africa, Asia, and Oceania. ● Track the Question Behind the Exploration: The variation in life expectancy across continents raises questions about global health equity. How do these continents compare? ● Articulate the Results: The box plot indicates that Africa has the lowest mean life expectancy, with a median around 62 years, and shows an outlier indicating at least one country with a significantly higher life expectancy. Asia has the highest median life expectancy of these, around 72, with the widest interquartile range. Oceania’s mean is close to 70 years with a narrow range, suggesting almost no variation within the continent. However, Oceania has several outliers. ● Respond with Next Steps: We need to explore why life expectancy varies across these continents to understand and address the disparities reflected in these data.

Try It Yourself: Interpret a Violin Plot with a Categorical Variable (3.4) 1.

2.

3.

4.

The upper middle-income group has more variation in GNI per capita. GNI per capita for upper middleincome countries ranges between about $7000 and $30,000, but only ranges from about $2500 to $13,000 for lower-middle-income countries. In addition, the interquartile range is much wider for upper middleincome groups (as evidenced by the bigger box among the box plots).

13,000 2500 STAR Framework: ● Scope the Data’s Context: The violin plot compares countries’ Gross National Income (GNI) per capita (y-axis) between lower-middle-income and upper-middle-income groups (x-axis and color), illustrating their income distribution. ● Track the Question Behind the Exploration: Economic disparity between different income groups is a significant global issue. This visualization prompts us to consider the breadth and depth of this divide.

How does the distribution of countries’ GNI per capita compare between the lower-middle-income group and the upper-middle-income group? ● Articulate the Results: The lower-middle-income group has a narrower distribution with a median GNI per capita significantly lower than the upper-middle-income group, which exhibits a wider distribution and a higher median. The upper-middle-income group's distribution also has a longer tail toward higher GNI per capita, indicating the presence of countries with substantially higher incomes within this group. ● Respond with Next Steps: To address economic inequality, it is important to delve into the structural differences that contribute to such disparities in GNI per capita. This information can be crucial for reforms.

Try It Yourself: Interpret a Scatterplot (3.4) 1.

No. The least happy country does not have the lowest GGEI score.

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Data Science for All

2.

3.

4.

5.

Rather, the lowest GGEI scoring country is relatively happy, with a World Happiness Report Score of about 66.

An acceptable range is between 0.69-0.71.

0.55 STAR Framework: ● Scope the Data’s Context: The scatterplot visualizes the relationship between the Global Green Economy Index (GGEI) (y-axis) and the World Happiness Report score (x-axis) for countries worldwide, offering insights into how environmental sustainability is associated with national wellbeing. ● Track the Question Behind the Exploration: We probe the connection between the happiness of a population and its country's green economic practices. Could countries that are overall happier care more about a green planet? Is there an association between the GGEI and the World Happiness Report score, and what does it look like? ● Articulate the Results: The scatterplot depicts an upward trend; higher values of World Happiness Report scores are associated with higher values of GGEI. So, countries that are happier overall are also investing more in green economies. ● Respond with Next Steps: The trend invites further exploration into how societal happiness correlates with sustainable environmental policies. We do not know if one causes the other. It is possible that countries whose citizens are happier care more about making the planet better, but this needs further investigation.

Try It Yourself: Explore a Line Chart (3.4) 1.

2.

3.

4.

Overall, the under-5 mortality rate significantly decreased between 1880 and 2020. There were over 300 deaths per 1000 births in the late 1880s, but the value is close to 6 as 2020. Despite an initial decrease, child mortality rates saw another rise around 1917, but then immediately went back down. Throughout the rest of the 1900s, the number decreased almost every year. As of the 2000s, it has remained below 10 deaths per 1000 births, demonstrating tremendous progress since the late 1800s.

1917 An acceptable range is 1941 – 1944.

STAR Framework: ● Scoping the Data’s Context: The line chart displays the United States child mortality rate (deaths per 1000 births; y-axis), plotted over the years (x-axis) from 1880 to 2020. ● Tracking the Question Behind the Exploration: The chart seeks to explore how child mortality rates have changed over time in the U.S. ● Articulating the Results: The chart shows a significant decrease in child mortality rates over the 140-year span, with a notable peak and dip in the late 1910s, before continuing its downward trend. ● Responding with Next Steps: Further investigation could include examining specific interventions or public health policies that correlated with major changes in the trend, as well as comparing these rates to other countries.

Try It Yourself: Describe a Bubble Chart (3.5) 1.

2.

3.

4.

The variability in GGEI increases as the proportion of people with internet access increases. There does not appear to be a strong relationship between internet access and the GGEI score. The range of GGEI scores among countries with larger proportions of people with internet access varies greatly, ranging from about 0.25 to 0.8 as the proportion of people with internet access approaches 100%.

C. GNI per capita D. 75–100% STAR Framework: ● Scope the Data’s Context: This bubble chart plots the Global Green Economy Index (GGEI) (yaxis) against the proportion of people in a country with internet access (x-axis), with bubble sizes representing GNI per capita values of each country.

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● ● ●

Track the Question Behind the Exploration: Do countries with higher economic status also have higher digital connectivity and green economic practices? What is the nature of the relationship between these three variables?

Articulate the Results: The plot suggests a trend where higher internet access level suggests a wider range of GGEI values. Additionally, larger bubbles, indicating higher GNI per capita, are generally found toward the right, suggesting that wealthier countries have higher internet access.

Respond with Next Steps: The visualization points to the potential for digital infrastructure to support green economic growth, particularly in wealthier nations, and prompts further investigation into how internet access can be leveraged for sustainable development in countries at all income levels.

Try It Yourself: Interpret a Heatmap (3.5) 1.

2.

3.

4.

The high-income group has the lowest values of mortality rates for children under 5. The lower-middleincome group contains the country with the highest under-5 mortality rate.

Bright orange Papua New Guinea STAR Framework ● Scoping the Data’s Context: The heatmap illustrates the under-5 mortality rate (color) by income group (x-axis) and country (y-axis) in Oceania. ● Tracking the Question Behind the Exploration: This visualization seeks to explore how economic status correlates with child mortality rates in Oceania, aiming to understand the impact of income on child health outcomes in this specific region. ● Articulating the Results: The pattern shows that higher-income countries, like Australia and New Zealand, have much lower under-5 mortality rates, while lower-middle-income countries have higher rates, indicating a disparity based on economic status. ● Responding with Next Steps: The next steps could include analyzing the healthcare systems and social policies of countries with lower mortality rates to identify successful strategies that could be replicated in areas with higher mortality rates.

Try It Yourself: Interpret a Geographic Map (3.5) 1.

2.

3.

Venezuela, Bolivia, and Ecuador Chile STAR Framework: ● Scope the Data’s Context: The map visualizes the Gross National Income (GNI) per capita across South American countries, with a color gradient from yellow (higher GNI per capita) to violet (lower GNI per capita). ● Track the Question Behind the Exploration: The map underscores the economic disparities within South America. How does the GNI per capita vary across South America? ● Articulate the Results: The distribution shows that countries such as Chile and Argentina are shaded yellow, indicating they have a higher GNI per capita, while countries such as Venezuela and Bolivia are shared violet representing a lower GNI per capita. This suggests economic variations within South America. ● Respond with Next Steps: Acknowledging these disparities is critical for policymakers to tailor economic development programs. Further analysis could involve exploring the underlying factors contributing to these income differences to address economic inequality across the continent.

Try It Yourself: Navigate the Dangers of Cherry-Picking (3.6) 1.

2.

Figure 3.48A does seem to be cherry-picking. Kenji might ask his teammate if there is any reason that he wanted to shed Kuwait in a good light during their presentation.

STAR Framework: ● Scoping the Data’s Context: Figure 3.48A shows carbon emissions per capita (y-axis) for the UAE and Kuwait for the years (x-axis) from 2010 to 2015, while Figure 3.48B extends this timeline to

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● ●

●

2021. Both charts use lines to represent each country, with time on the x-axis and emissions per capita on the y-axis.

Tracking the Question Behind the Exploration: These figures compare the carbon emissions trends of the two countries. The difference in time frames between the two charts demonstrates how selecting specific periods can influence the perceived trend.

Articulating the Results: Figure 3.48A shows lower emissions for Kuwait than for the UAE with a particularly noticeable difference in 2011-2012. However, Figure 3.48B, with extended data, shows that, despite the drop in 2011-2012, Kuwait’s emission level remains almost the same, while the UAE demonstrated a more persistent decrease after 2015 and has maintained lower emission levels than Kuwait for several years in a row.

Responding with Next Steps: To avoid cherry-picking data, consider the full context and range of available data.

Applying the Concepts Activity: Interpret a Histogram

1.

2.

3.

4.

60 to 85 The distribution is slightly skewed to the left.

22 STAR framework: • Scope the Data’s Context: This histogram depicts the distribution of life expectancy rates worldwide.

Each bar represents the number of countries with life expectancy rates within certain intervals.

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• •

•

Track the Question Behind the Exploration: The underlying question here could be: “What is the typical life expectancy across the globe, and how does it vary among different countries?”

Articulate the Results: The data shows a somewhat multimodal distribution with peaks around the 60, 70-75 and 82, suggesting that there are significant groups of countries with average life expectancies in these ranges. The distribution is slightly skewed to the left, so there are few countries with life expectancies below 60, as low as 52..

Respond with Next Steps: These results prompt us to investigate what factors might contribute to higher life expectancies in the groups around the 60s, 70s and 80s, and what might be causing lower life expectancies in some countries. Further examination could involve studying healthcare systems, socioeconomic status, and lifestyle factors in different countries.

Activity: Describe a Density Plot

1.

2.

3.

Right skewed Most countries have under-5 mortality rates below 30, with most frequent values being around 10. The number of countries with the mortality rate above 30 is significantly lower and decreases with the mortality rate.

STAR framework: • Scope the Data’s Context: The density plot illustrates the distribution of the under-5 mortality rate across countries worldwide. The plot represents the density of the mortality rates, showing where the values are concentrated over the interval.

• Track the Question Behind the Exploration: The critical question we're examining is: "What is the global distribution of the under-5 mortality rate, and where do most countries fall within this distribution?"

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• •

Articulate the Results: The plot shows a pronounced peak at the lower end of the under-5 mortality rate, indicating that a majority of countries have lower rates of child mortality. As the rate increases, the density diminishes, suggesting that high child mortality rates are less common.

Respond with Next Steps: Considering these results, the next steps would involve investigating the factors contributing to lower under-5 mortality rates in the countries represented by the peak and understanding why some countries have higher rates. This could include analysis of healthcare access, economic conditions, and nutrition.

Activity: Work with a Box Plot

1.

2.

3.

4.

From 2 to 14 years.

9 14 STAR framework: • Scope the Data’s Context: The box plot represents the distribution of mean years of schooling worldwide. It summarizes the central tendency and dispersion of the data, showing the median, quartiles, and potential outliers.

• Track the Question Behind the Exploration: What is the typical amount of schooling people receive on average across different countries, and how much does this vary?

• Articulate the Results: The median years of schooling, indicated by the line inside the box, appear to be around 9 years. The box, representing the interquartile range, suggests that the middle 50% of countries have mean years of schooling roughly between 6 to 12 years. There are no visible outliers or extreme values in this plot.

• Respond with Next Steps: With this distribution in mind, we could next explore the relationship between mean years of schooling and other factors such as economic development, employment 43 © 2026 Pearson Education, Inc.

Chapter 3: Making Sense of Data Through Visualization

rates, or quality of life indicators. Further research could also segment these findings by region, gender, and urban vs. rural settings to understand the disparities in educational attainment.

Activity: Interpret a Violin Plot

1.

2.

3.

Below 20,000 4 STAR framework: • Scope the Data’s Context: This violin plot shows the distribution of Gross National Income (GNI) per capita across countries worldwide. It combines elements of a box plot with a density plot to show the distribution of the data and its density.

• Track the Question Behind the Exploration: The question at hand may be: "What does the distribution of GNI per capita look like across different nations, and are there noticeable deviations or anomalies?"

• Articulate the Results: The plot reveals that most countries have a GNI per capita on the lower end of the scale, with a long tail extending to higher GNIs. The thick part of the violin indicates a higher density of countries with similar GNI per capita values. The box plot within the violin shows the median and interquartile range, and the whiskers extend to the more extreme data points. Outliers are present and represented as individual points beyond the whiskers.

• Respond with Next Steps: The data invites further investigation into why most countries have lower GNI per capita and what factors contribute to the wide distribution observed. We might analyze how economic policies, industry distributions, and international trade agreements impact GNI per capita.

Additionally, examining the outliers could provide insights into unique national circumstances leading to exceptionally high or low GNIs.

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Activity: Ponder a Word Cloud 1.

2.

you, call, your, free, now (“the” is also acceptable) STAR framework: • Scope the Data’s Context: The word cloud visualizes the most frequently occurring words. The size of each word corresponds to its relative frequency in the spam texts.

• Track the Question Behind the Exploration: What are the common keywords used in spam text messages, and what patterns can be discerned from their frequency?

• Articulate the Results: The dominant words like "free", "text", "claim", "prize", and "call" suggest a trend in spam texts towards offers that prompt immediate action, often implying urgency or reward.

This pattern indicates tactics that spammers use to entice recipients to engage.

• Respond with Next Steps: Recognizing these patterns, the next steps could include developing spam filters that target these keywords or educating users about the characteristics of spam messages to prevent fraudulent activities. Additionally, investigating the effectiveness of such keywords in user response rates could further enhance spam detection strategies.

Activity: Interpret a Column Chart

1.

2.

3.

4.

Proportion of Countries on the Continent Africa Asia She is incorrect because there are fewer countries in Oceania than in the other continents.

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Chapter 3: Making Sense of Data Through Visualization

Activity: Navigate a Treemap

1.

2.

3.

4.

Africa Europe Samoa and Tonga Moldova

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Data Science for All

Activity: Contemplate a Box Plot with a Categorical Variable

1.

2.

3.

4.

Oceania 72 Oceania STAR framework: • Scope the Data’s Context: This box plot displays the distribution of World Happiness Report scores across four continents: Africa, Asia, Europe, and Oceania. It depicts the median, quartiles, and outliers for happiness scores on each continent.

• Track the Question Behind the Exploration: The question we're examining is: "How does reported happiness vary across Africa, Asia, Europe, and Oceania according to the World Happiness Report?"

• Articulate the Results: The median happiness score is highest in Oceania and lowest in Africa, with Asia and Europe in the middle. The span of scores is widest in Asia, indicating greater variability in happiness scores. Oceania shows the least variability, suggesting more consistency in happiness levels across Oceania countries.

• Respond with Next Steps: These findings encourage us to investigate what factors contribute to the differences in happiness scores across continents. We could look into economic, social, and political conditions, as well as cultural attitudes towards happiness. Additionally, further research could include more continents and explore trends over time.

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Chapter 3: Making Sense of Data Through Visualization

Activity: Interpret a Violin Plot with a Categorical Variable

1.

2.

3.

4.

Lower middle income. The violin for lower middle income is longer than the upper middle income’s violin.

77 71 STAR framework: • Scope the Data’s Context: The violin plot represents life expectancy within two different income groups: lower middle income and upper middle income. This visual combines the box plot of data points and the density plot of these data points.

• Track the Question Behind the Exploration: The guiding question is: "How do life expectancy distributions differ between lower middle income and upper middle income groups?"

• Articulate the Results: The violin plot indicates that upper middle income groups tend to have a higher life expectancy, with a more concentrated range of data points, whereas lower middle income groups have a lower median life expectancy with a more distributed distribution.

• Respond with Next Steps: The next steps would involve exploring the factors contributing to the differences in life expectancy between these income groups. It would be important to analyze the impact of healthcare access, nutrition, and living conditions. Further research could also compare these findings against high and low income groups for a broader economic perspective.

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Activity: Interpret a Scatterplot

1.

2.

3.

There is a positive association between these variables, although it seems to become weaker with increasing GNI per capita values. here are no extreme outliers from the main trend, but values of several countries can be pointed out. For example, Brunei Darussalam and the United States have GNI per capita close to $65,000 and life expectancies of 75 and 77, respectively. That is several years lower than other countries of the comparable GNI per capita values.

83 STAR framework: • Scope the Data’s Context: This scatterplot visualizes the relationship between Gross National Income (GNI) per capita and life expectancy. Each point represents a country, with its position determined by its GNI per capita and life expectancy.

• Track the Question Behind the Exploration: The central question is: "Is there an association between a country's wealth, as indicated by GNI per capita, and the health of its citizens, as indicated by life expectancy?"

• Articulate the Results: The scatterplot, along with the fitted line, suggests a positive association between GNI per capita and life expectancy. As GNI per capita increases, life expectancy tends to increase as well.

• Respond with Next Steps: To delve deeper, we could conduct a statistical analysis to quantify the strength of the association. Further research may involve looking at other factors that could influence life expectancy, such as healthcare access, education, and lifestyle, and conducting a multivariate analysis to see how these factors interact with GNI per capita.

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Chapter 3: Making Sense of Data Through Visualization

Activity: Explore a Line Chart

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2.

3.

It roughly doubled from 40 to 80.

1918 STAR framework: • Scope the Data’s Context: The line chart illustrates life expectancy in the United States over time, from the early 1900s projected through to the year 2020.

• Track the Question Behind the Exploration: The question that this visualization addresses is: "How has life expectancy in the United States changed throughout the 20th and into the 21st century, and what are the projected trends into the future?"

• Articulate the Results: There is a clear overall upward trend in life expectancy from 1900 to the present, with a notable dip around the 1920s. The projection suggests that this upward trend is expected to continue into the future.

• Respond with Next Steps: Given this historical trend and future projection, we should consider investigating the factors contributing to the increase in life expectancy, such as advancements in healthcare, nutrition, and living standards. Additionally, we might explore the implications of an aging population for various sectors like healthcare, social security, and the workforce.

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Activity: Describe a Bubble Chart

1.

2.

3.

There is a positive association between these variables.

They tend to have World Happiness Scores from 40 to 80.

STAR framework: • Scope the Data’s Context: This bubble chart plots the relationship between the mean years of schooling and World Happiness Scores across various countries. Additionally, the size of each bubble represents the population of the country.

• Track the Question Behind the Exploration: The primary question being explored is: "Is there an association between the average education level, as measured by mean years of schooling, and the perceived happiness within a country, and how does the population size factor into this?"

• Articulate the Results: There appears to be a general trend that countries with higher mean years of schooling also report higher happiness scores. The distribution and size of the bubbles indicate that this trend is observed across countries with varying population sizes.

• Respond with Next Steps: Further analysis could involve a more detailed statistical examination of the relationship between education and happiness while controlling for population size. Additionally, qualitative studies might explore why education is associated with happiness, such as through increased economic opportunities or broader social benefits. It would also be beneficial to consider other potential contributing factors to happiness beyond education and population.

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Activity: Interpret a Heatmap

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2.

3.

GGEI is higher for higher income groups and lower for income groups.

High values of GGEI are shown in blue and are lighter (The answer depends on the chosen color scheme).

Bolivia

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Activity: Interpret a Geographic Map

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2.

3.

Haiti Canada STAR framework • Scope the Data’s Context: This geographic map displays life expectancy across countries in North America. The color gradient represents different ranges of life expectancy, with each country shaded according to its average life expectancy data.

• Track the Question Behind the Exploration: The exploration seeks to answer: "What are the life expectancy levels across different countries in North America, and how do they compare?"

• Articulate the Results: The map shows a variation in life expectancy across the continent. Countries with higher life expectancies are shaded in darker colors (depending on the chosen color scheme), and it appears that Canada has a higher life expectancy compared to the United States and Mexico.

• Respond with Next Steps: To understand the factors influencing these differences, we could look into healthcare systems, lifestyle, economic development, and other social determinants of health in each country. Further research could compare these findings with other continents or explore subnational differences within each country.

Activity: Navigate the Dangers of Cherry-Picking 1.

2.

Figure 3.82A represents a time in world history when the child mortality rate increased due to worldwide events.

Figure 3.82B shows the trend of a decreasing child mortality rate in the United States over a longer period.

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Chapter 4: Exploratory Data Analysis

Chapter 4: Exploratory Data Analysis Chapter Review Questions Section 1: Central Tendency 1.

Explain central tendency.

Answers may vary. Example: Central tendency is the typical value a variable assumes in a dataset.

2.

What does the median of a dataset represent?

It represents the value below which 50% of the data falls.

3.

Why is a 90% trimmed mean considered useful?

It reduces the impact of extreme values in the dataset.

4.

What does it mean to say that a dataset has a mode of 100,000?

100,000 is the most frequent value.

5.

What is another term for “the average”?

The mean

6.

What does a trimmed mean aim to achieve?

It aims to exclude the most extreme data values for a more representative mean.

7.

Which measure of central tendency is defined by the middle value in an ordered dataset?

The median

8.

In a dance class, 5 students practiced for the following hours: 5, 6, 7, 8, and 9. What is the mean practice time?

7 hours

9.

In a literature class, students read a varying number of books. If the counts are 2, 3, 3, 4, 4, and 5, what is the median number of books read?

3.5 books

10. During his thesis presentation, Raj, a sociology major, highlighted the most repeated score in his survey data. What statistic was he referring to?

The mode 11. In a geography class, what statistic would students be taught to identify the central tendency of population distribution?

The median

Section 2: Variability 12. How do we find the range of a variable in a dataset?

The range can be found by solving for the difference between the maximum and minimum values of the dataset.

13. During an annual math competition, what would contestants have to calculate as the square root of a variable’s variance?

The standard deviation

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14. What does variance measure for a variable in a dataset?

It measures the average squared distance of the values from the mean.

15. Explain how standard deviation is roughly calculated.

Standard deviation is roughly calculated by solving for the average distance from each value to the mean.

16. What does a large standard deviation indicate?

A wide variability of data around the mean.

17. How is variance calculated?

It is calculated as the average squared distance from the mean.

18. Why is the standard deviation more interpretable than the variance?

Standard deviation is in the same units as the original variable.

19. When comparing groups, why is variability important?

It shows how spread out the values are within each group.

20. What does a small range in a dataset indicate about its data points?

It indicates that the data points are close to each other.

21. Which measure of variability is not in the same units as the original variable?

Variance 22. Why is the variance considered non-interpretable?

Its unit is the square of the units of the original variable.

23. Students completed projects in [14, 7, 9, 10, 12] days in a computer science class. What is the variance in project completion days squared?

5.84 days squared 24. In an engineering class, students took [15, 6, 10, 5, 15] hours to design their models. What is the standard deviation? Round to the nearest whole number.

4 hours 25. In an environmental science class, students recorded the temperatures over 5 days as: 10, 15, 20, 25, and 30°C. What is the temperature range?

20°C 26. In a business class, 5 startups raised $2 million, $3 million, $4 million, $5 million, and $6 million. What is the range of funds raised? $4 million 27. In a sports class, athletes ran a distance in seconds as follows: 50, 55, 60, 65, and 70. What is the range of their running time?

20 seconds

Section 3: Shape 28. How is a left-skewed distribution characterized?

A long tail on the left side.

29. What does it mean if a distribution is symmetric?

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It mirrors itself on either side of the center.

30. What does a right-skewed distribution look like? What does that tell us about the data?

A right-skewed distribution has a longer tail on the right side. The right tail represents the larger values.

Most observations in a right-skewed distribution will be small.

31. Describe shape, as it relates to the overall appearance of data when graphed.

Shape describes how a quantitative variable in a dataset looks when graphed using a histogram.

32. What term is used to describe a distribution with the bulk of observations being small and a few much larger?

Right skewed.

33. In a symmetric distribution, what is typically true about the mean, median, and mode?

They are all approximately equal.

34. Which shape descriptor is most likely to identify a distribution with a few observations much smaller than the rest?

Left skewed.

35. How is a histogram with two distinct peaks most likely to be described?

Bimodal 36. If a variable’s values appear at regular frequencies, and the mean, median, and mode occur near the same value, how would its shape be defined?

Symmetric

Section 4. Resistant Central Tendency and Variability 37. Why is the mean considered a non-resistant measure of central tendency?

Its value is affected by outliers.

38. What does a resistant statistic indicate about the statistic’s value?

The statistic's value is not sensitive to extreme observations.

39. How does the trimmed mean compare to the mean in terms of resistance to outliers?

More resistant than the mean 40. Why is it important to consider the shape, center, and variability of a dataset together?

To get a complete picture of the dataset's characteristics 41. What is a benefit of using the median as a measure of center?

It is resistant to outliers.

42. What is the disadvantage of using the range as a descriptive statistic?

It is non-resistant.

43. In the context of descriptive statistics, what does “spread” refer to?

It refers to the measurement of how much variability is present in the data.

44. Which descriptive statistic provides the most information about the tails of a distribution?

The range 45. What does the standard deviation measure?

The average distance of the data points from the mean 56 © 2026 Pearson Education, Inc.

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Section 5. Data Associations 46. In a joint project between the math and psychology departments, what would students have used to determine the relationship between their numeric stress levels and their GPA? Correlation coefficients 47. In her environmental studies, what kind of visualization would Emily have used to show the association between air pollution levels and rates of respiratory diseases?

Scatterplots 48. What does the correlation coefficient measure?

The strength and direction of the linear relationship between two quantitative variables.

49. A correlation coefficient of -1 or 1 indicates what kind of relationship between two variables?

A strong relationship 50. In exploratory data analysis, what kind of relationship would you explore with side-by-side box plots?

Quantitative-by-Categorical 51. If a scatterplot shows a pattern where one variable increases as another decreases, how would the correlation coefficient likely be described?

Negative 52. Which graphical representation would be most appropriate to compare the distribution of a quantitative variable across different levels of a categorical variable?

Side-by-side box plots 53. In a dataset, if the mean and standard deviation of a quantitative variable are different for different values of a categorical variable, what might this suggest?

There is a potential quantitative-by-categorical relationship.

54. What methods are most useful when examining the relationship between two categorical variables?

Computing conditional counts or proportions 55. What type of chart is best suited for visualizing the distribution of a categorical variable across different values of another categorical variable?

Bar chart 56. What does a 2-way table of counts help to analyze?

The relationship between two categorical variables 57. What kind of relationship is being investigated when looking at how the summary measures of a quantitative variable change with the levels of a categorical variable?

Quantitative-by-Categorical 58. At an architecture symposium, Alex presented a visualization that represented the minimum, first quartile, median, third quartile, and maximum of the urban density data. What kind of visualization was it?

A box plot 59. In a sociology class, students were taught to use a table to understand the relationship between two categorical variables like gender and voting preference. What kind of table did they use?

Categorical-by-Categorical

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Section 6. Identifying Outliers 60. What is the purpose of identifying outliers in a dataset?

To determine extreme observations that may influence the analysis 61. According to the Quartile Method, what constitutes a mild outlier?

An observation’s value that falls below Q1 - 1.5*IQR or above Q3 + 1.5*IQR 62. When using the Mean/SD Method, what are the conditions for an observation’s value to be considered a regular outlier?

It must fall above mean + 3SD or below mean - 3SD 63. How is the interquartile range (IQR) calculated?

By subtracting Q1 from Q3 64. At an economics conference, Professor Thompson presented a visualization to display income distribution, highlighting a few outliers of extremely high income. What kind of visualization would she have used?

A box plot 65. In a sports science seminar, participants discussed how NBA players’ heights are anomalous when compared to the heights of the general population of athletes because they stand (literally) above the rest.

What term would be used to describe these observations?

Outliers 66. In a psychology experiment, the research team had to reconsider their findings, because a data point might have skewed the results. What would they call that observation?

An outlier 67. What can the identification of outliers help to determine in data analysis?

The presence of skewness in the data 68. Why might outliers be considered “mild” according to the Mean/SD Method?

Because they fall between 2 and 3 standard deviations from the mean 69. In the Quartile Method of identifying outliers, what does an observation's value falling below Q1 1.5*IQR indicate?

It is a mild outlier.

70. In a mathematics class, if the interquartile range of a student’s scores is 10 points and the third quartile is 75 points, what is the value a score must exceed to be considered a mild high outlier using the Quartile Method?

90 71. In a mathematics class, if a dataset consists of the numbers 5, 6, 7, 8, and 9, what is the third quartile?

8 72. In a sociology class analyzing social behaviors, if the mean behavioral score is 50 and the standard deviation is 8 points, what value must a score fall below to be considered a mild low outlier using the Mean/SD Method?

Below 34.

73. In a biology class studying plant growth, if the mean height of plants is 150 cm and the standard deviation is 15 cm, what value must a plant’s height exceed to be considered a regular high outlier using the Mean/SD Method?

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195 cm 74. Psychology students are conducting a study on stress levels during exams, rated on a scale of 1 to 10.

Calculate the upper and lower thresholds for determining a mild outlier using the Quartile Method based on the following dataset: [5, 7, 8, 9, 10].

Answer: Lower threshold for mild low outlier: 4.00 Upper threshold for mild high outlier: 12.00 75. In a biology experiment, plant growth is measured in centimeters. Calculate the upper and lower thresholds for determining a regular outlier using the Quartile Method based on the following data in centimeters per quarter of growth: [20, 25, 30, 35, 40].

Answer: Upper threshold for regular high outlier: 65.00 Lower threshold for mild low outlier: 10.00 76. Economics students estimate the price (in dollars) of a basket of goods in a market study. Calculate the upper and lower thresholds for determining a regular outlier using the Quartile Method based on the following dataset: [200, 220, 240, 260, 280].

Answer: Lower threshold for regular low outlier: 100.00 Upper threshold for regular high outlier: 380.00 77. A sociology research group studies the number of social interactions students have daily. Calculate the upper and lower thresholds for determining a mild outlier using the Mean/SD Method based on the following dataset: [15, 18, 20, 22, 25].

Answer: Upper threshold for mild high outlier: 27.62 Lower threshold for mild low outlier: 12.38 78. An environmental science class tracks the number of recyclable items collected weekly. Calculate the upper and lower thresholds for determining a regular outlier using the Mean/SD Method based on the following dataset: [50, 55, 60, 65, 70].

Answer: Upper threshold for regular high outlier: 83.72 Lower threshold for regular low outlier: 36.28 79. During a climate change seminar, researchers noted an exceptionally hot year in the temperature data over the past century. How would the researchers describe that data point?

An outlier 80. A computer science class observes the number of lines of code written for a project. Calculate the upper and lower thresholds for determining a mild outlier using the Mean/SD Method based on the following dataset: [1000, 1200, 1300, 1400, 1500].

Answer: Lower threshold for mild low outlier: 895.29 Upper threshold for mild high outlier: 1664.71 81. A music class measures various musical performances' length (in seconds). Calculate the upper and lower thresholds for determining a regular outlier using the Mean/SD Method based on the following dataset: [200, 220, 240, 260, 280].

Answer: Upper threshold for regular high outlier: 334.87 Lower threshold for regular low outlier: 145.13

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82. In his biology research, Dr. Lee should use what kind of visualization to identify potential outliers in a single variable representing plant growth?

Box plots

Explaining the Concepts 1.

The standard deviation, median, and IQR are statistics that would help conclude which company has better overall salaries. If the distribution of company A’s salaries is right-skewed, i.e. there are outliers in company A with significantly higher salaries, the company’s average salary could be inflated, making the average salaries the same at each company while company A’s average salary is much lower than company B’s average salary if the outliers in company A were ignored.

2.

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The 2nd histogram would not be considered as skewed to the right. In a skewed-to-the-right histogram, most observations are below the mean and there is a noticeably larger tail on the right. Here, the histogram is roughly normal and symmetric when we ignore these outliers. Exploring potential outliers is important since outliers can drastically change the value of certain non-resistant statistics (e.g. mean, maximum, and minimum). Additionally, describing the shape of the distribution can determine whether the Quartile Method or Mean/SD Method will remove more outliers.

3.

A statistic used to summarize or describe the data should provide a clear overview of the data. If the descriptive statistic is resistant, i.e. robust to outliers, then it provides a good summary of the data and won’t noticeably change if the data contains outliers.

A famous example of using non-resistant descriptive statistics is when universities deflate the average number of students in courses to improve a university’s ranking. Another example is when companies post salaries in a job posting. The company might post an artificially inflated average salary to attract job applicants.

4.

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4.1.

This graph shows that there is no association between the two variables X and Y. The value of X does not affect the value of Y (and vice-versa).

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4.2.

The value of the quantitative variable is independent of the value of the categorical variable. The means and standard deviation of the quantitative variables are roughly similar across the 4 different values for the categorical variable.

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4.3.

This graph tells us there is no association between the two categorical variables 1 and 2. The distribution of values for this categorical variable 1 is the same for different values of categorical variable 2 (X, Y, and Z).

5.

If a dataset used a special number for missing values that were either extremely large or extremely small, then the Quartile Method and the Mean/SD Method for identifying outliers in a dataset might fail to capture implausible or error values in the dataset. Imagine a dataset about houses, where each property is an observation and the age of property is a variable. If missing ages are recorded as 999, then the sample mean and the higher quartiles will be inflated. If the missing ages are recorded as 0 or negative numbers, then the sample mean and the higher quartiles will be deflated. Understanding variables in a dataset can determine which methods are most appropriate for exploratory data analysis. Cleaning the data based on the understanding of the variables allows one to correctly use the appropriate exploratory data analysis method.

Try It Yourself Try It Yourself: Calculate the Mean, Mode, Median, and 90% Trimmed Average (4.1) Mean of user rating: 7.1 90% trimmed average of user rating: 7.2 Median of user rating: 7.5 Mode of user rating: 7.8

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Try It Yourself: Calculate the Range, Variance, and Standard Deviation (4.2) Range of user rating: 9.7 Variance in user rating: 2.3 Standard deviation of user rating: 1.5

Try It Yourself: Describe the Shape of the Histogram (4.3) lower left-skewed down higher lower lower

Try It Yourself: Investigate Resistance (4.4) Mean: 544.5 80% Trimmed Mean: 562.0 Median: 628 Range: 927 Variance: 81,940.5 Standard Deviation: 286.3

Try It Yourself: Plot Correlations between Two Quantitative Variables (4.5) 1.

2.

3.

more, 0.58, closely, better.

0.03, no, 0.

STAR Framework: ● Scope the Data’s Context: The scatter plot depicts the relationship between critic_rating and user_rating within the Video Games dataset, highlighting how these two assessments compare. ● Track the Question Behind the Exploration: The analysis seeks to determine the degree of alignment between critics' evaluations and user feedback on video games. ● Articulate the Results: A moderate positive correlation of 0.58 is shown, indicating that higher critic ratings tend to be associated with higher user ratings, but with a considerable amount of scatter. ● Respond with Next Steps: For game developers, understanding this relationship can aid in managing expectations and targeting audiences. Further analysis could explore if certain genres of games show stronger correlations between critic and user ratings.

Try It Yourself: Match Box Plots with Histograms (4.5) Role-Playing: Figure 4.18 Adventure: Figure 4.19 Strategy: Figure 4.17 Explanation: Role-Playing Genre: Figure 4.18’s histogram matches the box plot for the Role-Playing genre, with a median rating that aligns closer to 8. The histogram shows a more symmetrical distribution, which corresponds with the box plot's fewer outliers, centered median, and a higher Q1 (the bottom of the box).

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Adventure Genre: Figure 4.19’s histogram matches the box plot for the Adventure genre, with a lower median than the Role-Playing genre. Even more than the Strategy genre, this histogram has frequent low ratings, which match the distribution and lower outliers in the box plot.

Strategy Genre: Figure 4.17’s histogram matches the box plot for the Strategy genre, with a median the same as the Adventure genre. However, it has a less left-skewed distribution compared to the Adventure genre.

Try It Yourself: Associate Two Categorical Variables (4.5) 1.

2.

3.

4.

5.

6.

Yes Sports M Action Action STAR Framework: ● Scope the Data’s Context: The side-by-side bar chart compares the frequency of different maturity ratings within video game genres, providing insights into the distribution of content suitability across Action, Sports, Shooter, and Role-Playing categories. ● Track the Question Behind the Exploration: The main question is how video game maturity ratings, such as “E” for Everyone, “M” for Mature, “T” for Teen, and “E10+” for Everyone 10 and older, are distributed across various game genres. ● Articulate the Results: Sports games predominantly have an “E” rating, indicating they are suitable for all ages, while Shooter games have a higher count of “M” ratings, suggesting content aimed at mature audiences. Action games show a balanced distribution of “T” and “M” ratings, reflecting their broader appeal across different age groups. ● Respond with Next Steps: Game developers and marketers can utilize this data to target appropriate audiences. Additional research could involve examining the impact of these ratings on sales and player engagement within each genre.

Try It Yourself: Determine Outliers Using the Quartile and Mean/SD Methods (4.6) 1.

The following table has the correct answers.

Type

Method

Lower Threshold

Upper Threshold

Mild Outliers

Quartile Method

-0.81

1.64

Regular Outliers

Quartile Method

-1.72

2.55

Mild Outliers

Mean/SD Method

-3.01

4.49

Regular Outliers

Mean/SD Method

-4.89

6.37

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2.

B. When “global_sales”, which only has positive values, gets a negative lower threshold in outlier detection, it means there are no outliers on the lower end (but there still might be outliers above the upper threshold).

3. a. b. c. d.

Mild Outliers (Quartile Method): 766 Regular Outliers (Quartile Method): 425 Mild Outliers (Mean/SD Method): 167 Regular Outliers (Mean/SD Method): 91

Applying the Concepts Activity: Calculate the Mean, Mode, Median, and 90% Trimmed Average 1. 77.5 2. 78.0. The 90% trimmed average is higher due to the exclusion of extreme scores, both low and high, which are included in the mean calculation.

3. 78.0. This signifies that half of the movies have a score higher than this value, and half lower.

4. 76.0. This indicates that this value is the most frequently occurring score.

Activity: Calculate the Range, Variance, and Standard Deviation 1. 72.0. This represents the difference between the highest and lowest critical ratings.

2. 156.1. This indicates the degree to which these ratings differ from the average.

3. 12.5. This shows the average extent to which scores vary from the mean.

4. There is a notable spread in how films are appraised by critics.

Activity: Describe the Shape of the Histogram 1. The mean IMDb rating is higher than the median.

2. The histogram is a right-skewed distribution. This means there are a number of films with particularly high ratings that are pulling the mean up.

3. Outliers are pulling the value of the mean higher.

4. This suggests that while most films receive moderate to high ratings, there exists a subset of films that are rated much higher.

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Activity: Investigate Resistance

Statistic

Value

Value with New Outlier

Resistant?

Mean

68.0

80.0

No

90% Trimmed Mean

52.2

52.6

Yes

Median

23.5

23.6

Yes

Mode

4.4

4.4

Yes

Range

936.7

10000.0

No

Variance

12045.1

130593.0

No

Standard Deviation

109.8

361.4

No

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Activity: Plot Correlations between Two Quantitative Variables

1.

2.

3.

4.

5.

6.

No Yes No No correlation -0.03 They are not related.

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7.

8.

9.

Weak positive “A correlation coefficient of 0.25 indicates that shorter runtimes appear to be weakly associated with lower user ratings.

This does not support the friend’s hypothesis that people like shorter movies.

Activity: Match Box Plots with Histograms 3.

Drama: Figure 4.29 Comedy: Figure 4.30 Animation: Figure 4.28

4.

STAR Framework: ● ● ● ●

Scope the data's context: The box plots visualize IMDb movie runtimes for Drama, Animation, and Comedy genres, offering a snapshot of how film durations compare across these categories.

Track the question behind the exploration: We examine the variability and central tendency of movie runtimes within each genre to discern patterns and deviations specific to these types of films.

Articulate the results: Dramas show the longest and most variable runtimes with several outliers, while comedies and animations are shorter on average with the most consistent duration.

Respond with next steps: These insights can inform industry decisions on film production and scheduling; further analysis might explore runtime correlations with commercial success or shifts in genre-specific runtime trends over time.

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Activity: Associate Two Categorical Variables

1.

2.

3.

4.

5.

The distribution of certificates is dependent on the genre.

Drama Drama: R, Action: UA, Comedy: U, Animation: U Action Animation

Activity: Determine Outliers Using the Quartile and Mean/SD Methods 1.

Type

Method

Lower Threshold

Upper Threshold

Mild Outliers

Quartile Method

7.10

8.70

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Regular Outliers

Quartile Method

6.50

9.30

Mild Outliers

Mean/SD Method

7.38

8.51

Regular Outliers

Mean/SD Method

7.10

8.80

2.

f imdb_rating, which only has positive values, had a negative lower threshold in outlier detection, there would be no outliers on the lower end.

3.

4.

• High Mild Outliers (Quartile Method): 13 • High Regular Outliers (Quartile Method): 0 (1 is acceptable) • High Mild Outliers (Mean/SD Method): 29 • High Regular Outliers (Mean/SD Method): 13 • Low Mild Outliers (Quartile Method): 0 • Low Regular Outliers (Quartile Method): 0 • Low Mild Outliers (Mean/SD Method): 0 • Low Regular Outliers (Mean/SD Method): 0 STAR Framework ● Scope the data's context: The table presents criteria for identifying outliers in IMDB ratings from the IMDb movie dataset using two different statistical methods, which is essential for analyzing the future success of movies. ● Track the question behind the exploration: We aim to determine acceptable ranges for movie ratings to identify unusually high or low performers, setting the stage for a more nuanced outcomes analysis. ● Articulate the results: The Quartile Method indicates regular outliers as film ratings above 9.30, and no such movies are in the dataset. Those above 8.70 are mild outliers, with 13 in the dataset.

The Mean/SD Method is less conservative with 13 regular outliers above 8.80 and 29 mild outliers above 8.51. No movies are rated below any outlier threshold. ● Respond with next steps: Applying these thresholds to the ratings data can refine the understanding of movie success and prompt a deeper dive into the factors contributing to these ratings outliers.

Chapter 5: Data Management Chapter Review Questions Section 1: Asking Questions of Data 1.

Why is it important to use queries in data science?

To help resolve questions to achieve goals

2.

What does a query represent in data science?

A request for information from data

3.

What are keys in the context of data tables?

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Regular Outliers

Quartile Method

6.50

9.30

Mild Outliers

Mean/SD Method

7.38

8.51

Regular Outliers

Mean/SD Method

7.10

8.80

2.

f imdb_rating, which only has positive values, had a negative lower threshold in outlier detection, there would be no outliers on the lower end.

3.

4.

• High Mild Outliers (Quartile Method): 13 • High Regular Outliers (Quartile Method): 0 (1 is acceptable) • High Mild Outliers (Mean/SD Method): 29 • High Regular Outliers (Mean/SD Method): 13 • Low Mild Outliers (Quartile Method): 0 • Low Regular Outliers (Quartile Method): 0 • Low Mild Outliers (Mean/SD Method): 0 • Low Regular Outliers (Mean/SD Method): 0 STAR Framework ● Scope the data's context: The table presents criteria for identifying outliers in IMDB ratings from the IMDb movie dataset using two different statistical methods, which is essential for analyzing the future success of movies. ● Track the question behind the exploration: We aim to determine acceptable ranges for movie ratings to identify unusually high or low performers, setting the stage for a more nuanced outcomes analysis. ● Articulate the results: The Quartile Method indicates regular outliers as film ratings above 9.30, and no such movies are in the dataset. Those above 8.70 are mild outliers, with 13 in the dataset.

The Mean/SD Method is less conservative with 13 regular outliers above 8.80 and 29 mild outliers above 8.51. No movies are rated below any outlier threshold. ● Respond with next steps: Applying these thresholds to the ratings data can refine the understanding of movie success and prompt a deeper dive into the factors contributing to these ratings outliers.

Chapter 5: Data Management Chapter Review Questions Section 1: Asking Questions of Data 1.

Why is it important to use queries in data science?

To help resolve questions to achieve goals

2.

What does a query represent in data science?

A request for information from data

3.

What are keys in the context of data tables?

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Variable that ensure observations are identifiable 4.

What can be a consequence of not having unique keys in a table?

Duplicate observations

5.

Define data management. Why is it valuable in data science?

Data management is the process of organizing, storing, and accessing data to retrieve information. It helps data scientists to resolve the questions they are investigating.

6.

Why are keys crucial when merging tables?

To uniquely identify matched observations

7.

What does being identified mean about an observation?

The observation can be uniquely determined using a key.

8.

What is the reason that a table might not have a unique key for each observation?

Answers may vary. There is a possibility of duplicate observations in the table, for example, the same student may be listed several times.

9.

What is the relationship between a table (or multiple tables) and a dataset?

A dataset is a collection of one or more tables.

10. What is special about a primary key?

A primary key is a unique identifier within a specific table.

11. What is a limitation of natural language tools compared to data scientists’ tools?

They often don't have access to internal or personal tables.

Section 2: Selecting Variables 12. What is the primary objective in selecting variables from a table?

To choose variables relevant to the analysis 13. What does selecting variables allow researchers to avoid?

Overloading memory with irrelevant data 14. How can we query the values of a variable from a table?

By selecting the observations of a variable 15. Why is it beneficial to access distinct values of a variable?

It helps to understand the diversity within a variable.

16. Why is the ability to select variables helpful?

It helps to focus on relevant data.

Section 3: Filtering and Ordering Observations 17. What is the purpose of filtering observations in a table?

To select specific observations relevant to a query 18. Suppose you are studying finish times in the most recent NYC Marathon. You wish to focus on race participants who finished in the top 10%. What would you do with your observations?

Answers may vary. Example: You would filter the race participants to show those with finish times in the top 10%.

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19. How do the filtering tools for text variables differ from those for quantitative variables?

Text variables use letter-based operators, while quantitative variables use mathematical operators.

20. In filtering text variables, how is the not equal operator useful?

To query observations that do not fit a certain criterion 21. How can wildcard characters be used in text variable filtering?

To represent unknown characters in pattern matching 22. When would you choose to filter by quantitative variables?

To restrict observations based on numerical criteria 23. How can ordering observations in a dataset be useful?

It can help to sort the observations.

24. What are the common representations of a missing value in a table?

Answers may vary. Example: NULL 25. Why might you sort by a variable in descending order?

To identify the highest values of a variable 26. When might you need to use text variable filtering with wildcard characters?

When you want to find patterns in text 27. What mathematical operator would you use for: a. Equal to? = b. Greater than or equal to? >= c. Less than? < 28. What is a null value?

A null value is an observation with an empty value for a given variable.

29. You are reviewing a list of customers for your company, making sure their contact information is complete.

But you notice that several of them are missing email addresses. What would be appropriate to enter in place of these missing values?

Answers may vary. Examples: “NULL,” “Unknown,” “--,” “N/A”

Section 4: Summarizing and Structuring Data 30. What is a common use of counts in summarizing text variables?

To measure the frequency of each value 31. What kind of descriptive statistics can summarizing queries provide for categorical data?

Counts but not averages, maximums and minimums 32. What kind of descriptive statistics can summarizing queries provide for quantitative data?

Averages, counts, maximums, and minimums 33. What does the grouping variable represent when computing descriptive statistics?

The categories for aggregation 34. When summarizing grouped text data, what is the main advantage of using counts?

They show the size of each group.

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35. Why is it important to summarize data before performing more complex analysis?

To simplify the data for easier understanding 36. Provide an example wherein you might want to create a new variable for your data.

Answers will vary. For example, you have a dataset on flights operated by an airline with numerous parameters listed. Among them are flight distance and the amount of fuel burned. You might want to take a look at the fuel consumption per mile for each flight, so you create a new variable defined as: (fuel burned)/(flight distance).

Section 5. Merging Tables 37. What is the purpose of merging tables?

To combine data from different sources 38. What does an inner join accomplish when merging tables?

Returns only the observations with matching keys in both tables 39. How does a left join differ from an inner join?

It includes all observations from the left table and the matched observations from the right table.

40. What might be a reason to perform a full join on two tables?

To ensure no data is excluded from the analysis 41. Which join type would you use if you need all observations from both tables in the results?

Full join 42. What is a Venn diagram?

A Venn diagram is a visual representation of the relationships between different sets of items. It shows their distinct and common attributes.

43. When dealing with two tables A and B, why might some observations from Table A not appear in a join with Table B? Observations in Table A do not have corresponding key values in Table B. 44. What is the benefit of using Venn diagrams to illustrate table joins?

They illustrate which observations in the tables are included in the join, based on key values.

45. What do you learn from a full join that results in many observations with null values?

There are many unmatched observations between the tables.

46. What type of join returns only observations that match in both tables being joined?

Inner join 47. What type of join returns all observations from the left table, plus matched observations from the right table?

Left join 48. What type of join returns all observations from the right table, plus matched observations from the left table?

Right join 49. What type of join combines tables and keeps all observations, including those without matches?

Full join

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50. What type of join is most likely to return the fewest number of observations?

Inner join

Explaining the Concepts 1.

We store data in many tables because it allows us to select variables relevant to our query without storing irrelevant variables in the computer’s working memory. Data can also be stored in one large table but this leads to inefficiencies and makes it hard for users to operate and understand such a table. We can keep track of unstructured data by storing the paths to the locations of files for unstructured data in a table. This can prevent us from storing the data directly in a table. Amazon and Instagram likely store data using a large number of tables. They likely have tables for user, product, and seller information.

2.

Results are based on Products.csv from TiY_Create_a_New_Variable.

Query 1: Sort by price, filter the top 5, then find the products with the SupplierID greater than 10. Result:

Query 2: Extract all products with a SupplierID greater than 10, sort the values by Price, then take the first 5 items. Result:

Query 3: Extract all products with a SupplierID greater than 10, take the first 5 items, then sort the values

by Price. Result:

3.

It would be better or faster to join two tables first, and then perform other operations if we wanted to find the total number of purchases made by all customers in every city. In TiY: Merge Two Tables (5.5), we need to join the Order and Customers tables first and then group the joined tables by city.

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It would be better or faster to join two tables after first performing other operations if we wanted to find the total number of purchases made by a customer in a certain city during a specific year. In TiY: Merge Two Tables (5.5), we could remove all rows from the Order table that do not have the specified year.

4.

5 examples of unique keys can include: 1. Social Security Number - every citizen is given a unique SSN when they are born or naturalized (though there have been rare instances where two people share a SNN).

2. Phone ID - each physical cell phone comes with a unique ID 3. Bank Account number - each account number at a bank should be unique. Otherwise, customers might access other customers’ accounts.

4. Student ID number - each student at a university should have a unique student ID number to access their individual records 5. Purchase order number - each purchase on a website/store should have a unique ID that allows one to find all information related to that purchase.

5.

Data security is crucial in data management. We do not want anyone to be able to access tables, including students’ contact information, grades, or payment information. We protect people’s privacy by using unique IDs that do not contain any personal information, i.e. names or addresses.

Try It Yourself Try It Yourself: Select Variables (5.2) 1.

2.

Canada Tsawwassen

Try It Yourself: Select Distinct Values of Variables (5.2) 1.

2.

21 ● Argentina (answer: yes) ● Armenia (answer: no) ● Austria (answer: yes) ● Belgium (answer: yes) ● Brazil (answer: yes) ● Cuba (answer: no) ● Mexico (answer: yes) ● New Zealand (answer: no) ● Norway (answer: yes)

Try It Yourself: Filter Text Variables (5.3) 1.

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5 13

Try It Yourself: Filter with the Not-Equal Variable (5.3) 1.

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88 86 78

Try It Yourself: Filter Using a Wildcard (5.3) 1.

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1 7

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Chapter 5: Data Management

3.

1

Try It Yourself: Filter a Quantitative Variable (5.3) 1.

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10 11

Try It Yourself: Filter Missing Values Variable (5.3) 1.

2.

1 Ireland

Try It Yourself: Filter Multiple Conditions Simultaneously (5.3) 1.

2.

3.

1 2 4

Try It Yourself: Order by a Variable (5.3) 1.

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Teatime Chocolate Biscuits Tarte au sucre

Try It Yourself: Summarize a Quantitative Variable (5.4) 1.

2.

3.

28.87 or $28.87 263.50 or $263.50 2.50 or $2.50

Try It Yourself: Summarize a Quantitative Variable by a Categorical Variable (5.4) 1.

2.

32.37 or $32.37 20.68 or $20.68

Try It Yourself: Create a New Variable (5.4) 1846 or ¥1846

Try It Yourself: Merge Two Tables (5.5) 1.

2.

3.

11/27/1996 8/26/1996 or 08/26/1996 STAR Framework: ● Scope the Data’s Context: Zahava requires a consolidated view of customer names and their corresponding order dates, necessitating a join of the Customers table with the Orders table using their mutual CustomerID key, focusing only on current customers with existing orders. ● Track the Question behind the Exploration: Query 5.13 finds the intersection of data where customer names from the current customer base align with their respective order dates, indicating active customer engagement. ● Articulate the Results: Results display a list of customer names alongside the dates when they placed orders, revealing the frequency and recency of orders by each customer. ● Respond with Next Steps: Further analysis could involve investigating patterns in ordering frequency, identifying trends over time, or segmenting customers based on order history for targeted marketing campaigns.

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Applying the Concepts Activity: Select Variables 1.

2.

3.

4.

A Brazil UK Canada

Activity: Select Distinct Values of Variables 1.

2.

3.

B 16 ● Australia (yes) ● Belgium (no) ● Brazil (yes) ● Canada (yes) ● China (no) ● Chile (no)

Activity: Filter Text Variables 1.

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3.

C 3 4

Activity: Filter with the Not-Equal Operator 1.

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D 25 26

Activity: Filter Using a Wildcard 1.

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3.

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C 9 17 3

Activity: Filter a Quantitative Variable 1.

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B 72 386

Activity: Filter Missing Values 1.

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“A, B, C, D. All four queries meet the request.

0

Activity: Filter Multiple Conditions Simultaneously 1.

2.

3.

C 2 2

Activity: Order by a Variable 1.

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Chapter 5: Data Management

2.

3.

Cheryl Saylor Yoshi Nagase

Activity: Summarize a Quantitative Variable 1.

2.

3.

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B 24.6 120 1

Activity: Summarize a Quantitative Variable by a Categorical Variable 1.

2.

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D 18.9 35.6 12.5

Activity: Create a New Variable 1.

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B 390 or ₺390

Activity: Merge Two Tables 1.

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3.

4.

5.

D Beverages Confections Seafood STAR framework: ● Scope the data's context: Zahava is looking to obtain a complete, matched list of products and their corresponding categories from the 'Products' and 'Categories' tables, ensuring that every product is linked to a category and each category has associated products. ● Track the question behind the exploration: To fulfill this requirement, we must identify the type of SQL JOIN that ensures products and categories are mutually inclusive in the results, displaying only matched pairs. ● Articulate the results: An INNER JOIN between the 'Products' and 'Categories' tables accurately meets Zahava's criteria, as it would yield a list where each product is paired with its category without any unlinked records. This is corroborated by the provided answers, which correctly relate individual products to their categories using their names and IDs. ● Respond with next steps: With the precise list of product-category pairs, Zahava can proceed to analyze the product assortment or streamline the product catalog; further analysis could involve assessing the range and distribution of products within each category for inventory management.

Chapter 6: Understanding Uncertainty, Probability, and Variability Chapter Review Questions Section 1: Variability and Uncertainty 1.

What does variability in data primarily represent?

The range of different outcomes in a dataset

2.

Describe uncertainty in the context of data science.

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Chapter 5: Data Management

2.

3.

Cheryl Saylor Yoshi Nagase

Activity: Summarize a Quantitative Variable 1.

2.

3.

4.

B 24.6 120 1

Activity: Summarize a Quantitative Variable by a Categorical Variable 1.

2.

3.

4.

D 18.9 35.6 12.5

Activity: Create a New Variable 1.

2.

B 390 or ₺390

Activity: Merge Two Tables 1.

2.

3.

4.

5.

D Beverages Confections Seafood STAR framework: ● Scope the data's context: Zahava is looking to obtain a complete, matched list of products and their corresponding categories from the 'Products' and 'Categories' tables, ensuring that every product is linked to a category and each category has associated products. ● Track the question behind the exploration: To fulfill this requirement, we must identify the type of SQL JOIN that ensures products and categories are mutually inclusive in the results, displaying only matched pairs. ● Articulate the results: An INNER JOIN between the 'Products' and 'Categories' tables accurately meets Zahava's criteria, as it would yield a list where each product is paired with its category without any unlinked records. This is corroborated by the provided answers, which correctly relate individual products to their categories using their names and IDs. ● Respond with next steps: With the precise list of product-category pairs, Zahava can proceed to analyze the product assortment or streamline the product catalog; further analysis could involve assessing the range and distribution of products within each category for inventory management.

Chapter 6: Understanding Uncertainty, Probability, and Variability Chapter Review Questions Section 1: Variability and Uncertainty 1.

What does variability in data primarily represent?

The range of different outcomes in a dataset

2.

Describe uncertainty in the context of data science.

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The unpredictability of the value of a future observation 3.

Explain why the standard deviation is a useful measure of variability.

To measure the spread or variability of the data

4.

What does a histogram tell us in the context of variability?

The frequency of occurrence of data values

5.

Why is understanding variability important in data science?

It enables the assessment of predictability and uncertainty in data

6.

How does the presence of duplicate records affect data variability?

Does not affect variability

7.

If data have low variability, what does that tell us about the observations?

Observations are closely clustered around the mean

8.

What role does uncertainty play in data analysis?

It underscores the need for thorough data exploration

9.

Explain how range and variability relate to each other.

A larger range typically indicates more variability

10. How is variability in a variable’s values related to uncertainty about that variable?

The more variability there is in a variable’s values, the more uncertainty we have about that variable.

11. How is variability in a variable’s values related to the predictability of that variable?

The more variability there is in a variable’s values, the less predictable it is.

12. How is uncertainty about a variable related to the predictability of that variable?

The more uncertainty we have about a variable, the less predictable it is.

13. What characteristics of a histogram would indicate less uncertainty and more predictability for the variable plotted?

A histogram that is narrower (or more condensed) would demonstrate less variability in the plotted variable, and therefore less uncertainty and more predictability.

14. In a scatterplot of two quantitative variables, what characteristics would indicate that knowledge of one variable increases the predictability of the other?

A noticeable relationship between these two variables, e.g., a linear relationship, would indicate that the predictability of the second variable is higher with knowledge of the first.

15. In a scatterplot of two quantitative variables, what characteristics would indicate that knowledge of one variable does not affect the predictability of the other?

No visible relationship between the two variables would indicate that knowledge of the first variable does not increase the predictability of the second variable.

Section 2: Probability 16. Define a random trial in data science.

An observation leading to an uncertain outcome 17. Give an example of a random trial involving data.

Answers may vary. Example: An example would be recording data about touchdowns scored in a football season.

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Chapter 6: Understanding Uncertainty, Probability, and Variability

18. What does the sample space of a random trial represent?

The collection of all possible outcomes that can occur 19. How many outcomes are in the sample space for a roll of two 6-sided dice? One is green and one is red.

36 20. What are the two rules for probability?

All probabilities within a sample space must sum to 1; All probabilities are between 0 and 1.

21. What is the sum of all probabilities in a sample space?

1.0 22. What are the valid values for probabilities?

0 through 1, inclusive 23. A randomly selected person’s blood type must be A, B, AB, or O. Together, what do these four blood types represent?

Sample space 24. What is the sample space of this random trial?

HH, HT, TH, TT 25. Give an example of an outcome in this random trial.

Answers may vary. Receive $20 for 2 Heads flips 26. What is the sample space for this random trial?

0-10 27. Give an example of an outcome in this random trial.

Answers may vary. Work 5 hours and receive $100

Section 3: Sampling Methods 28. What is the main goal of random sampling in data collection?

To represent the population of interest accurately 29. What is a potential limitation of using a convenience sample?

Non-random selection based on accessibility may not be representative of the population.

30. What is stratified random sampling primarily used for?

Collecting data from every stratum of the population 31. A hospital wants to survey religious participants in their city about what they seek from a hospital chaplain, so they go to the church nearest the hospital and survey every member. What sampling method is this?

Convenience sample 32. In the context of sampling, what is 'scope of inference'?

The range of conclusions that can be drawn from the sample 33. What is an observational study?

When data are collected from participants via a survey or a poll

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34. Each student at a school has a student identification number. Counselors have a computer that generates 100 random identification numbers, and the students associated with those numbers are asked to take a survey. What sampling method is this?

Simple random sample 35. What are the limitations of an observational study?

They’re difficult to be truly random.

36. Why is random sampling important in an observational study?

Removes bias 37. What are the benefits of stratified random sampling?

Minimal time and resources necessary, reduces influence of the surveyor to be closer to the population of interest 38. When is a convenience sample useful?

When there is limited time or limited resources; need data quickly 39. A boss puts employees’ names in a hat and chooses without looking to get a sample of 5 employees. What sampling method is this?

Simple random sample 40. A large university surveys 500 students by taking a random sample of 100 first-year students, 100 secondyear students, 100 third-year students, 100 fourth-year students, and 100 fifth-year students. What sampling method is this?

Stratified sampling 41. A TV network surveys its studio audience members one day in December. What sampling method is this?

Convenience sample 42. Given that Jalen only surveys students that are easy for him to access, what kind of sample would he be collecting?

Convenience sample 43. Given that Jalen is only surveying students in advanced classes, what concerns would you have about the sample he is taking?

It’s not representative of the population of his school.

44. Because each sample of size 50 is equally likely to be selected, what kind of sample would he be collecting?

Simple random sample 45. What kind of sample scheme would that be?

Stratified random sample 46. When working with statistics and data, why is it important to understand whether a sample is taken randomly and is representative of the population?

To more effectively answer our research questions

Section 4: Simulation 47. The probabilities associated with certain events in a random trial can be estimated using which process?

Simulation 48. What is the primary purpose of simulation in probability?

To replicate a real-world process for analysis 83 © 2026 Pearson Education, Inc.

Chapter 6: Understanding Uncertainty, Probability, and Variability

49. In a simulation, what does a larger number of repetitions help achieve?

Better estimation of probabilities 50. What is essential for conducting a successful simulation?

Knowledge of the random trial and associated probabilities 51. How does simulation relate to the concept of variability?

It demonstrates the effects of variability 52. Why is simulation an important tool in understanding random trials?

It allows for the repetition of trials to understand long-run behavior 53. In ten flips of a fair coin, are we guaranteed 5 Heads and 5 Tails?

No. The outcome of this coin flip is random. Even though the probability of Heads is 0.5, we are not guaranteed exactly 5 Heads in 10 flips.

54. How can we use simulation to visualize the distribution of sample proportions of Heads in 30 flips of a coin?

Flip a coin 30 times, many times (or have the computer do this). For each sample of 30 flips, record the proportion of Heads. Then plot a histogram of these recorded proportions.

Section 5: Working with Probabilities and Common Fallacies 55. Explain the law of averages.

The law of averages is the belief that a particular outcome or event will, over specific periods, occur at a frequency similar to its probability.

56. Your friend is spinning a wheel with five equal sections, two of which are blue and three of which are red.

After six spins, she has consistently landed on red. She then says that since she has landed on red so many times, she is more likely to land on blue during her next spin. Is your friend correct? Why or why not?

Incorrect - Gambler’s fallacy 57. Explain the appeal to probability fallacy.

Assuming something is certain to happen because it will probably be the case 58. Murphy’s Law is an application of the appeal to probability. Explain Murphy’s Law. Anything that can go wrong will go wrong 59. Explain the false positive paradox. test might yield more false positives than true positives when testing for a rare condition.

60. In the base rate fallacy, what do people ignore the base rate in favor of?

Information regarding a specific case 61. In the confusion of the inverse fallacy, with what is a conditional probability equated with?

Its inverse 62. Your friend tells you that the probability of it raining today given that it rained yesterday must be the exact same as the probability of there having been rain yesterday given that it is raining today. Which fallacy is exhibited in this statement?

Confusion of the inverse 63. Give an example of the Gambler’s Fallacy.

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Answers may vary. After seeing a coin land on Heads 30 times in a row, believing that you are “due” to see a Tails come up soon.

64. Your friend believes that the probability of you having a disease given that a test for that disease is positive is the same as the probability of a test for that disease being positive given that you have the disease. Which fallacy is exhibited in this statement?

Confusion of the inverse.

Explaining the Concepts 1.

In the first question, the date is fixed while the date is not fixed in the second question. Johnny Carson was born on October 23rd, 1925. The first question asks if any audience members’ birthday is October 23rd. The 2nd question asks whether there is a pair of people in the audience whose birthday is on January 1st OR is there a pair of people in the audience whose birthday is January 2nd OR is there a pair of people in the audience whose birthday is January 3rd, repeated until December 31st. The probability of the second question is much larger than the first question.

2.

A sample is representative of our population if each member of the population has an equal probability of being included in the sample. This aims to ensure that the sample represents the population without bias and enables a sample to be representative of the population without every detail of the population being included in our sample. Since some details of the population are not captured in the sample, it is important to document the demographics of our sample and who it represents. Documentation can inform us about what details might be missing in the sample.

3.

A random sample is appropriate in scenarios where the statistic of interest does not vary widely based on different subgroups. A stratified random sample is more appropriate in scenarios where a sample should have an equal amount of individuals from different groups.

Examples of when a simple random sample would be better are: if we want to determine the average income in a zip code or if we want to determine the average GPAs of all students within a specific major.

In both cases, we do not expect wide variability based on different subgroups.

Examples of when a stratified random sample would be better are: if we want to determine the average income in a state or if we want to determine the average height of students in an elementary school. In both examples, the statistics of interest vary widely based on which subgroups are predominately in the sample. Stratified random sampling ensures that a sufficient number of individuals from different groups (e.g. more affluent zip codes vs less affluent zip codes in the first example or students in younger grades vs students in older grades) are included in the sample.

4.

Ensuring randomness is a challenge for performing a simple random sample or a stratified random sample. Many samples might seem random but might be samples of convenience. Limited access to individuals makes it difficult to perform a simple random sample or a stratified random sample. A lack of detailed information where individuals cannot be divided into groups or a small number of individuals in groups presents a challenge for performing a stratified random sample.

Try It Yourself Try It Yourself: Assess Variability in Histograms (6.1) 1.

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C A B Die 2 appears to vary the least, based on the shape of its histogram, while Die 1 appears to vary the most, based on the shape of its histogram.

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Chapter 6: Understanding Uncertainty, Probability, and Variability

5.

STAR Framework: ● Scope the Data’s Context: The histogram visualizes the outcomes of 10,000 rolls for three different six-sided dice, with the x-axis representing the roll outcomes (1 to 6) and the y-axis showing the count of each outcome. The histograms are separated into three panels, one for each die. ● Track the Question Behind the Exploration: The visualization seeks to determine if the dice are balanced or loaded by comparing the frequency of each outcome and investigating potential biases in the dice that might affect games or experiments using them. ● Articulate the Results: Die 1 outcomes are fairly balanced; Die 2 shows a clear bias, with the outcome of 6 occurring significantly more; and Die 3 has a bias toward even outcomes. ● Respond with Next Steps: Further investigation could involve statistical tests to quantify the biases in Dice 2 and 3 and a study of the manufacturing process to identify causes for the imbalance in these dice.

Try It Yourself: Relate Variability to Uncertainty and Predictability (6.1) 1.

2.

3.

4.

During Dungeons and Dragons, Elliott sometimes wants a coin to land showing Heads, and so wants to sneakily use a coin that he can predict an outcome of Heads with more certainty. For these situations, Elliott should use Coin 2. This coin also has the least variability of the three coins.

To be the most fair to all players of the game, only coins and dice whose outcomes cannot be predicted accurately should be used. Coin 1 has the most variability in its outcomes and would be the hardest to predict, so it should be used.

Coin 3 has less variability than Coin 1 because we would predict a value of Tails for a new coin toss with more certainty.

STAR Framework: • Scope the Data’s Context: The histograms represent the counts of flip outcomes (heads or tails) for three different coins, based on 10,000 trials per coin.

• Track the Question Behind the Exploration: This visualization aims to understand the fairness and predictability of each coin's outcomes to inform decisions in a game setting.

• Articulate the Results: Coin 2 shows a strong bias toward heads, suggesting less variability and more predictability. Coin 1 displays a more balanced distribution of outcomes, indicating higher variability and less predictability. Coin 3 is slightly biased toward tails but is more balanced than Coin 2, suggesting moderate variability.

• Respond with Next Steps: Further analysis could involve testing the coins with additional flips to confirm the initial findings.

Try It Yourself: Estimating Likelihood of Values in Data (6.1) 1.

2.

3.

4.

The shape of the distribution of the number of full-time undergraduates is right-skewed, which means that the sample average number of full-time undergraduates is greater than the sample median number of fulltime undergraduates.

The proportion of schools with fewer than 5000 full-time undergraduates is 0.78, which means that it would not be surprising to encounter a new school with fewer than 5000 full-time undergraduates.

The proportion of schools with more than 15,000 full-time undergraduates is 0.04, meaning it would be surprising to encounter a new school with more than 15,000 full-time undergraduates.

STAR Framework: ● Scope the Data’s Context: The histogram shows the distribution of full-time undergraduate enrollment across various colleges, with the number of full-time undergraduates on the x-axis and the count of schools on the y-axis. ● Track the Question Behind the Exploration: The aim is to understand the typical enrollment size of full-time college undergraduates and to identify common and outlier enrollment figures. ● Articulate the Results: The distribution is right-skewed, with most schools having fewer than 5000 full-time undergraduates, indicating that small-sized colleges are more common than large ones.

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●

Respond with Next Steps: Considering the skewness, we might explore the reasons behind this distribution pattern and investigate the characteristics of schools with very high versus low fulltime undergraduate populations (e.g., public versus private, urban versus rural, etc.).

Try It Yourself: Predictability from Histograms (6.1) 1.

2.

3.

Because Darc’s scores demonstrate the least variability, Miles and Elliot should feel the most confident about predicting Darc’s score in their next game of pinball.

Because Ember’s scores demonstrate the most variability, Miles and Elliot should feel the least confident about predicting Ember’s score in their next game of pinball.

Despite Darc having the highest average pinball score, Qpawnz is more likely than Darc to have a score over 1250 million, with an estimated probability of 0.04. Ember has the lowest average pinball score but is the most likely to score over 1250 million, with an estimated probability of 0.08.

Try It Yourself: Predictability with Two Variables (6.1) 1.

2.

3.

The pinball scores in Figure 6.11 are the same as those in Figure 6.12, with span values of around 100 million points to approximately 1400 million points.

The variability in pinball scores in Figure 6.11 appears to be similar for each value of sleep. The variability in pinball scores in Figure 6.12 appears to be similar for each value of caffeine. However, it is much lower for a given value of caffeine than for a given value of sleep.

Predicting the score of an upcoming pinball game is harder knowing how much sleep the person got the previous night than knowing how much caffeine they’ve consumed at the arcade because of the greater variability displayed in Figure 6.11. The relationship between score and caffeine in Figure 6.12 results in a reduction of the variability in scores if we know how much caffeine was consumed. This improves its predictability if we know how much caffeine was consumed.

Try It Yourself: Describe Outcomes in a Sample Space (6.2) 1.

2.

There are 36 possible outcomes for the random trial of rolling these two six-sided dice.

There are 11 outcomes that involve rolling at least one 6. This is the same as the number of outcomes that include rolling at least one 2.

Try It Yourself: Values of Variables as Samples Spaces (6.2) 1.

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Variable: Number of siblings; Values: {0, 1, 2, …}; Shape: right-skewed Variable: Daily time spent on social media; Values: 0 minutes–1440 minutes; Shape: right-skewed Variable: Distance from hometown; Values: 0 miles–8000 miles; Shape: right-skewed Variable: Average SAT score; Values: 0–2400; Shape: symmetric Variable: Number of books read over the summer; Values: {0, 1, 2, …, 180}; Shape: right-skewed Variable: Number of social media followers; Values: {0, 1, 2, …}; Shape: right-skewed Variable: Height; Values: 36 inches–108 inches; Shape: symmetric

Try It Yourself: Estimation from Different Sampling Methods (6.3) 1.

2.

3.

Only in the stratified random sample did we collect a sample representing the distribution of class standing in the population. It would be reasonable to think that the mean social media time estimates for each class standing (and overall) are most accurate in this sampling method.

The balanced stratified random sample included a disproportionately high number of third-years and fourth-years, and a disproportionately low number of first-years and second-years. We can see that this lowered the overall mean social media time relative to the other sampling methods because third-years and fourth-years appear to spend less time on social media.

By chance, the simple random sample included a much higher proportion of third-years than is present in the population. Because third-years tend to spend less time on social media and there were so many in the sample, the overall mean social media time was lower here than for the stratified random sample

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Chapter 6: Understanding Uncertainty, Probability, and Variability

Try It Yourself: Simulate Flipping a Loaded Coin (6.4) 1.

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3.

The proportion of Heads is not 0.7 in all of these situations. Once again, we’re in an uncertain situation where the flip's outcome can vary. Because of this uncertainty, we can’t predict the outcome of each flip and are not guaranteed to see precisely 70% Heads in any number of flips.

As the number of flips increases, the proportion of Heads gets closer to 0.7, and the proportion of Tails gets closer to 0.3. To ensure our simulations help estimate probabilities, we need to perform a large number of them.

STAR Framework: ● Scope the Data’s Context: The visualization displays the proportions of head-to-tail outcomes from flipping a loaded coin with a 70% chance of landing on heads and a 30% chance of landing on tails. It shows results from 5, 10, 100, and 1000 flips. ● Track the Question Behind the Exploration: How does increasing the number of coin flips affect the observed proportion of outcomes relative to the expected probabilities? ● Articulate the Results: The proportion of heads does not consistently equate to the expected 70% in smaller samples but tends to stabilize around the expected probability of 70% as the number of flips increases. ● Respond with Next Steps: To improve the reliability of simulations as tools for estimating true probability, we should conduct more flips, which would likely result in the observed proportions converging closer to the expected probabilities of 70% heads and 30% tails.

Try It Yourself: Simulate Samples (6.4) 1.

2.

3.

4.

As the sample size decreases, the distributions of sample proportions get wider and wider, but all still have the same center of 0.5.

The greater variability in the distribution of sample proportions for samples of size 30 means we have more uncertainty about the sample proportion in this situation. This means there are fewer values of the sample proportion that would be surprising to see relative to the other sample size situations.

Elliott reports that after one year and 1000 flips of his coin, he has recorded a sample proportion of Heads of 0.6. This sample proportion is surprising because none of the sample proportions fall at or above this value on the distribution for a sample size of 1000.

STAR Framework: ● Scope the Data’s Context: The visualization shows overlaid histograms for the distribution of sample proportions of heads from flipping a fair coin, with sample sizes of 30, 100, and 1000. ● Track the Question Behind the Exploration: The focus is on how the sample size affects the variability and predictability of sample proportions in repeated simulations. ● Articulate the Results: As the sample size increases, the distribution of sample proportions becomes narrower, indicating less variability and increased certainty in the sample proportion.

The center of the distributions remains consistent across sample sizes. ● Respond with Next Steps: To assess the predictability of sample proportions, we could conduct further simulations with varying sample sizes.

Applying the Concepts Activity: Assess Variability in Histograms 1.

2.

3.

4.

5.

6.

7.

Die 1 appears to be loaded so that odd values appear more often than even values.

Die 2 appears to be loaded so that values from 1 to 10 appear more often than values from 11 to 20.

Die 3 appears to be balanced across all 20 possible outcomes.

Die 4 appears to be heavily loaded towards the value of 20.

Die 5 appears to be loaded so that the values of 18, 19, and 20 appear much more often than the rest.

Based on the histograms, the die with the most variability is Die 3, followed by Die 1, and Die 2, followed by Die 5, followed by Die 4.

STAR Framework:

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Data Science for All

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Scope the Data's Context: The histograms show the distribution of roll outcomes for five different 20-sided dice, with each die rolled 100,000 times to observe the proportion of each outcome.

Track the Question behind the Exploration: The histograms are analyzed to determine whether each die is fair or loaded, and if loaded, which outcomes are favored.

Articulate the Results: Die 1 is loaded towards odd values, Die 2 is loaded towards lower values, Die 3 is balanced, Die 4 is loaded towards the value 20, and Die 5 is loaded towards the highest three values. The variability in outcomes decreases from Die 3 to Die 4, indicating different levels of predictability or fairness.

Respond with Next Steps: Consider further testing to confirm the initial findings.

Activity: Relate Variability to Uncertainty and Predictability

1.

2.

3.

4.

During Dungeons and Dragons, Elliott sometimes wants a 20-sided die to show a value of 20, and so wants to sneakily use a die that he can predict an outcome of 20 with more certainty. For these situations, Elliott should use Die 2 This die also has the least variability of the three coins.

To be the most fair to all players of the game, only coins and dice whose outcomes cannot be predicted well should be used. Die 1 has the most variability in its outcomes and would be the hardest to predict, so it should be used.

Die 2 has less variability than Die 3 because we would predict a value of 18, 19, or 20 for a new roll of the die with more certainty.

STAR framework. ● Scope the Data's Context: The histograms represent the outcomes of rolling three different 20sided dice. Die 3 has increased probabilities for 18, 19, and 20, Die 2 is heavily weighted towards a 20, and Die 1 has a uniform distribution. Each panel corresponds to one die, with the x-axis showing possible roll outcomes and the y-axis showing the proportion of 10,000 rolls. ● Track the Question behind the Exploration: The histograms assess how the roll outcome probabilities of the three dice compare, with a focus on predictability and fairness. ● Articulate the Results: Die 3's histogram has increased heights for 18, 19, and 20, showing a bias towards higher values but less so than Die 2. Die 2's histogram has one tall bar at 20, reflecting 89 © 2026 Pearson Education, Inc.

Chapter 6: Understanding Uncertainty, Probability, and Variability

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high predictability for that outcome. Die 1's histogram shows equal height bars, indicating uniform probability across all outcomes.

Respond with Next Steps: To ensure fairness, Die 1 would be recommended for unbiased gameplay.

Activity: Estimating Likelihood of Values in Data

1.

2.

3.

4.

The shape of the distribution of the body mass is right-skewed, which means that the sample median body mass is less than the sample average body mass.

The proportion of penguins with body mass between 3500 g and 5500 g is 0.7117, which means that it would not be surprising to encounter a new penguin with body mass between 3500 g and 5500 g.

The proportion of penguins with body mass of at least 6000 g is 0.012, which means that it would be surprising to encounter a new penguin with body mass of at least 6000 g.

STAR framework. ● Scope the Data's Context: The histogram visualizes the body mass of Palmer penguins. The xaxis indicates the body mass in grams, and the y-axis represents the count of penguins within each mass range. ● Track the Question behind the Exploration: The distribution's shape and the proportion of penguins within specific body mass ranges are analyzed to understand common body masses and the likelihood of encountering penguins of certain mass ranges. ● Articulate the Results: The histogram is right-skewed, indicating that more penguins have a body mass on the lower end of the scale than the higher end. About 71.17% of penguins fall between 3500 g and 5500 g, making this a common weight range. Conversely, only 1.2% weigh at least 6000 g, making such a weight rare.

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Respond with Next Steps: Future studies might investigate the ecological or biological factors contributing to this distribution and the rarity of heavier penguins. Further data collection could help understand the health and distribution of the Palmer penguin population.

Activity: Predictability from Histograms

1.

2.

3.

Because Ember’s scores demonstrate the least variability, we should feel the most confident about predicting Ember’s score in the next game of skeeball they play.

Because Darc’s scores demonstrate the most variability, we should feel the least confident about predicting Darc’s score in the next game of skeeball they play.

Darc has the lowest average skeeball score and the lowest probability of scoring over 400 points with an estimated probability of 0.001. Qpawnz is more likely to have a score over 400 points with an estimated probability of 0.018 Ember has the highest average score, and is the most likely to have a score over 400 points with an estimated probability of 0.063.

Activity: Predictability with Two Variables

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Chapter 6: Understanding Uncertainty, Probability, and Variability

1.

2.

The number of treats in Scatterplot 1 are the same as those in Scatterplot 2, and span values of around 0 treats to around 60 treats.

The variability in total treats in Scatterplot 1 appears to be similar for each value of hour of the day. The variability in total treats in Scatterplot 2 appears to be similar for each value of age, but is much lower for a given value of age than it is for a given value of hour of the day.

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3.

Predicting the total treats for an upcoming week in Jake’s life is harder knowing the hour of the day he’ll be getting the treats than it is knowing his age because of the greater variability displayed in Scatterplot 1.

The relationship between total treats and age visible in Scatterplot 2 results in a reduction of the variability in total treats, if we know Jake’s age in weeks. This, in turn, improves its predictability if we know his age in weeks.

Activity: Describe Outcomes in a Sample Space 1.

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52 possible outcomes 1/52 4 51 possible outcomes 3 0.0045 4-5 times.

Activity: Values of Variables as Sample Spaces 1.

2.

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Variable: Number of subscribers; Values: {0, 1, 2, …}; Shape: right-skewed Variable: Duration/Length; Values: 0 seconds - 60 seconds; Shape: symmetric Variable: Number of views; Values:{0, 1, 2, …}; Shape: right-skewed Variable: Number of likes; Values:{0, 1, 2, …}; Shape: right-skewed Variable: Number of comments; Values: {0, 1, 2, …}; Shape: right-skewed Variable: Length of title; Values: {0, 1, 2, …, 100}; Shape: left-skewed

Activity: Estimation from Different Sampling Methods Table. Descriptive statistics for a simple random sample.

Class Standing

GPA

Number of Observations

First-year

3.76

27

Second-year

3.53

25

Third-year

3.10

24

Fourth-year

2.96

24

Overall

3.35

100

Table. Descriptive statistics for a stratified random sample: simple random sample of 30 freshmen, simple random sample of 30 Second-years, simple random sample of 20 Third-years, simple random sample of 20 Fourth-years.

Class Standing

GPA

Number of Observations

First-year

3.75

30

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Chapter 6: Understanding Uncertainty, Probability, and Variability

Second-year

3.49

30

Third-year

3.13

20

Fourth-year

2.96

20

Overall

3.39

100

Table. Descriptive statistics for a balanced stratified random sample: simple random sample of 25 freshmen, simple random sample of 25 Second-years, simple random sample of 25 Third-years, simple random sample of 25 Fourth-years.

Class Standing

GPA

Number of Observations

First-year

3.76

25

Second-year

3.41

25

Third-year

3.06

25

Fourth-year

3.07

25

Overall

3.33

100

1.

2.

3.

Only in the stratified random sample did we collect a sample that exactly represents the distribution of class standing in the population. It would be reasonable to think that the estimates of average GPA for each class standing (and overall) are most accurate in this sampling method.

The balanced stratified random sample and simple random sample both included a disproportionately high number of Third-years and Fourth-years, and a disproportionately low number of First-years and Secondyears. We can see that this lowered the overall mean GPA relative to the other sampling methods because Third-years and Fourth-years appear to have lower GPAs. By chance, the simple random sample resulted in a distribution of class standing that was very similar to that of the balanced stratified random sample. This is not guaranteed in a probability-based sampling method such as a simple random sample. So, we must carefully consider our choice of sampling method in order to obtain a sample that is representative of the population.

Activity: Simulate Flipping a Loaded Coin (Roulette Wheel)

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Visualization from R:

Visualization from Python:

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Chapter 6: Understanding Uncertainty, Probability, and Variability

Answers: 1. The proportion of Red is not 18/38 in all of these situations. Once again, we’re in an uncertain situation where the outcome of the spin can vary. Because of this uncertainty, we can’t predict the outcome of each spin and are not guaranteed to see exactly 18/38 Red in any number of spins.

2. As the number of spins increases, the proportion of Red gets closer to 18/38, the proportion of Black gets closer to 18/38, and the proportion of Green gets closer to 2/38. In order to ensure our simulations are useful for estimating probabilities, we need to make sure to perform a large number of them.

3. STAR Framework: ● Scope the Data's Context: The bar charts display the results of simulating a roulette wheel, with proportions of red, black, and green outcomes recorded for 5, 10, 100, and 5000 spins. ● Track the Question behind the Exploration: The investigation is centered on how the observed proportions of colors align with the expected probabilities (18/38 for red and black, 2/38 for green) as the number of spins increases. ● Articulate the Results: The proportion of red does not consistently match the expected probability in smaller numbers of spins due to variability, but as the number of spins increases, the observed proportions align more closely with the expected values. ● Respond with Next Steps: To improve the reliability of simulation as a tool for estimating true probability, we should conduct a large number of them. This would likely result in the observed proportions getting closer to the expected probabilities.

Activity: Simulate Samples Visualization from R:

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Visualization from Python:

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2.

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4.

As the sample size increases the distributions of sample proportions get narrower and narrower, but all still have the same center of 18/38.

The greater variability in the distribution of sample proportions for samples of size 30 means we have more uncertainty about the sample proportion in this situation. This means that there are fewer values of the sample proportion that would be surprising to see relative to the other sample size situations.

This sample proportion is surprising because none of the sample proportions fall at or above this value on the distribution for our simulation with a sample size of 1,000.

STAR Framework: ● Scope the Data's Context: The histograms represent the distribution of the sample proportions of landing on an odd space in samples of roulette wheel spins, derived from 10,000 simulated samples of sizes 30, 100, and 1,000. ● Track the Question behind the Exploration: The goal is to understand how the sample size influences the variability and precision of the sample proportion of landing on an odd number. ● Articulate the Results: As sample size increases, the distribution of sample proportion becomes more concentrated around the center, which is around 18/38, indicating less variability and increased precision of the sample proportion estimate. ● Respond with Next Steps: Given the surprising result of a 0.7 sample proportion from our dataset of 1,000 spins, it may be necessary to examine whether our data is representative or if there are outliers or biases affecting the proportion.

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Chapter 7: Drawing Conclusions from Data

Chapter 7: Drawing Conclusions from Data Chapter Review Questions Section 1: Introduction to Statistical Inference 1.

After conducting research and crafting a hypothesis, what is the next immediate step in the scientific method?

Collecting data

2.

Describe what it means to have a reproducible data collection strategy.

A reproducible data collection strategy means that the data collection process can be repeated, if necessary.

3.

What is a good way to estimate the mean value of a particular variable for a population?

Calculate the mean value of the variable for a representative sample.

4.

What is the first step of the scientific method?

To ask a question.

5.

After conducting background research into a question, what is the next step in the scientific method?

Constructing a hypothesis about the answers to the question

Section 2: Data Collection and Study Design 6.

An ecologist wishes to study the lengths of leopard sharks. What is the population of interest?

All leopard sharks

7.

How does data collection typically occur?

Data collection typically occurs through experiments or observational studies.

8.

Describe how a sample relates to a population of interest.

Answers may vary. Example: A sample is representative of the population of interest and is used as a more feasible group on which to conduct statistical analyses.

9.

The mean of a sample is an estimate for the mean of what?

The mean of a sample is an estimate for the mean of the population of interest.

10. What is a confounding variable?

A confounding variable is an “extra” variable. It relates to both a response variable and an associated explanatory variable, and is often overlooked, confounding statistical analyses.

11. What kind of assignment helps remove the possibility of other variables explaining differences in the response variable?

Random assignment 12. What is the primary goal of conducting experiments in data science?

To limit the effects of confounding variables and identify causal relationships between variables.

13. What is the explanatory variable in this study?

Sleep quality

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Chapter 7: Drawing Conclusions from Data

Chapter 7: Drawing Conclusions from Data Chapter Review Questions Section 1: Introduction to Statistical Inference 1.

After conducting research and crafting a hypothesis, what is the next immediate step in the scientific method?

Collecting data

2.

Describe what it means to have a reproducible data collection strategy.

A reproducible data collection strategy means that the data collection process can be repeated, if necessary.

3.

What is a good way to estimate the mean value of a particular variable for a population?

Calculate the mean value of the variable for a representative sample.

4.

What is the first step of the scientific method?

To ask a question.

5.

After conducting background research into a question, what is the next step in the scientific method?

Constructing a hypothesis about the answers to the question

Section 2: Data Collection and Study Design 6.

An ecologist wishes to study the lengths of leopard sharks. What is the population of interest?

All leopard sharks

7.

How does data collection typically occur?

Data collection typically occurs through experiments or observational studies.

8.

Describe how a sample relates to a population of interest.

Answers may vary. Example: A sample is representative of the population of interest and is used as a more feasible group on which to conduct statistical analyses.

9.

The mean of a sample is an estimate for the mean of what?

The mean of a sample is an estimate for the mean of the population of interest.

10. What is a confounding variable?

A confounding variable is an “extra” variable. It relates to both a response variable and an associated explanatory variable, and is often overlooked, confounding statistical analyses.

11. What kind of assignment helps remove the possibility of other variables explaining differences in the response variable?

Random assignment 12. What is the primary goal of conducting experiments in data science?

To limit the effects of confounding variables and identify causal relationships between variables.

13. What is the explanatory variable in this study?

Sleep quality

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14. What is the response variable in this study?

Class performance 15. What is a confounding variable in this study?

Being an athlete.

16. What is the population of interest in this study?

All students in the teacher’s class 17. What is the sample in this study?

The 20 randomly selected students

Section 3: The Language of Statistical Inference 18. What is a strategy to use when it’s not feasible to perform an experiment by hand many times?

Perform simulation using a computer 19. What is a null hypothesis?

A null hypothesis typically represents the explanation of results due to chance alone.

20. In statistical inference, what does a p-value indicate?

The probability of observing the results (or more extreme results) if the null hypothesis is true.

21. What is the significance level in hypothesis testing?

The probability of rejecting the null hypothesis when it is true.

22. What is an alternative hypothesis?

An alternative hypothesis represents the explanation in hypothesis testing that contradicts the null hypothesis and reflects researcher’s suspicion about a population characteristic.

23. How is the strength of evidence against the null hypothesis judged in a hypothesis test?

The p-value is used to assess the strength of evidence against the null hypothesis. A lower p-value represents stronger evidence.

24. What does it mean if a data scientist sets a significance level at 0.01?

If a data scientist sets a significance level at 0.01, it means that they accept a 1% probability of falsely rejecting the null hypothesis and that they will reject the null hypothesis only if the p-value is less than 0.01.

25. What does it mean when results are statistically significant?

Results are statistically significant when they are unlikely to occur due to chance alone, assuming that the hypothesis under investigation is true.

26. When can we reject our null hypothesis?

We can reject our null hypothesis when the hypothesis test’s p-value is below the test’s significance level.

27. How do researchers quantify evidence against the null hypothesis in an experiment?

They calculate the p-value from comparing the observed value to the statistics based on the null hypothesis being true.

28. How do we use a p-value to make a decision in statistical inference?

We compare it to the predefined significance level: if the p-value is less, we have statistically significant evidence to reject the null hypothesis in favor of the alternative. Otherwise, we don’t have strong enough evidence to do so.

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Chapter 7: Drawing Conclusions from Data

29. Explain how the p-value is calculated.

We gather or simulate data assuming the null hypothesis is true multiple times to get a statistical distribution. From it, we calculate the probability of obtaining results at least as surprising as our observed statistic. That probability is the p-value.

30. What does a small p-value mean?

A small p-value means that we have strong evidence against the null hypothesis.

31. What does a p-value of 0.05 tell us?

Assuming that the null hypothesis is true, the probability of observing a result at least as extreme as ours is 0.05.

Section 4: Exploratory Data Analysis to Begin Inference 32. Describing data is part of which type of data analysis?

Exploratory

Section 5: Drawing Conclusions in an Observational Study 33. In hypothesis testing, rejecting the null hypothesis when it is actually true is known as what kind of error?

A Type I error (false positive).

34. How is a standardized statistic used in a hypothesis test?

It provides means to assess how surprising the data are based on their distribution characteristics.

35. What is extrapolation?

Extrapolation is the process of making conclusions about variable values beyond the range represented in our data.

36. What does the scope of inference mean in statistical inference?

The scope of inference tells us the population about which we can make inferences.

37. What are t-tests commonly used for comparing?

T-tests are commonly used for comparing means.

Section 6: A/B Testing as a Case of Experiments 38. Why would someone implement an A/B test?

Answers may vary. Example: An A/B test might be implemented when evaluating two treatments in a controlled setting.

39. In A/B testing, why is it important to show both treatments to individuals simultaneously?

It ensures that participants are exposed to the same potential confounding variables.

40. How do you know which is the winning treatment in an A/B test?

By conducting a two-sample t-test on the results of the A/B testing.

41. What is a false positive in the context of experimental results?

Assuming one of the treatments will perform better on the full population, when it does not.

42. When does a false negative occur in an experiment?

When both treatments are assessed to perform equally in the test, while one of them would have

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performed significantly better than the other on the whole population.

43. In A/B tests, how are participants assigned to different treatments? Explain why this is important.

By random assignment. This minimizes the influence of confounding variables.

44. What are common metrics used in an A/B test?

Answers may vary. Examples: average view duration, completion rate, sharing, click-through rate.

45. What is the goal of an A/B test?

To measure which of the two treatments performs better in a real situation.

46. What are the two phases of A/B testing?

Testing and implementation 47. What is the first part of the testing phase in an A/B test?

Choosing the size of the representative subset exposed to the testing phase and randomly assigning participants to the groups 48. Describe randomization and explain why it is important in A/B testing.

Answers may vary. Example: In randomization, participants are assigned to groups without bias. It may be used so that only a subset of people are involved in the A/B test.

49. Describe the two groups being compared in an A/B test.

Each group is exposed to the corresponding treatment without any knowledge of the other. The exposure occur simultaneously for both groups.

50. Describe the implementation phase of A/B testing.

In the implementation phase, the more successful treatment from the testing phase is applied to a broader audience, or even to the entire target population.

51. What is another name for A/B testing?

Split testing and bucket testing.

52. What is a false positive in A/B testing?

A false positive is when the results of the testing phase reveal that one treatment is indicated to perform better than the other, but the winning treatment does not perform better upon implementation.

53. A/B testing allows data scientists to make what kind of inference about how to improve digital outcomes?

Causal 54. What type of statistical test can be used to determine whether the difference in the performance of two treatments is likely due to our hypothesis rather than random chance?

Two sample t-test 55. What are the potential challenges with overly complex experiments?

Answers may vary. Example: They may obscure the specific factors contributing to the results.

56. In the context of experiments, under what circumstances does a Type I error occur?

A Type I error occurs when the test suggests that one of the treatments will perform better than the other on the full population but it does not.

57. What is meant by the scope of inference in experimental design?

The range of subjects the experiment's results can be generalized to.

58. What is the primary goal of the testing phase in A/B testing?

To identify which treatment performs better and whether there is a significant difference.

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Chapter 7: Drawing Conclusions from Data

59. What does statistical significance imply in the context of an experiment's results?

There is strong evidence toward a causal relationship between the variables.

Explaining the Concepts 1.

A Type II error occurs when the null hypothesis is not rejected when it should be. Increasing the sample size will reduce the probability of a Type II error. As our sample size increases, our estimate of the parameter gets closer to its true value.

2.

A confounding variable is an “extra” variable that can affect the variables that we are analyzing. We can draw cause-and-effect conclusions in experiments where we can remove confounding variables. In observational studies we cannot account for confounding variables, thus we cannot draw cause-and-effect conclusions from observational studies.

3.

Random sampling is a process of randomly collecting data to represent a population of interest. In random sampling, every member of a population has an equal chance of being included in the sample. Random sampling is used to create a sample that can be used to study a larger population. Random assignment is the process of randomly assignment individuals into groups. Random assignment is used in A/B testing to assign individuals to each treatment group.

4.

Controlled experiments are constructed to minimize the effect of confounding factors. As we add more variables to an A/B test, we create more opportunities for confounding factors to affect our analyses. To maintain high credibility of the test results we will need to either run many simple A/B tests consequently or run a test with multiple treatments simultaneously. The first option will extend the test duration, because each option requires a test. The second option requires a larger sample as each option requires a dedicated group, which should be large enough to remain representative of the population.

5.

A unanimous decision to find a defendant guilty is similar to our specification of a small significance level in a hypothesis test since both situations require strong evidence to make a Type I error. We can decrease our significance level to demand more evidence from our data.

In the criminal trial analogy, a Type I error would be finding the person guilty when in fact they are innocent. A Type II error would be finding the person innocent when in fact they are guilty.

Try it Yourself Try It Yourself: Estimate the Probability of an Event (7.3) 1.

2.

a. See Figure 7.7. b. 2 c. 0.02 d. Low e. Against STAR Framework: ● Scope the Data's Context: The data consist of the number of times heads appeared in 100 flips of a coin, with the coin being claimed as fair. The histogram is derived from 100 trials to form an expected distribution of heads in such a series of flips. ● Track the Question Behind the Exploration: We explore whether the coin is indeed fair by analyzing the frequency of heads appearing. The tension here is between the observed frequency of heads (60 out of 100 flips) and the expected distribution for a fair coin. ● Articulate the Results: The histogram peaks around 50 heads, which is the expected outcome for a fair coin in 100 flips. Our result of 60 heads is much greater than the peak and appears outside the

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●

common range of outcomes. This suggests that the result provides strong evidence against the coin's fairness.

Respond with Next Steps: More trials with Michael’s coin would help in solidifying the assessment of the coin's fairness.

Try It Yourself: Make a Statistical Inference (7.3) 1.

2.

reject; below; in groups; null fail to reject; above; either; do not

Try It Yourself: Evaluate a Hypothesis (7.5) 1.

2.

3.

0.65 0.79 Reject the null hypothesis

Try It Yourself: Perform a t-Test on Observational Data (7.5) 1.

2.

3.

-2.167 0.017 a

Try It Yourself: Perform a t-Test Using a Different Alternative Hypothesis (7.5) 1.

2.

3.

10.577 1.000 STAR Framework: ● Scope the Data's Context: The dataset consists of records from students at a school. Hazel and Everett have hypothesized that the academic performance at this school, as measured by the

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Chapter 7: Drawing Conclusions from Data

● ●

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mean GPA, is lower than their own institution's standard, which they consider to be reflected by a GPA of 3.5.

Track the Question Behind the Exploration: The central question of this exploration is to determine if the mean GPA at the other school is indeed lower than 3.5.

Articulate the Results: The results of the t-test are quite interesting. A t-statistic of 10.577 indicates a significant difference from the null hypothesis. At the same time, a p-value of essentially 1 means that there is practically no evidence against the null hypothesis in favor of the alternative hypothesis, which posits that the mean GPA is less than 3.5. Does it mean that one of the values is incorrect? How can it be?

These results are due to Hazel and Everett’s suspicion (alternative hypothesis) being the opposite of what’s actually true. That is, their interest is in whether the mean GPA is less than 3.5 when in fact the mean GPA of their sample was far greater than 3.5. In general, it’s important to keep the direction of the alternative hypothesis in mind for a more complete interpretation of the test statistic and computation of the p-value.

Respond with Next Steps: Drawing conclusions from these results, we cannot assert that the data support Hazel and Everett's hypothesis about the lower mean GPA at the other school. However, it's important to approach these conclusions with an understanding of what these numbers represent and what they do not. While the statistical analysis does not indicate a lower mean GPA, it does not inherently provide reasons for this. Further investigation would be necessary to understand the underlying circumstances. Factors such as differences in curriculum difficulty, grading policies, student support services, and student demographics could all play a role.

Additionally, this analysis does not necessarily imply a qualitative judgment about the overall educational quality of the school.

Try It Yourself: Draw Conclusions from a Correlation (7.5) 1.

2.

3.

4.

5.

d 0.00 -0.42 a. The p-value may vary with simulation but it is beneath 0.001 in most cases.

STAR Framework: ● Scope the Data's Context: The dataset includes gpa and avg_course_difficulty for a sample of students, exploring the relationship between academic performance and perceived course difficulty. ● Track the Question Behind the Exploration: The question addresses whether a lower GPA is associated with a higher perceived course difficulty. ● Articulate the Results: Our analysis indicates a significant negative correlation (correlation coefficient of -0.42) between GPA and course difficulty, suggesting that higher course difficulty is associated with lower GPAs. ● Respond with Next Steps: Further investigation could look into the factors contributing to this relationship, such as subject matter, teaching methods, or student study habits, and could also examine if this trend holds true across different schools or academic programs.

Try It Yourself: Identify Type I and II Errors (7.5) 1.

2.

New Drug: a. False Negative: Type II Error b. False Positive: Type I Error c. True Negative d. True Positive GPA: 1. True Positive 2. False Positive: Type I Error 3. True Negative 4. False Negative: Type II Error

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Try It Yourself: Identify an A/B Tests’ Null and Alternative Hypotheses (7.6) 1.

2.

d b

Try It Yourself: Perform t-Tests on A/B Testing Data (7.6) 1. 2011 2. 1980 3. 4.4 4. 4.3 5. d 6. b 7. Two-sample test 8. 0.023 9. b 10. STAR Framework: ● Scope the Data's Context: The data from an Italian restaurant's website A/B test includes website visitors' session duration categorized by the image treatment (pizza or pasta). ● Track the Question Behind the Exploration: The study aims to determine whether a pizza image on the restaurant's website will cause people to read the menu page longer than the pasta image. ● Articulate the Results: The pizza image was viewed by 2011 users with a mean session duration of 4.4 minutes. The pasta image was seen by 1980 users, with a 4.3-minute average session duration.

Statistical tests showed significant differences favoring the pizza image (session duration p = 0.023). ● Respond with Next Steps: The results suggest prioritizing pizza imagery on the website could enhance user engagement. Further testing with varied imagery and layout modifications could optimize these findings.

Try It Yourself: Identify Type I and Type II errors in A/B Testing Results (7.6) 1.

2.

3.

4.

True Negative False Positive: Type I Error False Negative: Type II Error True Positive

Applying the Concepts Activity: Estimate the Probability of an Event 1.

2.

16.67 Very surprising

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Chapter 7: Drawing Conclusions from Data

3.

4.

5.

6.

7.

1 time 0.001 The die is not fair.

STAR Framework: ● Scope the Data's Context: We observe the outcome of rolling a single six-sided die 100 times. The variable of interest is the count of six appearing on a die roll, with the data sourced from 1000 trials to establish a distribution of expected counts for a fair die. ● Track the Question behind the Exploration: The question is whether the die is fair based on the frequency of rolling a six. The core tension here is the discrepancy between the observed frequency (29 times in 100 rolls) and the expected frequency if the die were fair. ● Articulate the Results: The histogram indicates that rolling six approximately 15 to 18 times in 100 rolls is the most common outcome in a fair scenario. Our observation of 29 occurrences is much higher and not within the range of expected outcomes in the histogram's distribution, suggesting that our die significantly deviates from what we'd expect with a fair die. ● Respond with Next Steps: Given the results, we might consider conducting additional trials to increase the reliability of our assessment.

Activity: Make a Statistical Inference 1.

2.

3.

4.

5.

6.

Null hypothesis: A shortened day does not affect student well-being.

Alternative hypothesis: A shortened day improves student well-being.

Fail to reject the null hypothesis Recommend maintaining the traditional 8-hour day since the p-value of 0.08 does not provide enough evidence for the alternative hypothesis Reject the null hypothesis Recommend moving to a shortened 6-hour since the p-value of 0.003 provides evidence supporting the alternative hypothesis.

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Activity: Evaluate a Hypothesis 1.

2.

3.

4.

5.

Null hypothesis: The proportion of all students taking the least difficult classes is 0.10.

Alternative hypothesis: The proportion of all students taking the least difficult classes is less than 0.10.

0.10 0.05 Fail to reject

Activity: Perform a t-Test on Observational Data 1.

2.

3.

4.

5.

6.

7.

8.

9.

Null hypothesis: The average hitting percentage is equal to 30%.

Alternative hypothesis: The average hitting percentage is greater than 30%.

Her hitting percentage will be calculated as an average over 50 sets, so a few sub-30% games alone do not determine the result. If she achieves higher percentage in other games, the average may well be above 30% .38, which is 38% (or 38.26%) .23, which is 23% (or 22.52%) 2.59 0.006 Reject the null hypothesis because the p-value is lower than the threshold of 0.05.

Answer may vary. Indy might be hesitant to generalize these results to all of her future games since this is a small sample of games.

Activity: Draw Conclusions from a Correlation 1. Alternative Hypothesis: Higher ice cream sales are associated with higher rates of drowning.

2. Null Hypothesis: Higher ice cream sales are not associated with higher rates of drownings.

3. 0.00 4. 0.50 5. P-value is 0.0. We can reject the null hypothesis.

6. 0.91 7. Yes 8. 0.56 9. Yes 10. Temperature is a confounding variable, because it was ignored in the initial analysis while in fact it has stronger correlations with each of the variables than they have with each other.

11. Star Framework ● Scope the Data's Context: We attempted to understand the relationship between daily ice cream sales, drownings, and temperature. ● Track the Question behind the Exploration: We initially sought to explore the relationship between ice cream sales and drownings and then considered the impact of temperature on both as a confounding variable. ● Articulate the Results: The analysis of the first dataset (icecream1.csv) revealed a moderate positive correlation (correlation coefficient of 0.50) between ice cream sales and drowning incidents, initially suggesting a direct relationship between higher ice cream consumption and an increased number of drownings. However, further investigation using the second dataset (icecream2.csv) revealed a strong positive correlation (correlation coefficient of 0.91) between temperature and ice cream sales and a moderate positive correlation (correlation coefficient of 0.56) between temperature and drowning incidents. Higher temperatures not only increase ice cream sales but also potentially elevate the likelihood of drowning incidents, primarily by increasing swimming or water-related activities. The key finding here is that temperature acts as a confounding variable, impacting ice cream sales and drowning rates, thereby explaining the observed correlation between these seemingly unrelated variables. This analysis underscores the importance of identifying and accounting for confounding variables in statistical studies to avoid misleading conclusions. ● Respond with Next Steps: This analysis highlights the importance of considering external factors in correlational studies and the caution needed in interpreting such relationships as causal. Future studies could explore other external factors influencing such relationships.

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Chapter 7: Drawing Conclusions from Data

Activity: Perform a t-Test Using a Different Alternative Hypothesis 1.

2.

3.

4.

-2.124 0.019 b STAR Framework • Scope the Data's Context. We asked about students' dietary habits, specifically focusing on the average number of times they consume fruits or vegetables daily. Hazel and Everett's hypothesis suggests a belief that students at the neighboring school have a lower daily intake of fruits and vegetables compared to their own school's average of 2.21 servings.

• Track the Question behind the Exploration. The exploratory question is whether the mean daily consumption of fruits and vegetables at the other school is less than 2.21 servings.

• Articulate the Results. We obtained a t-statistic of -2.124 and a p-value of 0.019. The negative tstatistic indicates that the mean number of servings consumed at the other school is lower than 2.21.

Furthermore, the p-value is less than the significance level of 0.05, which supports rejecting the null hypothesis. This suggests that the other school's average fruit and vegetable consumption is statistically significantly lower than 2.21 daily servings.

• Respond with Next Steps. While the statistical analysis indicates a lower mean consumption of fruits and vegetables, it's crucial to note that this study is observational. As such, it can reveal associations but not causality. The lower average consumption could be influenced by various factors, such as availability of fresh produce, cultural dietary preferences, economic factors, or even differences in how servings are measured or reported. Further investigation might involve understanding the context behind these dietary habits. For instance, examining the school's cafeteria offerings, local availability of fruits and vegetables, and students' economic backgrounds could provide valuable insights.

Additionally, educational initiatives focusing on healthy eating habits could be considered to encourage higher consumption of fruits and vegetables.

Activity: Identify Type I and Type II Errors 1.

2.

Court case: 1. True Positive 2. False Positive - Type I Error 3. True Negative 4. False Negative – Type II Error Correlation: 1. True Positive 2. False Positive - Type I Error 3. True Negative 4. False Negative - Type II Error

Activity: Identify an A/B Test’s Null and Alternative Hypotheses 1.

2.

Null hypothesis: The time spent viewing the portfolio and number of contacts made do not vary significantly between portfolios A and B. Alternative hypothesis: The time spent viewing the portfolio and number of contacts made are significantly higher for one of the portfolios.

Activity: Perform t-Tests on A/B Testing Data 1.

2.

3.

4.

5.

50 50 43.7 32.5 The null hypothesis is that there is no difference in how long a person spends on his site based on whether they see portfolio A or B

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6.

The alternative hypothesis is that there is a difference in how long a person spends on his site based on whether they see portfolio A or B 7. two-sample t-test 8. 0.000 9. We can reject the null hypothesis.

10. STAR Framework: ● Scope the Data's Context: Matthew, a computer science graduate, conducted an A/B test on his website to assess two portfolio presentations (A and B). The data includes the duration each visitor viewed his portfolio. ● Track the Question behind the Exploration: The test aimed to determine if Portfolio A leads to a different viewing time than Portfolio B. ● Articulate the Results: Portfolio A was viewed by 50 participants with an average viewing time of 43.7 seconds. Portfolio B had 50 viewers with an average viewing time of 32.5 seconds. A two-tailed t-test showed a significant difference in viewing times (p = 0.000). This indicates that Portfolio A attract more attention of website visitors. ● Respond with Next Steps: Further investigation could explore combining portfolio elements to optimize viewer engagement.

Activity: Identify Type I and Type II Errors in A/B Testing Results 1.

2.

3.

True Negative False Negative - Type II Error False Positive - Type I Error

Chapter 8: Machine Learning Chapter Review Questions Section 1: Artificial Intelligence 1.

What does artificial intelligence aim to create?

Human-like intelligence.

2.

Describe machine learning.

Machine learning is a subfield of AI. It focuses on developing statistical models that enable computer systems to learn from data and then improve their performances on given tasks.

3.

What does a machine learning model represent?

An algorithm for analyzing data

Section 2: Three Steps in the Machine Learning Process 4.

What does unsupervised learning find in data?

Overall patterns and structure.

5.

Explain how the cookie factory analogy applies to machine learning.

Answers may vary. Example: A cookie factory aims to produce the best cookies from varying ingredients each day, using knowledge obtained from exploring different recipes. A machine learning system aims to make the most correct predictions or inference from the inserted data, using analysis of the previously ingested data.

6.

In the cookie factory analogy, what do different recipes represent?

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6.

The alternative hypothesis is that there is a difference in how long a person spends on his site based on whether they see portfolio A or B 7. two-sample t-test 8. 0.000 9. We can reject the null hypothesis.

10. STAR Framework: ● Scope the Data's Context: Matthew, a computer science graduate, conducted an A/B test on his website to assess two portfolio presentations (A and B). The data includes the duration each visitor viewed his portfolio. ● Track the Question behind the Exploration: The test aimed to determine if Portfolio A leads to a different viewing time than Portfolio B. ● Articulate the Results: Portfolio A was viewed by 50 participants with an average viewing time of 43.7 seconds. Portfolio B had 50 viewers with an average viewing time of 32.5 seconds. A two-tailed t-test showed a significant difference in viewing times (p = 0.000). This indicates that Portfolio A attract more attention of website visitors. ● Respond with Next Steps: Further investigation could explore combining portfolio elements to optimize viewer engagement.

Activity: Identify Type I and Type II Errors in A/B Testing Results 1.

2.

3.

True Negative False Negative - Type II Error False Positive - Type I Error

Chapter 8: Machine Learning Chapter Review Questions Section 1: Artificial Intelligence 1.

What does artificial intelligence aim to create?

Human-like intelligence.

2.

Describe machine learning.

Machine learning is a subfield of AI. It focuses on developing statistical models that enable computer systems to learn from data and then improve their performances on given tasks.

3.

What does a machine learning model represent?

An algorithm for analyzing data

Section 2: Three Steps in the Machine Learning Process 4.

What does unsupervised learning find in data?

Overall patterns and structure.

5.

Explain how the cookie factory analogy applies to machine learning.

Answers may vary. Example: A cookie factory aims to produce the best cookies from varying ingredients each day, using knowledge obtained from exploring different recipes. A machine learning system aims to make the most correct predictions or inference from the inserted data, using analysis of the previously ingested data.

6.

In the cookie factory analogy, what do different recipes represent?

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Chapter 8: Machine Learning

Machine learning models 7.

Describe supervised learning.

Supervised learning is a subcategory of machine learning, wherein a response variable is identified beforehand. The data set contains values that are used to make predictions for new observations.

8.

Describe unsupervised learning.

Unsupervised learning is a subcategory of machine learning, wherein data modeling is used to find patterns and groupings in a data set. No response variable is identified beforehand.

9.

What does the “supervised” part of supervised learning mean?

It means that a response variable is identified.

10. What are the three primary steps in machine learning?

Ingesting the data; analyzing the data; producing model-based results.

11. In which phase are structured or unstructured data fed to a machine learning system?

Feeding structured or unstructured data to a machine learning system occurs during data ingestion.

12. What distinguishes supervised learning from unsupervised learning?

In supervised learning, a response variable is identified in advance, and predictions are made for new observations. In unsupervised learning, no response variable is identified, and the machine learning system aims to identify patterns or groupings in the data.

13. After ingesting data and performing analysis, describe a possible course of action for a machine learning algorithm.

Producing model-based results.

14. Suppose you are using an app to track the health of your houseplants. One feature that the app has involves pattern recognition, wherein it suggests when certain plants are likely to develop root rot or become too dry. What step in the machine learning process does this example represent?

Producing model-based results.

Section 3: Characteristics of Machine Learning Method 15. What are classification models used to predict?

Categories 16. What are regression models used to predict?

Numeric values 17. What tool could help visualize two quantitative variables before applying machine learning?

Scatterplot 18. What do the leaf nodes in a decision tree represent?

The nodes in a decision tree show the predicted value of a given response variable.

19. Which model type predicts a data point's category based on its features?

Classification 20. Why would a classification model not be able to predict the age of a car?

Answers may vary. Example: Classification models identify groups or patterns in data, rather than making specific quantitative predictions.

21. Explain the interpretability versus complexity tradeoff when selecting a model.

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The more complex a machine learning model is, the less interpretable it becomes.

Section 4: Machine Learning Method Evaluation 22. What does the process of training a model involve?

It involves learning from training data to make predictions.

What does the R-squared value of a linear regression model indicate?

The amount of variance in the response variable explained by the model.

23. What data is used to train a machine learning model?

Training data 24. Explain how accuracy is calculated in the context of machine learning.

Accuracy is calculated by finding the proportion of observations for which the predictions were correct.

25. Explain how precision is calculated in the context of machine learning.

Precision is the proportion of positive predictions that are correct. Precision = TP/(TP+FP) 26. Explain how recall is calculated in the context of machine learning.

Recall is the proportion of actual positives that were predicted correctly. Recall = TP/(TP + FN) 27. Describe overfitting.

Overfitting refers to excessive fitting to the training data.

28. What happens when a model is underfitted?

It fails to capture key patterns in the data and makes poor predictions on any data.

29. What is model validation?

Model validation is the process of designating part of a data set as training data and the remainder as test data.

30. Explain the “No Free Lunch” theorem.

No one machine learning model excels at all types of problems.

31. How do we compare the performances of machine learning models?

Using evaluation metrics 32. When is a complex model suitable?

When high accuracy is critical 33. What are the factors to consider when selecting the best model to solve a problem?

Answers may vary. Example: comparing performance metrics.

34. What metric indicates how well a classification method identifies positives?

Recall 35. When is precision important?

When the false positives are costly 36. Why is testing on new data important?

To evaluate real-world performance 37. In the context of machine learning, what does accuracy measure?

The proportion of predictions which are actually correct 111 © 2026 Pearson Education, Inc.

Chapter 8: Machine Learning

38. Give examples of metrics that are not useful when evaluating a classification model’s performance.

Mean squared error and R-squared are not useful when evaluating a classification model’s performance.

They are useful for evaluating a regression model’s performance.

39. Describe a confusion matrix.

Answers may vary. Example: A confusion matrix is a useful tool for comparing the number of positive and negative classes and whether they were correctly predicted.

40. Describe a false positive.

Answers may vary. Example: A false positive occurs if a model incorrectly assigns a value of the positive label to an observation of the negative label.

41. Explain a true positive.

Answers may vary. Example. A true positive occurs if the model correctly classifies a value of the positive class.

42. What’s another term that can be used to describe the true positive rate?

The true positive rate is also referred to as recall.

43. When does overfitting occur?

Overfitting occurs when models fit too closely to previously trained data and perform poorly on new data.

44. Consider the following confusion matrices for various machine learning methods. Calculate the accuracy, precision, and recall for each model. Round your answers to two decimal places.

Model

True Positive

True Negative

False Positive

False Negative

Decision Trees

2496

3705

1526

2758

Random Forests

2818

3524

1708

2436

k-Nearest Neighbors

3080

3125

2106

2174

Logistic Regression

2316

4288

943

2938

Answer: Model

Accuracy

Precision

Recall

Decision Trees

0.59

0.62

0.48

Random Forests

0.60

0.62

0.54

k-Nearest Neighbors

0.59

0.59

0.59

Logistic Regression

0.63

0.71

0.44

45. Consider the following model results. Calculate the accuracy, precision, and recall for each model. Round your answers to two decimal places.

Model

True Positive

True Negative

False Positive

False Negative

Decision Trees

496

705

526

758

Random Forests

818

524

708

436

k-Nearest Neighbors

680

125

106

174

Logistic Regression

316

288

43

938

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Answers: Model

Accuracy

Precision

Recall

Decision Trees

0.48

0.49

0.40

Random Forests

0.54

0.54

0.65

k-Nearest Neighbors

0.74

0.87

0.80

Logistic Regression

0.38

0.88

0.25

46. Which of the following three models is overfitted, well-fitted, and underfitted? What are your conclusions about them?

Model

Data set

Accuracy

Model 1

Training

1.00

Model 1

Testing

0.32

Model 2

Training

0.80

Model 2

Testing

0.78

Model 3

Training

0.61

Model 3

Testing

0.60

Answers: ● Model 1 was overfitted. It fit the training data perfectly but did not predict the testing data well. ● Model 2 was well-fitted. It fit the training data well and continued to predict the testing data equally well. ● Model 3 was underfitted. It fit neither the training nor the testing data well.

Section 5: Deep Learning 47. Explain why neural networks are hard to interpret.

Answers may vary. Example: Neural networks are very complex and do not derive clear to interpret relationships between the variables. They are good options for working with large quantities of unstructured data.

48. What enables deep learning models to process data effectively?

Neural networks combined with extensive computational power.

49. Describe neural networks.

Neural networks are a machine learning method that pass large amounts of data through various layers.

50. Describe how deep neural networks differ from traditional neural networks.

A deep neural network is more computationally intensive than a traditional neural network.

51. What are the common challenges with training deep neural networks?

Answers may vary. Example: Requiring large amounts of data 52. What is the reason to use more layers in a deep neural network?

Using more layers in a deep neural network aims to allow the algorithm to identify more patterns.

53. Give three examples of deep learning.

Answers may vary. Examples: facial recognition, virtual assistants, chatbots

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Chapter 8: Machine Learning

Section 6: Use High-Quality Data in Machine Learning 54. Why is data volume important for machine learning?

A larger quantity of data improves upon a machine learning system’s ability to identify patterns.

55. What issue may arise when data comes from a narrow demographic?

Lack of diversity which leads to biased predictions and poor model performance for a wider population.

56. What data qualities help ensure the success of machine learning?

Data freshness; data volume; data diversity; data quality.

Explaining the Concepts 1.

2.

3.

4.

5.

Evaluating our machine learning models on the test data allows us to determine how well our models can generalize to new data that it has not seen before. Evaluating on training data would determine how well our models can memorize data that it has seen before.

Regression machine learning methods predict a quantitative, numerical value while classification machine learning methods predict a qualitative, categorical value. Predicting a stock's value or predicting a home's price are examples that can be addressed by a regression machine learning method. Predicting if a stock will move up or down or if an email is spam or not are examples of problems that classification machine learning methods can address.

The No-Free Lunch theorem states that one model is not universally superior for every type of data or problem. This means that no one machine learning algorithm will always work the best across all situations and we must try different methods.

Deep learning is a powerful subset of machine learning. Unlike many machine learning methods, deep learning requires large training datasets, needs advanced computational resources for training, takes longer to train, and is less interpretable.

We can use the training data and test data to identify if we should use a more complex or simple machine learning model. This will prevent underfitting and overfitting. If a model performs poorly on the training data, then the model underfits, i.e. it fails to capture the details of the training data. In this case, we might want to use a more complex model. When a machine learning model performs well on the training data but does not predict the test set well, the model overfits, i.e. it excessively captures the details of the training data but cannot generalize the unseen data. In this case, we might want to use a less complex model.

Try it Yourself Try It Yourself: Identify Artificial Intelligence (8.1) These use artificial intelligence: Email spam filters Google Translate Autocorrect when text messaging Plagiarism detection software

Try It Yourself: Identify Data Ingestion, Data Analysis, and Results (8.2) 1.

2.

3.

4.

5.

6.

7.

Step 1: Ingesting Data Step 1: Ingesting Data Step 1: Ingesting Data Step 1: Ingesting Data Step 2: Analyzing the Data Step 2: Analyzing the Data Step 3: Producing Model-Based Results 114 © 2026 Pearson Education, Inc.

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8.

9.

Step 3: Producing Model-Based Results Step 2: Analyzing the Data

Try It Yourself: Reading a Decision Tree (8.3) 1.

2.

3.

4.

5.

Based on Figure 8.11, without knowing anything else about a new penguin, we would predict the species of that penguin to be Adelie.

Based on Figure 8.11, the decision tree created a split on the flipper_length_mm variable.

Based on Figure 8.11, if a penguin has a flipper length below 207 mm, then we would predict the penguin’s species to be Adelie; otherwise we would predict the penguin’s species to be Gentoo.

Based on the bottom nodes (leaves) of the tree in Figure 8.11, we would correctly make 144 Adelie predictions and 118 Gentoo predictions, for a total of 262 correct predictions for the penguins in our dataset.

STAR Framework: ● Scope the Data’s Context: The decision tree is trained on the Palmer penguins’ dataset to predict penguin species. It uses flipper length in millimeters as the deciding factor. ● Track the Question Behind the Exploration: Can we predict the species of a penguin based on its flipper length? ● Articulate the Results: The decision tree predicts that a penguin with a flipper length of less than 207 mm is an Adelie, while a penguin with a flipper length of 207 mm or more is a Gentoo. This model makes 144 correct predictions for Adelie penguins and 118 correct predictions for Gentoo penguins. ● Respond with Next Steps: To improve the model, we could consider including more variables, such as bill measurements or body mass. Additionally, we might explore the data for other patterns or anomalies and validate the model’s performance with a separate test dataset to ensure its generalizability.

Try It Yourself: Regression versus Classification (8.3) 1.

2.

3.

4.

5.

6.

Classification Regression Regression Classification Classification Regression

Try It Yourself: Inference versus Prediction (8.3) 1.

2.

3.

Answer: This is a regression problem because our response variable, director salary, is quantitative. We are interested in inference because we want to know which factors are associated with director salary.

Answer: This is a classification problem because our response variable, outcome of the book, is categorical (success or failure). We are primarily interested in prediction.

This is a regression problem because our response variable, percent change in near-surface air temperature, is quantitative. We are interested in inference because we want to know how ocean surface temperatures are associated with nearby, near-surface air temperature.

Try It Yourself: Interpretability versus Complexity (8.3) 1.

2.

3.

The variable that appears at the top of all these trees is the shelf_location variable.

The third tree makes more correct predictions than the other two decision trees The first tree is the most interpretable and least complex because it contains a single split. It’s relatively simple to answer a single question about shelf location to make a prediction. The third tree is the least interpretable and most complex because it contains the most splits. This means more variables are being used to improve predictions. Asking so many questions to produce a prediction makes it more challenging to characterize relationships between the explanatory variables and sales_cat. The second tree is in the middle of the other two for interpretability and complexity.

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Chapter 8: Machine Learning

4. ● ● ● ●

STAR Framework: Scope the Data’s Context: The decision trees are based on the car seats dataset and aim to predict the sales_cat value using various explanatory variables. The initial split in all trees is based on shelf_location.

Track the Question Behind the Exploration: The guiding question for these models is how different variables, starting with shelf location, predict the sales category of car seats as either “High” or “Low”.

Articulate the Results: The first tree, with a single split, is the simplest model, predicting “High” sales for good shelf locations. The third tree, with multiple splits, offers the most detailed predictions by considering more variables and their interactions. It achieves the highest number of correct predictions.

Respond with Next Steps: The next steps involve evaluating the trade-off between complexity and interpretability. For practical use, it is crucial to determine if the increased accuracy of the more complex model justifies its reduced interpretability. Further validation, possibly with a different dataset, would ensure the models’ robustness and help to finalize the choice of the model for deployment.

Try It Yourself: Calculate Accuracy, Precision, Recall, and Specificity Using a Confusion Matrix (8.4) 1.

2.

3.

4.

5.

0.73 0.67 0.67 0.77 STAR Framework: ● Scope the Data’s Context: The confusion matrix provides the outcomes, on test data, of a decision tree model used to classify car seat sales as either “High” or “Low”. ● Track the Question Behind the Exploration: The model’s effectiveness is measured by how well it predicts actual sales categories on test data. The key metrics of interest are accuracy, precision, recall, and specificity. ● Articulate the Results: The model’s accuracy, which reflects the overall correctness of predictions, is 73%. Precision, which shows how reliable the model’s positive predictions are, stands at 67%. Recall, indicating the model’s ability to find all positive instances, is also 67%.

Specificity, measuring the model’s performance in correctly identifying negatives, is 77%. ● Respond with Next Steps: While the model shows a reasonable level of accuracy and specificity, there is room for improvement in precision and recall. To enhance the model’s predictive power, we could consider additional features, more complex models, or tuning the model’s parameters.

Moreover, it would be useful to conduct further tests, such as using a different test set, to confirm these performance metrics.

Try It Yourself: Overfitted, Underfitted, and Well-Fitted Models (8.4) 1. a. b. c. d.

2.

Model

A cell is true positive (TP) if Predicted = 1 and Actual = 1 A cell is true negative (TN) if Predicted = 0 and Actual = 0 A cell is false positive (FP) if Predicted = 1 and Actual = 0 A cell is false negative (FN) if Predicted = 0 and Actual = 1

Dataset

Accuracy

1

Training

1.00

1

Testing

0.42

2

Training

0.70

2

Testing

0.69

3

Training

0.51

Testing

0.50

3 3.

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a. b. c.

Model 1 was overfitted. It fit the training data and did not predict the test data well.

Model 2 was well-fitted. It fit the training data and predicted the test data well.

Model 3 was underfitted. It did not fit the training data and did not predict the test data well.

Applying the Concepts Activity: Identify Artificial Intelligence ● ● ● ●

Spam caller labeling on your cell phone Talk-to-text on your cell phone Amazon Echo Dot Text suggestions in your email

Activity: Identify Data Ingestion, Data Analysis and Results Step 1: Ingesting Data: ● Calendar Data: The app syncs with the athlete’s game, training and personal calendar to track practices, games and workouts. ● Eating Habits: Uses online journal tools or third-party apps to track food and drink details throughout every day. ● Physical Activity Monitoring: With consent, the app scans the athlete’s personal calendar and wearable monitor data for activity outside of practices, games, and workouts. ● User Input: Optional daily or weekly emotional self-check surveys.

Step 2: Analyzing the Data: ● Muscle Level Prediction: Machine learning algorithms analyze the ingested data to predict muscle levels based on upcoming games, current eating and exercise habits, and emotional tone from the surveys. ● Pattern Recognition: Identifies recurring behaviors or triggers that lead to following the diet and exercise plans less closely. ● Feedback Loop: Continuously learns from the effectiveness of implemented actions and user feedback, refining its recommendations over time.

Step 3: Producing Model-Based Results: ● Alerts and Recommendations: If a period of lower activity is predicted, the app sends a notification with tailored strategies for exercising. ● Resource Allocation: The app suggests rescheduling personal activities, allocating more time to specific exercises, or prioritizing particular types of food.

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Activity: Reading a Decision Tree

1.

2.

3.

4.

5.

Based on the figure, without knowing anything else about a new college, we would predict that college to be private.

Based on the figure, the decision tree created a split on the f_undergrad variable.

Based on the figure, if a college has a f_undergrad value of at least 2995 then we would predict the college is public, otherwise we would predict the college is private.

Based on the bottom nodes (leaves) of the tree in the figure, we would correctly make 171 Public predictions and 495 private predictions for a total of 666 correct predictions for the colleges in our dataset.

STAR Framework: ● Scope the Data's Context: The decision tree is designed to classify colleges into 'private' or 'public' based on the number of full-time undergraduates (f_undergrad). ● Track the Question behind the Exploration: The question guiding this analysis is whether the number of full-time undergraduates at a college can be used to predict whether the college is private or public. ● Articulate the Results: The tree splits on the 'f_undergrad variable at the value of 2995. If a college has fewer than 2995 full-time undergraduates, it is predicted to be 'Private'; if it has 2995 or more, it is predicted to be 'public'. The model has made correct predictions for 171 public and 495 private colleges. ● Respond with Next Steps: We could consider integrating additional factors such as tuition costs, acceptance rate, or geographic location to refine our predictions. Also, assessing the model's accuracy with validation or on a separate test dataset would be valuable to ensure the reliability of the predictions.

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Activity: Regression versus Classification 1.

2.

3.

4.

5.

6.

Classification Regression Regression Classification Classification Regression

Activity: Inference versus Prediction 1.

2.

3.

This is a regression problem because our response variable, rank, is quantitative. We are interested in inference because we want to know which factors are associated with rank.

This is a classification problem because our response variable, elected, is categorical. We are primarily interested in prediction.

This is a classification problem because our response variable, at risk or not, is categorical. We are interested in prediction f because we want to know which areas are most at risk of fires breaking out based on what we know from past fire data.

Activity: Interpretability versus Complexity

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1.

2.

3.

4.

The variable that appears at the top of all of these trees is the flipper length variable.

The third tree. It makes more correct predictions than the other two decision trees.

The first tree is the most interpretable and least complex because it contains a single split. It’s relatively simple to answer a single question to make a prediction. The third tree is the least interpretable and most complex because it contains the most splits. Thismeans more variables are being used to improve predictions. Asking so many questions to produce a prediction makes it more challenging to characterize relationships between the explanatory variables and species. The second tree is in the middle of the other two for interpretability and complexity.

STAR framework: ● Scope the Data's Context: The decision trees use the penguins dataset to predict species. The initial split in all trees is based on the flipper length variable. ● Track the Question behind the Exploration: The question guiding these models is whether we can predict the species of a penguin—Adelie, Chinstrap, or Gentoo—based on certain characteristics, starting with flipper length. ● Articulate the Results: The first tree makes its predictions based on a single split on flipper length and is the simplest, thus the easiest to interpret. The third tree, with the most splits, uses additional variables to make its predictions, resulting in a more accurate but less interpretable model. The second tree strikes a balance between complexity and interpretability. ● Respond with Next Steps: The complexity of the decision trees increases with the number of splits.

While the third tree may provide the most accurate predictions, it also requires a deeper understanding of the interplay between variables. Deciding which tree to use in practice involves balancing the need for accuracy with the ease of interpretation. Further analysis could involve validation to test the robustness of each model and possibly gather more data to refine the predictions.

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Activity: Calculate Accuracy, Precision, Recall, and Specificity Using a Confusion Matrix Actual High

Actual Low

Predicted High

12

5

Predicted Low

21

43

1.

2.

3.

4.

5.

Accuracy: 0.679 Precision: 0.706 Recall: 0.364 Specificity: 0.672 STAR framework: ● Scope the Data's Context: A confusion matrix is used to evaluate a new decision tree model on the car seats test data. This matrix records the number of correct and incorrect predictions for two classes: 'High' and 'Low' sales. ● Track the Question behind the Exploration: We're examining the model's predictive accuracy, precision, recall, and specificity to understand its performance in classifying sales categories. ● Articulate the Results: The model has an accuracy of 67.9%, meaning it correctly predicts the sales category 67.9% of the time. Precision, which assesses the model's positive predictive value, is 70.6%, indicating that when it predicts 'High' sales, it is correct about 70.6% of the time. Recall, or the model's sensitivity, is 36.4%, suggesting it correctly identifies 36.4% of all actual 'High' sales.

Specificity is 67.2%, showing that it correctly predicts 'Low' sales 67.2% of the time. ● Respond with Next Steps: The model appears to struggle with recall, missing a significant number of 'High' sales. The next steps could involve re-evaluating the features used or exploring other modeling techniques to improve recall without compromising on precision and specificity. Additional testing with validation would also help to ensure these metrics hold across different subsets of the data.

Activity: Overfitted, Underfitted, and Well-Fitted Models 1. a. b. c. d.

2.

Model

A cell is true positive (TP) if Predicted = 1 and Actual = 1 A cell is true negative (TN) if Predicted = 0 and Actual = 0 A cell is false positive (FP) if Predicted = 1 and Actual = 0 A cell is false negative (FN) if Predicted = 0 and Actual = 1

Data set

Accuracy

1

Training

0.80

1

Testing

0.64

2

Training

0.72

2

Testing

0.65

3

Training

0.59

Testing

0.59

3 3.

a. b.

c.

Model 1 was overfitted. It fit the training data and did not predict the test data well.

Model 2 was well-fitted. It fit the training data and did not predict the test data well.

Model 3 was underfitted. It did not fit the training data and did not predict the test data well.

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Chapter 9: Supervised Learning Chapter Review Questions Section 1: Linear Regression with No Explanatory Variables 1.

What is the difference between the sample mean and the population mean?

The sample mean estimates the population mean which is unknown.

2.

In the Palmer penguins study, what does modeling help us estimate?

Population characteristics

3.

Describe the shape of a traditional linear regression model between a response variable and one explanatory variable.

Straight-line functions

4.

What characterizes the null model in linear regression?

Zero explanatory variables

5.

How does the null model predict the response variable?

Using the average of the response variable in our training data

6.

Why is it valuable to use multiple explanatory variables in modeling?

For more accurate predictions

Section 2: Linear Regression with a Categorical Explanatory Variable 7.

Eddie receives a text message and is trying to determine if it’s a phishing attempt. To ascertain whether this is a phishing attempt, Eddie checks to see if the sender is in his list of contacts. Identify the response variable and the explanatory variable.

Response variable: phishing attempt Explanatory variable: sender in Eddie’s list of contacts

8.

Laura wants to purchase tickets to a summer concert. She compares ticket prices in different cities across the country to determine where she can get the best deal. Identify the response variable and the explanatory variable.

Response variable: ticket price Explanatory variable: concert location

9.

Explain how modeling helps us understand our data better.

Answers may vary. Example: Modeling helps us to characterize relationships in the data.

10. What type of analysis specifically tests for differences in means between groups?

Two-sample t-test 11. Which statistical procedure is useful for testing a hypothesis about a population mean of interest?

T-test 12. Skyler wants to test a hypothesis regarding whether the mean weights of four different strawberry varieties are significantly different. What would be the correct procedure for Skyler to use?

ANOVA

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Chapter 9: Supervised Learning

Section 3: Linear Regression with a Quantitative Explanatory Variable 13. Annika wants to rent an apartment for the first time. She looks at the sizes of multiple apartments and compares the rent prices of each. Identify the response variable and the explanatory variable.

Response variable: rent price of apartment Explanatory variable: size of apartment 14. What does a correlation near -1 indicate about two quantitative variables?

Strong negative linear relationship 15. What term refers to estimating a value beyond the observed data range based on variable relationships?

Extrapolation 16. In a linear regression model, what does the estimated slope coefficient represent?

The expected change in the response variable for a one unit increase in the explanatory variable.

17. What does extrapolation refer to?

Predicting variable values beyond the observed data range 18. What are the primary tools for evaluating relationships between two quantitative variables?

Scatterplot and correlation coefficient 19. When graphed, a linear model for a response variable and one explanatory variable will be perfectly horizontal if the correlation coefficient is at what value?

0 20. Provide examples of possible explanatory variables for a car insurance agency building a linear regression model to predict the biannual dollar amount charged to new insurees.

Answers may vary. Examples: age of the driver, age of the driver’s vehicle, distance in miles between the driver’s home and workplace.

21. The simple linear regression model equation follows the form of a line known as what?

A regression line 22. Explain why it doesn’t make sense to use a model to predict the weight of a 0-inch tall person.

Answers may vary. Examples: The predicted value would represent the y-intercept. This is a form of extrapolation. This predicted value is not very meaningful in context.

23. A dataset consists of houses with between 0 and 5 bedrooms and their listing prices. A linear model predicting the listing price of a 10-bedroom house would be considered what?

Extrapolation 24. Why is extrapolation generally discouraged?

Answers may vary. Example: Models are built using data we have access to, and we do not know for sure if individuals in our population that differ greatly from individuals in the data set would be represented by the model.

Section 4: Multiple Linear Regression 25. What phenomenon does Simpson's paradox describe?

Reversal, disappearance, or emergence of an association when the population is divided into subpopulations.

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26. What was the benefit of fitting separate lines for each penguin species when we modeled the Palmer penguins data?

The separate lines have lower variability 27. Describe how our modeling of the penguin data is an example of Simpson’s Paradox.

Answers may vary. Example: After further investigation, the relationship between bill depth and bill length was reversed from weakly negative to strongly positive by including species in the model.

28. Describe one way in which adding variables to a linear model improves the model, and one way in which weakens it.

Adding variables to the model increases its accuracy but the relationship becomes difficult to visualize and less clear for interpretation.

29. What relationship do we expect to see in a graph of test R-squared versus the number of explanatory variables in a multiple linear regression model?

The test R-squared value may initially increase with complexity, showing improved fit, but at higher numbers of variables it may decrease due to overfitting.

30. How does the value of training R-squared change in a multiple linear regression model as more explanatory variables are included?

The R-squared value increases 31. Suppose we fit a multiple linear regression model to a dataset and obtain an R-squared value of 1. What relationship do we expect to see in a graph of the actual values of the response variable versus the predictions?

The predictions perfectly fit the data.

32. As displayed in Table 9.3 of this chapter, we can model the average bill depth of an Adelie penguin as 11.49 + 0.18*bill length. What would be the average bill depth of an Adelie penguin with a bill length of 40 mm?

18.69

Section 5: Nonparametric Regression Models 33. What does a decision tree split on?

Explanatory variable values 34. Explain how a decision tree uses training data to make a prediction in a regression problem.

In a decision tree, splits are identified using the explanatory variables in the data set that result in better predictions of the response variable than without those splits.

35. Explain how a random forest uses training data to make a prediction in a regression problem.

A random forest collects decision trees and their predictions to make a single, final prediction.

36. Explain how a k-nearest neighbors model uses training data to make a prediction in a regression problem.

In k-nearest neighbors, the nearest neighbors are the training observations that are closest to a new observation. A model is created specifying the value of k --the number of considered neighbors, and their values are used to make a prediction for a new observation.

37. How is training data used in machine learning?

Fitting the model's parameters 38. How does a random forest model improve on decision trees?

Answers may vary. Example: by reducing overfitting

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39. What is an advantage of using decision trees for data modeling?

They are highly interpretable.

40. What must be specified for a k-nearest neighbors model?

The number of neighbors to use 41. How are models typically evaluated and compared during the model-building process?

By their test error on test data 42. Which modeling approach is most straightforward to interpret?

Linear regression 43. When using a random forest, which metric indicates the usefulness of the explanatory variables?

Height on the included decision trees 44. What is an advantage of linear regression over k-nearest neighbors for prediction?

Linear regression has greater interpretability.

45. Suppose we fit a random forest model to a dataset and obtain an R-squared value of 1. What relationship do we expect to see in a graph of the actual values of the response variable vs the predictions?

Perfect model fit

Section 6: Classification Models 46. Give an example of a binary categorical response variable.

Answers may vary. Example: Passing or failing an exam 47. Give an example of a categorical response variable with more than two values.

Answers may vary. Example: votes in presidential elections only, votes for local elections and presidential elections, votes in local elections only, does not vote at all.

48. What is the difference between regression and classification in supervised learning?

Regression has a quantitative response variable while classification has a categorical response variable.

49. What type of model would be used to predict the probability of passing an exam (a binary variable) based on hours studied (a continuous variable)?

Logistic regression 50. What type of model would be most appropriate for predicting whether an email is spam?

Logistic regression 51. Explain why logistic regression and linear regression are parametric models.

With both models, it needs to be specified which explanatory variables to include in the model and estimate the coefficient parameters for them.

52. Explain how a decision tree uses training data to make a prediction in a classification problem.

Answers may vary. Example: A decision tree uses training data and creates splits, which introduce categories. A new observation can be classified into one of these categories.

53. Explain how a random forest uses training data to make a prediction in a classification problem.

Answers may vary. Example: A random forest uses training data and combines multiple decision trees and their respective predictions. A new observation can be classified into the random forest’s final prediction.

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54. Explain how a k-nearest neighbors model uses training data to make a prediction in a classification problem.

Answers may vary. Example: A k-nearest neighbors model uses training data to identify observations that are closest to a new observation, or its “nearest neighbors.” The nearest neighbors are averaged into the final prediction.

55. In k-nearest neighbors classification, what does it mean if k=5?

The prediction is based on the 5 closest points.

56. Which model is more prone to overfitting? Why? Decision tree due to their complexity.

57. Give some examples of models that are prone to underfitting and explain why.

Logistic regression, because they are simple models and can under-represent the real relationship.

K-nearest neighbors, because the decision boundary can be smoothed out if the k value is high.

Decision trees, if they are grown too small.

58. Describe and explain a scenario where using logistic regression is useful.

Answers may vary, but they should include a response variable of interest that is categorical and binary.

59. What is the key difference between linear regression and logistic regression?

Linear regression assumes a continuous response variable, while logistic regression is used to model binary outcomes.

60. What is an example of a binary classification problem?

Detecting spam emails 61. When might logistic regression be preferable to linear regression?

When the response variable is binary

Explaining the Concepts 1.

The sign of the correlation between these two variables and the estimated slope are the same. Both tell us if there is a linear relationship between the quantitative variables and if so, the direction of the relationship. The estimated slope tells us the average expected change in one variable for a 1-unit change in the other variable.

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2.

Same y-intercept: This constraint means that all species have the same bill depth when the length is 0.

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Same slope: This constraint means that all species have the same rate of the bill depth increasing when the length increases

3.

4.

5.

6.

7.

Neither of these is better than the graph and model originally displayed in Figure 9.13. Figure 9.13 captures the relationship between bill depth and length that are unique to each species.

The type of variables is the difference between these types of problems. Regression-type problems deal with quantitative variables while classification-type problems deal with qualitative variables.

Linear regression predicts the value of a quantitative variable while logistic regression predicts the value of a qualitative variable. The former is used for regression-style problems while the latter is used for classification-style problems.

A random forest is an ensemble of small decision trees that are each trained on different (overlapping) subsets of the training set. A random forest collects decision trees and their predictions to make a single, final prediction. Random forests reduce overfitting.

Evaluating models on test data is the best way to compare two models. This allows us to determine which fitted model can make better predictions or draw more accurate conclusions about new data.

Interpreting models that are not the best at prediction can still provide insights into the data. However, we should avoid interpreting models that are bad at prediction (i.e. have very low predictive accuracy). We should balance interpretability versus predictive power when creating models by setting a lower threshold which we define as a model having satisfactory predictive power that can enable us to interpret the model to learn about the data.

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Chapter 9: Supervised Learning

Try it Yourself Try It Yourself: Perform a Two-Sample t-Test (9.2) 1.

2.

Yes The average bill depth for males is 17.89 mm, and the average for females is 16.43 mm.

The t-test statistic is 7.31, and the p--value is 0.00.

Try It Yourself: Connecting Linear Regression and t-Test (9.2) 1.

2.

3.

7.31 7.31 the same; the same; the same

Try It Yourself: Compute a Correlation and Fit a Regression Line (9.3) 1.

2.

-0.58.

The estimated slope coefficient for this model is -0.08. This means that for every 1 mm increase in flipper length, we expect a 0.08 mm decrease in bill depth. The correlation and the slope coefficient are both negative. This makes sense as they correspond to the linear relationship between these variables.

Try It Yourself: Extrapolation (9.3) 1.

2.

3.

4.

2.25 inches 31.47 inches 121.47 inches (121.61 is valid based on precise coefficients).

STAR Framework ● Scope the Data's Context: The data comprises height measurements of a child recorded annually from ages 6 to 20. A linear regression model, 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐻𝑒𝑖𝑔ℎ𝑡 = 31.47 + 2.25 ∗ 𝐴𝑔𝑒, is used to analyze the relationship between age and height. ● Track the Question Behind the Exploration: We aim to understand the average annual growth in height and predict the child's height at different ages using the linear regression model. ● Articulate the Results: The average annual growth in height is calculated to be 2.25 inches.

According to the model, the child's height at birth would be 31.47 inches, and at the age of 40, the predicted height is 121.47 (or 121.61 with precise coefficients) inches (over 10 feet). Extrapolation is risky here for several reasons. First, growth rates for humans are not linear throughout their lifespan.

For instance, babies and adolescents may experience growth spurts, while growth eventually ends in adulthood. Second, the data only covers ages 6 to 20, so it may not represent growth patterns outside this range. This can lead to unrealistic or inaccurate predictions. ● Respond with Next Steps: Future investigations should consider nonlinear models and age-specific growth patterns for more accurate predictions beyond the observed data.

Try It Yourself: Exploring R-Squared in Multiple Linear Regression (9.4) 1.

2.

The estimated slope coefficient for the model with just body mass is -0.001. This means that for every 1 g increase in body mass, we expect a 0.001 mm decrease in bill depth.

As the number of explanatory variables increases, the R-squared value of the model increases. Including more explanatory variables improves the model’s performance as measured by R-squared. However, remember that this R--squared is computed on training data, not test data!

Try It Yourself: Fit, Evaluate, and Interpret Decision Trees (9.5) 1.

2.

The model with a Max Depth of 3 performs best because it has the highest R-squared value on the test data.

The species is at the top of the trees and appears most important. After this, sex, flipper_length, and body_mass (in this order) seem most important.

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3.

4.

Using the best performing tree model, the predicted bill depths for the two penguins in Table 9.4 are around 19.25 mm and 15.56 mm, respectively. This is a big difference in predicted bill depth between Adelie and Gentoo penguins, almost 2 standard deviations different. This is in alignment with the answer to question 2 because the species was the most crucial variable in our decision trees. Use the standard deviation value for the whole dataset from Table 9.2.

STAR Framework ● Scope the Data's Context: We use the penguins dataset, dividing it into training and testing subsets.

The response variable is bill depth, with other features as potential explanatory variables. We explore decision tree models at various maximum depths (1, 3, 5, 7, 10). Max depth is a control for the model's complexity, affecting the tree's depth. ● Track the Question Behind the Exploration: Our primary question is identifying which decision tree depth optimally predicts bill depth in penguins, balancing complexity and accuracy. We are also interested in the relative importance of different variables in these models. Additionally, we look for signs of underfitting at lower depths and overfitting at higher depths, as evidenced by changes in the R-squared value. ● Articulate the Results: The model with Max depth of 3 shows the best performance, indicated by the highest R-squared value on the test data. Species is the most important variable, followed by flipper_length, and body_mass. For two specific penguins, the predicted bill depths are notably different, aligning with the observed importance of the species variable. The test R-squared value initially increases with complexity, showing improved fit, but we anticipate a decrease at higher Max depths due to overfitting. ● Respond with Next Steps: Further investigation should focus on understanding the trade-off between model complexity and overfitting. Analyses at different Max depths can provide insights into the model's sensitivity to changes in complexity. Additionally, exploring the biological implications behind species differences and other variables' impact on bill depth can enrich our understanding. It would also be beneficial to visualize the test R-squared versus Max depths to confirm the expected upside-down U-shape pattern, thus providing a more comprehensive view of model performance against complexity.

Try It Yourself: Fit, Evaluate, and Interpret Random Forests (9.5) 1.

2.

3.

4.

The random forest model with Min n of 30 performs best because it has the highest R-squared value on the test data.

The species variable is the most important variable, having the largest variable importance value. The next most important is the sex variable. After this, the other variables are similar to each other in importance, i.e., it is less than 0.05 for each.

Using the best performing random forest model, the predicted bill depths for the two penguins in Table 9.5 are around 19 mm (zero decimal places). This is a very small difference in the predicted bill depth between the two flipper lengths. This is in alignment with the answer to question 2 because flipper length was not an essential variable in our models.

STAR Framework: ● Scope the Data's Context: We are working with the penguins dataset, focusing on bill depth as the response variable. Random forest models can be fitted with varying Min n values (30, 20, 15, 10, 5), where Min n represents the minimum number of observations required to further split a node. This parameter controls the complexity of the model, with larger Min n values leading to simpler models. ● Track the Question Behind the Exploration: The key question is determining which Min n value in a random forest model optimally predicts bill depth, reflecting a balance between model complexity and predictive accuracy. We're also examining the importance of variables such as species and sex. Additionally, we look for signs of underfitting and overfitting as reflected in the test R-squared values at different Min n levels. ● Articulate the Results: The random forest model with Min n of 30 shows the best performance, as indicated by the highest R-squared value. The variables species and sex emerge as the most important variables. The predicted bill depths for two similar penguins with different flipper lengths are almost identical, supporting the non-critical role of flipper length in the model.

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●

Respond with Next Steps: It would also be beneficial to visualize the test R-squared versus Min n to confirm an upside-down U-shape, with overfitting likely at lower Min n values (more complex models) and underfitting at higher values (simpler models). This visualization can provide insights into the optimal complexity for the random forest models in this context. Additionally, further investigation into why certain variables like species and sex are more predictive than others such as flipper length could offer deeper biological insights. Exploring the models at the points of overfitting and underfitting can also provide valuable lessons on the trade-offs between model complexity and predictive accuracy.

Try It Yourself: Fit and Evaluate k-Nearest Neighbors (9.5) 1.

2.

The model with 𝑘 of 25 performs best because it has the highest R-squared value on the test data.

STAR Framework: ● Scope the Data's Context: We return to the Palmer penguin dataset, focusing on bill depth as the response variable and bill length as the sole explanatory variable. We explore kNN models with different values of k (1, 3, 5, 10, 25). Here, k represents the number of nearest neighbors considered for making predictions. ● Track the Question Behind the Exploration: The central question is determining which k value in the kNN model optimally predicts bill depth, balancing between model robustness and predictive accuracy. We are particularly interested in how changes in k impact the model's R-squared value on test data, looking for signs of overfitting with low k values and underfitting with high k values. ● Articulate the Results: The kNN model with k of 25 shows the best performance, as indicated by the highest R-squared value. This suggests that considering a larger pool of neighbors leads to more accurate predictions for bill depth based on bill length. ● Respond with Next Steps: Future work should include visualizing the relationship between k and the test R-squared to confirm the expected pattern and identify the optimal k value. Specifically, we expect an upside-down U-shape, with overfitting likely at lower k values and underfitting at higher values. This can provide insight into the trade-off between robustness and accuracy in kNN models.

Additionally, exploring the effect of including more explanatory variables in the kNN model could offer a broader understanding of the factors influencing bill depth in penguins. Understanding why a certain range of k values provides the best fit can also offer valuable lessons on the nature of the data and the applicability of kNN models in similar contexts.

Try It Yourself: Compare Regression Models (9.5) 1.

The multiple linear regression model performs best due to its highest test R-squared value of 0.87.

Try It Yourself: Fit, Evaluate, and Interpret Logistic Regression (9.6) 1.

2.

3.

The model with four explanatory variables performed best because it had the highest test accuracy. If there is a tie among the models, report the model with the highest number of variables.

Using the logistic regression model with all four explanatory variables, make predictions about the stores in Table 9.6. Store 1 results in higher predicted probability of “High” sales than Store 2. Therefore, we recommend opening a store in the area with a(n) younger average population.

STAR Framework: ● Scope the Data’s Context: We’ve applied logistic regression to the car seats dataset, predicting sales_cat with four explanatory variables: advertising, population, shelf_location, and age. This model aims to understand the factors influencing high or low sales in different store situations. ● Track the Question Behind the Exploration: The primary question is identifying which factors increase the likelihood of a store achieving high or low sales. We're interested in the impact of advertising, population, shelf location, and urban setting on sales performance. ● Articulate the Results: The model with all four variables performed the best, indicating the importance of considering multiple factors for sales predictions. In this model, the coefficients for advertising and shelf_location are positive. This means that increases in advertising and placement of car seats on a 'Good' shelf are associated with a higher likelihood of high sales predictions. Conversely, the negative 132 © 2026 Pearson Education, Inc.

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●

coefficients for population and age suggest that higher populations and urban store locations are linked with a higher likelihood of low sales predictions. The interaction of these variables paints a complex picture of sales dynamics.

Respond with Next Steps: Further analysis could delve into the nuances of these relationships. For example, exploring why increased advertising and good shelf placement are related to high sales predictions could provide valuable marketing insights. Similarly, understanding the positive association of population and its average age with low sales could inform strategic planning for store locations and targeted advertising. Additionally, testing the model in different market conditions or with additional variables could further refine our understanding of sales determinants in the retail industry.

Try It Yourself: Compare Classification Models (9.6) All of the test accuracies are fairly high. The logistic regression model achieved the highest test accuracy of 0.81 and is the best model among these four.

Applying the Concepts Activity: Perform a Two-Sample t-Test 1.

2.

3.

Yes, there is a significant difference, at the 0.05 significance level, in sales between stores in the US versus outside.

The average sales for US stores is 7.87 thousand units and the average sales for non-US stores is 6.82 thousand units. The t-test statistic is 3.70. The p-value is 0.00.

The average sales is significantly higher in the US stores than in non-US stores.

Activity: Connecting Linear Regression and t-Tests 1.

2.

3.

The value of the t-test statistic in the two-sample t-test is similar to the t-test statistic in the regression output.

The p-value of the two-sample t-test is similar to the p-value for the US term in the regression model output.

So, the conclusions of the two procedures are the same.

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Activity: Compute a Correlation and Fit a Regression Line

1.

2.

3.

Correlation between sales and advertising is 0.27.

The estimated slope coefficient is 0.11.

This means that for every one thousand dollar increase in advertising (1 unit) we expect a 0.11 unit (thousands of units sold) increase in sales. The correlation and the slope coefficient are both positive, which makes sense as they both correspond to the linear relationship between these two variables.

Activity: Extrapolation 1.

2.

3.

4.

8.27 lbs 4.71 lbs 335.44 lbs STAR framework. ● Scope the Data's Context: We have a dataset tracking a child's weight from age 1 to 18 years. A linear regression model is used to relate weight to age. This model is aimed at understanding the average growth in weight as the child ages. ● Track the Question behind the Exploration: The questions focus on determining the average yearly weight gain, the estimated weight at birth, and the projected weight at age 40 using the linear model.

These inquiries aim to understand and predict weight changes over different life stages based on existing data. ● Articulate the Results: The linear regression model has equation: Average Weight = 4.71 + 8.27 * Age. The model suggests an average growth of 8.27 lbs per year. Extrapolating this model to birth (age 0) gives an estimated birth weight of 4.71 lbs. Projecting to age 40, the model predicts a weight of 335.44 lbs. However, extrapolation here is risky. Human growth rates are not linear throughout life 134 © 2026 Pearson Education, Inc.

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●

- babies and adolescents have growth spurts, and growth tends to plateau in adulthood. Our data only spans ages 1 to 18, so extrapolating beyond this range, especially to age 40, could result in unrealistic and inaccurate predictions.

Respond with Next Steps: Given the limitations of linear extrapolation in predicting human weight, it's advisable to use models that better capture the nonlinear nature of human growth. Exploring models that incorporate age-specific growth patterns or plateauing effects in adulthood would yield more realistic predictions. Additionally, gathering data across a broader age range could improve the model's reliability for predictions at older ages. Caution should be exercised when using the current model for predictions outside the 1 to 18 age range, especially for significant extrapolations like predicting weight at age 40.

●

Activity: Exploring R-Squared in Multiple Linear Regression

1.

2.

3.

The estimated slope coefficient for the model with just competitor price is 0.01.

This means that for every 1 dollar increase in competitor price we expect a 0.01 thousand unit increase in sales.

As the number of explanatory variables increases, the R-squared value of the model also increases. It appears that including more explanatory variables improves the model performance as measured by Rsquared. The most significant improvements occur upon adding price and shelf_location variables, which suggests higher correlations between sales and these variables. However, remember that this R-squared value is being computed on training data and not test data!

Activity: Fit, Evaluate, and Interpret Decision Trees Decision tree produced in R:

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Chapter 9: Supervised Learning

Decision tree produced in Python:

1.

2.

The model with max depth of 5 is performing best because it has the highest value of R-squared on the test data.

Shelf location is at the top of the trees and appears most important.

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3.

4.

5.

The predicted sales are around 10.72 (in thousands of units) and 5.22 (in thousand of units), respectively.

This appears to be a big difference in predicted sales between a good shelf placement and a medium shelf placement. Good shelf location is definitely recommended.

STAR framework. ● Scope the Data's Context: The analysis involves fitting decision tree models to the cars seats dataset with varying maximum depths (1, 3, 5, 7, 10), using 'Sales' as the response variable. This approach aims to identify the optimal model complexity for predicting sales, with Max Depth controlling the complexity of each tree. ● Track the Question behind the Exploration: The key question is to determine which maximum depth yields the best predictive performance as measured by the R-squared value on the test data.

Additionally, we're interested in the importance of different variables like shelf location, price, and competitor price in predicting sales. ● Articulate the Results: The decision tree model with a max depth of 5 shows the best performance, indicated by the highest R-squared value. Shelf location emerges as the most important variable, followed by price. The model's predictions for two scenarios with different shelf placements show a significant difference in sales, emphasizing the impact of shelf location on sales performance. ● Respond with Next Steps: Future analysis should include examining the test R-squared versus max depth graph to visualize the expected relationship. This would help to identify the points of underfitting (at low max depth) and overfitting (at high max depth). Understanding this relationship can guide decisions on model complexity, ensuring that the chosen model balances the trade-off between underfitting and overfitting. Additionally, exploring other variables or combinations of variables could provide further insights into sales determinants and help refine predictive models for similar retail datasets.

Activity: Fit, Evaluate, and Interpret Random Forests 1.

2.

3.

4.

The model with Min n of 5 is performing best because it has the highest value of R-squared on the test data.

Price and shelf_location appear to be the most important, followed by competitor_price and advertising.

Using a random forest model with Min n of 5, store 1 results in a prediction of higher sales, and so we would recommend pricing the car seats at $100 if we wanted to sell the greatest number of units (not necessarily considering profit).

STAR Framework. ● Scope the Data's Context: The car seats dataset is analyzed using random forest models with different 'Min n' values (30, 20, 15, 10, 5), controlling the complexity of the model. 'Min n' specifies the minimum number of training observations required for further splitting a node, with larger values leading to simpler models. The goal is to identify the optimal model complexity for predicting sales, using R-squared as the metric for model performance. ● Track the Question behind the Exploration: The primary question is to determine which 'Min n' value in the random forest model yields the best predictive performance as measured by R-squared on the test data. Additionally, the analysis seeks to identify the most important variables for predicting sales, with a specific focus on the impact of price and shelf location. ● Articulate the Results: The random forest model with 'Min n' of 5 demonstrates the best performance, as indicated by the highest R-squared value. Price and shelf location emerge as the most important variables, followed by competitor price and advertising. This suggests that these factors play a crucial role in determining sales. The model predicts higher sales for store 1, which has a lower price, indicating the strong influence of pricing on sales volume. ● Respond with Next Steps: Future analysis should include examining the test R-squared versus 'Min n' to visualize the expected upside-down U-shaped relationship. This would help identify the points of underfitting (at high 'Min n' values) and overfitting (at low 'Min n' values). Understanding this relationship can guide decisions on model complexity, ensuring that the chosen model balances the trade-off between underfitting and overfitting. Additionally, exploring the impact of other variables or different combinations of variables could provide further insights into the determinants of sales in the retail industry. Last, selling the greatest number of units may not be the most profitable. We need to explore whether store 2 with higher profits per unit contributes to a greater overall profit compared with store 1.

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Activity: Fit and Evaluate k-Nearest Neighbors 1.

2.

The model with k of 25 is performing best because it has the highest value of R-squared on the test data.

STAR framework. ● Scope the Data's Context: We applied kNN models to the cars eats data set, focusing on 'sales' as the response variable. The models were tested with varying values of k (1, 3, 5, 10, 25), where k represents the number of nearest neighbors considered for making predictions. ● Track the Question behind the Exploration: The key question is to determine which k value in the kNN model optimally predicts sales, balancing between model robustness and predictive accuracy.

We're particularly interested in how changes in k impact the model's performance, measured by Rsquared on the test data, and looking for signs of overfitting with low k values and underfitting with high k values. ● Articulate the Results: The kNN model with k of 25 shows the best performance, as indicated by the highest R-squared value. This suggests that considering a larger pool of neighbors leads to more accurate predictions for sales. ● Respond with Next Steps: Future work should include visualizing the relationship between k and test MSE to confirm the expected pattern and identify the optimal k value. Specifically, we expect an overturned U-shaped curve, with overfitting likely at lower k values and underfitting at higher values.

This can provide insights into the trade-off between robustness and accuracy in kNN models.

Additionally, exploring the effect of including more explanatory variables in the kNN model could offer a broader understanding of the factors influencing sales in car seats. Understanding why a certain range of k values provides the best fit can also offer valuable lessons on the nature of the data and the applicability of kNN models in similar contexts.

Activity: Compare Regression Models 1.

The multiple linear regression and random forest models performed the best, with the multiple linear regression model ultimately having the highest test R-squared value among these four models making it the best.

Activity: Fit, Evaluate, and Interpret Logistic Regression 1.

2.

3.

4.

5.

The best logistic regression model appears to be the model with four explanatory variables. It is the best because it has the highest test accuracy.

The best model was the one with f_undergrad, room_board, top_10_perc, and accept. The estimated coefficient for f_undergrad is negative which means that an increase in the number of full time undergraduates is associated with a lower likelihood of the college being private. All three of the other estimated coefficients are positive, which means increases in them are associated with an increased likelihood of the college being private.

Situation 2 results in a much higher predicted probability of being identified as a private school (above 50%), while Situation 1 would result in a classification as a public school.

If the goal is to be more like other private schools, then this new school should increase its recruitment of new students that are in the top 10% of high school students.

STAR Framework: ● Scope the Data's Context: The college dataset is analyzed using logistic regression to predict whether a college is private or public. The best model includes four explanatory variables: f_undergrad, room_board., top_10_perc, and accept. This model aims to understand how various factors influence the likelihood of a college being classified as private. ● Track the Question behind the Exploration: The primary question is to identify which factors are most predictive of a college's classification as private or public. We focus on the influence of the number of full-time undergraduates, room and board charges, the percentage of new students in the top 10% of their high school class, and the acceptance rate. ● Articulate the Results: The best model is the one with four variables, which has the highest test accuracy. The negative coefficient for f_undergrad suggests that an increase in the number of fulltime undergraduates is associated with a lower likelihood of being a private college. Conversely, the positive coefficients for room_board, top_10_perc, and accept mean that increases in these variables are associated with a higher likelihood of being a private college. This indicates that smaller colleges,

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●

or those with fewer full-time undergraduates, are more likely to be private, while factors like higher room and board charges, a higher proportion of top-performing students, and a higher acceptance rate contribute to a higher likelihood of a college being private.

Respond with Next Steps: Further exploration could involve investigating why these specific variables influence the classification of a college as private. For instance, understanding the economic, educational, and social factors that lead to higher room and board charges or a higher proportion of top students in private colleges could provide deeper insights. Additionally, examining these variables in the context of public colleges could offer a more comprehensive understanding of the higher education landscape. This model could also be used as a predictive tool for new or evolving colleges to gauge their potential classification based on these variables.

Activity: Compare Classification Models 1.

2.

All of the models have relatively high test accuracies.

The logistic regression model performed the best of all of them as evidenced by it having the highest test accuracy.

Chapter 10: Unsupervised Learning Chapter Review Questions Section 1: What Is Unsupervised Learning?

1.

Describe unsupervised learning.

Answers may vary. Example: Unsupervised learning is a subcategory of machine learning focused on overall data patterns and structure.

2.

What is anomaly detection?

Identifying events that deviate from normal behavior

3.

Explain why anomaly detection is an example of unsupervised learning.

Answers may vary. Example: Anomaly detection involves identifying rare observations, but data are not labeled beforehand as “normal” or “outlier.” Unsupervised learning is used to characterize the data.

4.

What is the relationship between anomalies and outliers?

Every anomaly is an outlier, but not every outlier is an anomaly.

5.

What question might you be looking to answer about your data, generally speaking, when completing a cluster analysis?

Answers may vary. Example: You might be looking to understand groupings in your data, to identify similarities.

6.

Describe the primary goal of unsupervised learning.

To discover underlying patterns and relationships in data

7.

What does a standardized score tell us? What else can we call it?

A standardized score says how many standard deviations a data point is from the mean of its dataset. It can also be called a z-score.

8.

How would you compute a standardized score?

You can use the following equation: standardized score = (value of observation - mean of dataset)/standard deviation of dataset.

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●

or those with fewer full-time undergraduates, are more likely to be private, while factors like higher room and board charges, a higher proportion of top-performing students, and a higher acceptance rate contribute to a higher likelihood of a college being private.

Respond with Next Steps: Further exploration could involve investigating why these specific variables influence the classification of a college as private. For instance, understanding the economic, educational, and social factors that lead to higher room and board charges or a higher proportion of top students in private colleges could provide deeper insights. Additionally, examining these variables in the context of public colleges could offer a more comprehensive understanding of the higher education landscape. This model could also be used as a predictive tool for new or evolving colleges to gauge their potential classification based on these variables.

Activity: Compare Classification Models 1.

2.

All of the models have relatively high test accuracies.

The logistic regression model performed the best of all of them as evidenced by it having the highest test accuracy.

Chapter 10: Unsupervised Learning Chapter Review Questions Section 1: What Is Unsupervised Learning?

1.

Describe unsupervised learning.

Answers may vary. Example: Unsupervised learning is a subcategory of machine learning focused on overall data patterns and structure.

2.

What is anomaly detection?

Identifying events that deviate from normal behavior

3.

Explain why anomaly detection is an example of unsupervised learning.

Answers may vary. Example: Anomaly detection involves identifying rare observations, but data are not labeled beforehand as “normal” or “outlier.” Unsupervised learning is used to characterize the data.

4.

What is the relationship between anomalies and outliers?

Every anomaly is an outlier, but not every outlier is an anomaly.

5.

What question might you be looking to answer about your data, generally speaking, when completing a cluster analysis?

Answers may vary. Example: You might be looking to understand groupings in your data, to identify similarities.

6.

Describe the primary goal of unsupervised learning.

To discover underlying patterns and relationships in data

7.

What does a standardized score tell us? What else can we call it?

A standardized score says how many standard deviations a data point is from the mean of its dataset. It can also be called a z-score.

8.

How would you compute a standardized score?

You can use the following equation: standardized score = (value of observation - mean of dataset)/standard deviation of dataset.

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Chapter 10: Unsupervised Learning

9.

Describe the process of dimensionality reduction.

Answers may vary. Example: In dimensionality reduction, we transform a dataset into a form that is more manageable dataset still contains representative properties of the original.

10. What is variable selection in dimensionality reduction?

Variable selection is a way of transforming an overwhelming dataset to make it more manageable, in which we identify variables or columns in the dataset that can be excluded from the rest of the analysis.

11. Explain variable projection.

Answers may vary. Example: Variable projection is the process of converting all variables to a smaller group of new variables that contain most of the original information.

12. What is the difference between variable selection techniques and variable projection (extraction) techniques?

In variable selection, variables are identified for exclusion from analysis. In variable projection, variables are combined.

13. Describe an example where you might choose to use variable projection instead of variable selection.

Answers may vary. Example: You might use variable projection when conducting a survey on hours devoted to types of physical exercise conducted per week, and you found that you received responses including time per week spent on running, jogging, speed-walking, walking, treadmill, weight lifting, yoga. You might combine the first five variables, so that your new variables were: cardio, weight lifting, yoga.

14. What technique can identify a handful of new variables containing similar information to the full set of original variables?

Principal components analysis 15. What is the goal of principal component analysis?

In principal component analysis, the goal is to identify new variables, created by combining the existing ones, that can be used in analysis instead of the original variables.

16. How does the “No Free Lunch” theorem apply to cluster analysis?

It applies in that there is no objectively “correct” clustering algorithm. Each clustering method will have different results on a different dataset; for this reason, it’s useful to document the data science process.

Section 2: Getting to Know Cluster Analysis 17. How does cluster analysis differ from supervised learning methods?

It groups observations based on similarity using multiple variables without focusing on one of them as the response variable.

18. In cluster analysis, how are observations assigned to clusters?

They are assigned in a way that observations in the same cluster are more similar to each other than those in other clusters.

19. In cluster analysis, how can the similarity between two observations be determined?

By computing the distance between them 20. What are several popular measures for defining how close two clusters are to each other?

Centroid linkage, single/minimum linkage, complete/maximum linkage, average linkage 21. In clustering, describe complete linkage.

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Complete linkage defines the distance between clusters as the maximum distance between any observations in the two clusters.

22. What is centroid linkage?

Centroid linkage uses the distance between the two centers of two given clusters to determine the distance between those clusters.

23. In clustering, how do we find the average linkage?

Find the average of the distances between all pairs of observations between the two clusters.

24. In clustering, what kind of data do average and complete linkage perform well on?

Cleanly separated globular clusters 25. Describe some of the pros and cons of using single linkage.

Single linkage can perform well on non-globular data. However, single linkage can perform poorly in the presence of noise.

26. When would you choose to use maximum linkage?

Answers may vary. Example: when working with cleanly separated globular clusters.

27. How does one scale a quantitative variable?

Subtract the mean from all of the variable values, then divide them all by the standard deviation.

28. Why can scaling quantitative variables be important in clustering?

It helps ensure that the variables are balanced with respect to the sizes of their values.

29. Is scaling of quantitative variables always better when doing clustering?

Scaling variables is standard practice to ensure that results are reliable and reasonable.

Section 3: Introduction to k-Means Clustering 30. Why does k-means clustering have its name?

The number of clusters is identified as k. A centroid-, or mean-based method is used wherein the means of the k clusters are computed, and observations are reassigned to clusters based on which mean is closest.

This is done iteratively.

31. What must be specified at the start when using k-means clustering?

The number of clusters 32. How does k-means clustering work?

By iteratively recomputing cluster means and reassigning observations based on distance to the means 33. How do we know when the k-means clustering process is complete?

When the cluster assignments are no longer changing 34. When visualizing the k-means clustering algorithm, what do the colored points represent?

Cluster means 35. Is the k-means clustering process the same or different every time? Why? They are different every time, because Step 1 in the process involves randomly assigning data points to clusters.

36. How does k-means clustering contribute to the overall goal of unsupervised learning?

Answers may vary. Example: k-means clustering iteratively works to identify groupings, or similarities, within a dataset.

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37. What do we mean when we say that k-means clustering is an iterative process?

The steps are repeated to create stronger results—in this case, more defined clusters.

Section 4: Introduction to Hierarchical Clustering 38. Compared to k-means clustering, what is an advantage of hierarchical clustering?

It does not require pre-specifying the number of clusters.

39. Compare and contrast k-means clustering and hierarchical clustering.

Answers may vary. Example: k-means clustering reassigns data points to clusters iteratively based on which mean is closest—it is a mean-based method. In hierarchical clustering, data points are reassigned to clusters iteratively by either merging to create larger clusters, or by dividing to create smaller ones.

40. What are the two ways in which hierarchical clustering can work, and do they differ from each other?

Agglomerative or divisive. In agglomerative, each observation starts in its own cluster, and pairs of clusters are merged iteratively. In divisive clustering, all observations start in one cluster, and splits are performed iteratively.

41. In hierarchical clustering, what does a dendrogram represent?

The full clustering process 42. In hierarchical clustering, at what point are cluster assignments made?

After deciding on the number of clusters and cutting the dendrogram 43. In hierarchical clustering, what connects first at the very bottom of the dendrogram?

The two closest observations 44. When reading a dendrogram from bottom to top, at what point do new individual observations connect to existing clusters?

When an individual observation is closer to the existing cluster than to other individual observations 45. What is represented by the horizontal cut line on a dendrogram?

Specification of the number of clusters 46. What does hierarchical clustering produce that shows the full clustering process?

A dendrogram

Section 5: Assessing the Quality of Clusters 47. Why is clustering considered to be more challenging than supervised learning?

Answers may vary. Example: It is considered to be more challenging because we do not know upfront how the observations in the data are grouped.

48. What is one way to objectively evaluate clustering results?

By assessing cluster distinctiveness through variability 49. What subjective assessment might be made when evaluating clustering results?

Using domain knowledge to judge cluster number, size, distinctiveness, and meaning 50. In the variability ratio plot, what is an elbow generally indicative of?

A transition point after which additional clusters improve the variability ratio to a lesser extent than before the transition

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51. Why might small cluster sizes be a concern when evaluating clustering results?

Because small clusters may not be meaningful groups 52. When evaluating clustering results, what does a lower variability ratio indicate?

Better delineated clusters 53. What does higher between-cluster variability indicate?

Clusters that are sufficiently distinct

Explaining the Concepts 1.

2.

3.

4.

5.

6.

7.

Classification (supervised learning) aims to predict a response variable while clustering focuses on grouping observations according to similar characteristics in a dataset in ways that reveal interesting relationships between variables.

Variables that are unrelated to the question at hand typically do not contain information useful to modeling or analysis. In Vista’s music dataset, album_name, track_name, popularity, and track_genre are examples of features that do not contain information useful to modeling or analysis.

We can group related variables in a dataset so that we have valuable information contained in fewer variables. For example, if two variables are strongly correlated, then we can remove one of the variables since together they do not provide additional information. In Vista’s music dataset, energy and liveness variables are related and contain similar information.

Anomaly detection aims to find outliers in a dataset that may demand immediate action or investigation.

These anomalies are data points that may have been generated by a method different from that used for the rest of the data. In a small dataset, it might be harder to characterize how the rest of the data was generated, making it harder to differentiate between an anomaly and “normal” data points. In a low quality dataset with a lot of noise, we might mistake an anomaly with just noise in our data. k-means clustering computes the means of clusters while hierarchical cluster builds a hierarchy of clusters. In k-means, the number of clusters must be chosen before-hand while hierarchical clustering does not require a specified number of clusters. Lastly, the clusters in k-means are independent while in hierarchical clustering there is a relationship between some clusters.

When clusters contain very different sizes, very small clusters that are far from all clusters might indicate that the corresponding data points are outliers or anomalies.

In cluster analysis, response variables are not provided or used. The goal is not to predict a value for each new example, rather the goal is to find groups among the data based on the examples’ characteristics and then possibly assign new example to an existing group.

Cluster analysis can be used to determine which courses in a major a student might enjoy. A student can use characteristics from a course, e.g. course reviews, times of the course, pre-requisites, to group courses and identify ones that they might like.

Try it Yourself Try It Yourself: Detect Anomalies (10.1) 1.

2.

Mean = 80.077, Standard deviation = 41.029 Standardized scores for each bid: 88

0.193

92

0.291

95

0.364

76

-0.099

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3.

4.

89

0.217

40

-0.977

180

2.435

15

-1.586

90

0.242

80

-0.002

85

0.120

91

0.266

20

-1.464

The bid of 180 has a standard deviation of 2.435 on the high end.

STAR Framework: ● Scope the Data’s Context: We analyzed eBay winning bids for the same product, a new bird cage, with values ranging from $15 to $180. ● Track the Question Behind the Exploration: The analysis aims to determine if there are any significantly low or high bids, which could indicate anomalies in the bidding behavior for this product. ● Articulate the Results: Upon standardizing the bids, it was found that most bids are close to the mean, but a bid of 180 is an outlier with a standardized score of 2.435, significantly higher than the rest. The mean and standard deviation of these bids are 80.077 and 41.029, respectively. ● Respond with Next Steps: Further investigation could involve analyzing the reasons behind such high bids. Are they driven by product scarcity, bidding wars, or other factors?

Try It Yourself: Recognize Unsupervised Learning (10.1) 1.

2.

3.

4.

5.

6.

Supervised learning Unsupervised learning Supervised learning Supervised learning Unsupervised learning Unsupervised learning

Try It Yourself: Calculate Similarity Using Distance (10.2) 1.

2.

There appear to be 3 clusters of penguins in the scatterplot of bill_depth versus bill_length.

The calculated distances and scatterplot groupings do match the species guesses by our colleague.

Cluster #1 consists of 5 penguins and has the maximum distance of 4.46 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 6.67.

Cluster #2 consists of 5 penguins and has the maximum distance of 4.24 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 2.10.

Cluster #3 consists of 5 penguins and has the maximum distance of 1.93 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 2.10.

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Try It Yourself: Choose Variables for Clustering (10.3) 1.

2.

3.

Penguins of the same species are thought to be similar to each other with respect to these physical characteristics. This is visible in the graphs because distances appear to be smaller for penguins of the same species. Distance is our metric for similarity; the smaller it is, the more similar the penguins are. We can see this in the blue box plots, which are generally lower than the red box plots.

The distributions of distances for penguins of the same and differing species have a larger overlap when the variables are not scaled. The distance values for the unscaled variables span values from 0 to around 3500.

This is evidence of the body mass variable overwhelming the distance calculations.

If we want penguins of the same species to appear more distinct from other species with respect to our distances, then we want there to be as little overlap as possible between the distributions of distances displayed here. Therefore, we should scale our variables.

When the variables are not scaled, there does not appear to be a difference in the distances across the different sets of variables. When the variables are scaled, the overlap in the distributions of distances appears to decrease as we include more variables in the calculations.

If we want penguins of the same species to appear more similar with respect to our distances, then we want there to be as little overlap as possible between the distributions of distances displayed here. Therefore, we want 4 variables involved in our distance calculation.

Try It Yourself: Compare Cluster Analysis Results (10.4) 1.

2.

3.

The data are fairly spread, but there appear to be 4 clusters or groups in these data with respect to energy and danceability.

The variability within the clusters appears to go down as the number of clusters increases.

4 clusters appear(s) to be the best across the different sets of clustering results, based on the appearance of the graph. However, this number of clusters does not correspond to the lowest within-cluster variability.

Try It Yourself: Examine the Elbow of a Cluster Analysis (10.5) 1.

2.

The elbow appears to be around a value of 5 for the number of clusters.

Because of the elbow’s placement, we would specify 5 clusters and continue to investigate and explore the data. The within-cluster variability (inertia) appears to drop in smaller increments after this number of clusters.

Applying the Concepts Activity: Detect Anomalies 1.

2.

Mean = 90.85, Standard deviation = 155.61 Standardized scores for each purchase: 35

-0.36

12

-0.51

50

-0.26

42

-0.31

600

3.27

5

-0.55

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3.

4.

70

-0.13

37

-0.35

90

-0.01

73

-0.11

45

-0.29

100

0.06

22

-0.44

The purchase of 600 has a standardized score of 3.27 on the high (low/high) end.

STAR framework ● Scope the Data's Context: The data consists of various credit card purchase amounts during a vacation, ranging from $5 to $600. We're considering these purchases as observational units, with the purchase amount being the variable. We aim to identify any outliers based on their standardized scores (z-scores). ● Track the Question behind the Exploration: The primary question is to determine if any of these credit card purchases are exceptionally high or low. This exploration helps in understanding spending patterns and identifying any anomalies which might suggest unusual behavior. ● Articulate the Results: The purchase of $600, with a standardized score of 3.27, is significantly higher than the mean (90.85) and standard deviation (155.61) of the dataset, marking it as an outlier on the high end. ● Respond with Next Steps: Further investigation could involve analyzing the context of this high purchase. Was it for a necessary or luxury item? Understanding the reason behind such outliers can provide insights into spending behavior during vacations.

Activity: Recognize Unsupervised Learning 1.

2.

3.

4.

5.

6.

Supervised Learning Unsupervised Learning Supervised Learning Supervised Learning Unsupervised Learning Unsupervised Learning

Activity: Calculate Similarity Using Distance 1.

2.

There appear to be 3 clusters of irises in Figure 10.18.

Cluster #1 consists of 5 irises and has the maximum distance of 0.94 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 0.86.

Cluster #2 consists of 6 irises and has the maximum distance of 0.72 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 0.57.

Cluster #3 consists of 4 irises and has the maximum distance of 0.98 between two different observations within it. Using single linkage, the distance to the nearest other cluster is 0.57.

Activity: Choose Variables for Clustering 1.

For the most part, distances appear to be smaller for irises of the same species.

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2.

3.

4.

5.

6.

7.

Distance is our metric for similarity; the smaller it is the more similar the irises. We can see this in the blue boxplots being generally lower than the red boxplots. There is still overlap between these distributions, and the overlap is greater for the situations in which we have fewer variables.

The distance values appear to be roughly similar between using scaled variables and non-scaled variables.

So, there does not appear to be any variables that are overwhelming our distance calculations.

The separation between the blue and red s most evident because there is no intersections between the two interquartile ranges. The distance increases.

If we want irises of the same species to appear more distinct from other species with respect to our distances then we want there to be as little overlap as possible between the distributions of distances displayed here.

If our main concern is distinctiveness of each iris species, we should use unscaled variables because they result in a lower overlap between the box plots.

The overlap in the distance distributions for one and two variables is much greater than that for three and four variables. This is true for both non-scaled and scaled variablesIf we want the irises of the same species to appear more distinct with respect to our distances then we want there to be as little overlap as possible between the distributions of distances displayed here. Therefore, we want 4 variables involved in our distance calculation.

Activity: Compare Cluster Analysis Results R Visualizations:

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Python Visualizations:

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Chapter 10: Unsupervised Learning

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1.

2.

3.

The data are fairly spread, but there appear to be 2-3 clusters or groups in these data with respect to sepal length and sepal width.

The variability within the clusters appears to go down as the number of clusters increases.

3 clusters appear(s) to be the best across the different sets of clustering results, based on the appearance of the graph. This number of clusters does not correspond to the lowest within-cluster variability.

Activity: Examine the Elbow of Cluster Analysis R Visualization:

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Python Visualization

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1. The elbow appears to be around a value of 3 for the number of clusters.

2. Because of the elbow’s placement, we would specify 3 clusters and continue to investigate and explore the data.

The within-cluster variability (inertia) appears to drop in smaller increments after this number of clusters.

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