Solution Manual For Business Analytics Data Analysis & Decision Making, 8th Edition by S. Christian

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 2: Describing the Distribution of a Variable

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 2: DESCRIBING THE DISTRIBUTION OF A VARIABLE

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

An airline analyst wishes to estimate the proportion of all American adults who are afraid to fly because of potential terrorist attacks. To estimate this percentage, the analyst decides to survey 1500 Americans from across the nation. Identify the relevant sample and population in this situation.

Answer: The relevant population consists of all people (adults?) in the U.S. who would consider flying commercially. This might be discovered by a survey of a lot of people (several thousand?) that first asks whether they ever have flown or might fly in the future and then asks whether the fear of terrorist attacks would change their minds.

Only the people responding “Yes” to the first question would remain in the sample for analyzing the second question.

C.2.

The number of children living in each of a large number of randomly selected households is an example of which data type? Be specific.

Answer: The “number of children” is a count, so this is a discrete data type.

C.3.

Does it make sense to construct a histogram for the state of residence of randomly selected individuals in a sample? Explain why or why not.

Answer: A histogram is typically relevant only for a continuous variable that is binned into discrete categories, so it isn’t relevant here. A column chart of counts in states is relevant, and it’s a close relative of a histogram.

C.4.

Characterize the likely shape of a histogram of the distribution of scores on a midterm exam in a graduate statistics course.

Answer: Depending on the difficulty of the exam, the shape could easily be skewed to the left, with the students who didn’t study comprising the long left tail. But it could also be symmetric or even skewed to the right, with a few “brains” comprising the long right tail.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 2: Describing the Distribution of a Variable

C.5.

A researcher is interested in determining whether there is a relationship between the weekly number of room air-conditioning units sold and the time of year. What type of descriptive chart would be the most useful in performing this analysis? Explain your choice.

Answer: This should be a time series graph, which is really a scatterplot of the number of air conditioner sales versus time with the “dots” connected.

C.6.

Suppose that the histogram of a given income distribution is positively skewed. What does this fact imply about the relationship between the mean and median of this distribution?

Answer: The mean will be larger than the median, maybe considerably larger. This is because the large incomes in the right tail pull up the average, but they have no effect on the median.

C.7.

“The midpoint of the line segment joining the first quartile and third quartile of any distribution is the median.” Is this statement true or false? Explain your answer.

Answer: This is true only if the distribution, or at least the middle 50% of it, is reasonably symmetric. If it’s skewed to the right, say, the median will be closer to the 1st quartile than to the 3rd quartile. This is often clear in boxplots, where the median is not in the middle of the box.

C.8.

Explain why the standard deviation would likely not be a reliable measure of variability for a distribution of data that includes at least one extreme outlier.

Answer: The standard deviation, like the mean, is highly sensitive to outliers.

Remember that in the definition of standard deviation, deviations from the mean are squared, so outliers on either side can really increase the standard deviation. As stated in the book, an accepted procedure when outliers are clearly present is to report measures such as the standard deviation with the outliers and without them.

C.9.

Explain how a box plot can be used to determine whether a distribution of values is essentially symmetric.

Answer: There are basically two indications: (1) the median will be about in the middle of the box, indicating that the middle 50% of the distribution is symmetric, and (2) the lines (whiskers) and outliers on either side of the box will be similar, indicating that the more extreme observations are distributed about the same on the low side as on the high side.

C.10. Suppose that you collect a random sample of 250 salaries for the salespersons employed by a large PC manufacturer. Furthermore, assume that you find that two of these salaries are considerably higher than the others in the sample. Before analyzing this dataset, should you delete the unusual observations? Explain why or why not.

Answer: It all depends. If these two salaries correspond to employees who are somehow outside the population of interest, such as top executives, it is fine to delete them. But if they aren’t, they shouldn’t be deleted. For example, they might reveal a “shady” company salary policy and hence could represent the most interesting finding of the study.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 3: Finding Relationships among Variables

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 3: FINDING RELATIONSHIPS AMONG VARIABLES

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

When you are trying to discover if there is a relationship between two categorical variables, why is it useful to transform the counts in a crosstabs to percentages of row or column totals? Once you do this, how can you tell if the variables are related?

Answer: When analyzing a possible relationship between two categorical variables, the joint counts (i.e., the counts of joint categories of the two variables) are all you have to go on. Specifically, it is useful to show these joint counts as percentages of row totals or percentages of column totals. If they are shown as percentages of row totals, for example, where the percentages in each row add to 100%, then the rows should be nearly the same if there is no relationship. Otherwise, you can study the differences between the rows to understand the type of relationship.

C.2.

Suppose you have a crosstabs of two “Yes/No” categorical variables with the counts shown as percentages of row totals. What will these percentages look like if there is absolutely no relationship between the variables? Besides this case, list all possible types of relationships that could occur. (There aren’t many.)

Answer: As in the previous problem, the two rows will be nearly the same, such as the following, if the variables aren’t related: Variable 1 No

Yes

Variable

No

65%

35%

2

Yes

65%

35%

But they could be as follows, indicating that those in the Yes category for Variable 2 are more likely to be in the Yes category for Variable 1 than those in the No category for Variable 2:

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 3: Finding Relationships among Variables

Variable 1 No

Yes

Variable

No

65%

35%

2

Yes

30%

70%

Or they could be as follows, which is the opposite.

Variable 1 No

Yes

Variable

No

60%

40%

2

Yes

35%

65%

And with only two categories per variable (Yes and No), these are the only two possible forms of dependence, corresponding (roughly speaking) to positive correlation and negative correlation.

C.3.

If you suspect that a company’s advertising expenditures in a given month affect its sales in future months, what correlations would you look at to confirm your suspicions?

How would you find them?

Answer: You can imagine two columns, one for monthly sales and one for monthly advertising. If you found the correlation between the two columns, it would indicate the effect of advertising on sales in that month. If you think the effect is delayed, you should create one or more lagged columns of advertising (easy with StatTools) that “pushes down” the advertising column by one or more rows. Then, you can check the correlation between the sales column and any of the lagged advertising columns.

C.4.

Suppose you have customer data on whether they have bought your product in a given time period along with various demographics on the customers. Explain how you could use pivot tables to see which demographics are the primary drivers of their “yes/no” buying behavior.

Answer: You can proceed exactly as in Example 3.5 (frozen lasagna triers). Put any demographic variable in the Rows area, put the “Have Bought” Yes/No variable in the Columns area, and summarize Have Bought by count in the Values area, displayed as percentages of row totals.

C.5.

Suppose you have data on student achievement in high school for each of many school districts. In spreadsheet format, the school district is in column A, and various student achievement measures are in columns B, C, and so on. If you find fairly low correlations (magnitudes from 0 to 0.4, say) between the variables in these achievement columns, what exactly does this mean?

Answer: Because the correlations mentioned are across various achievement measures, these low correlations indicate that the measures themselves don’t appear to be measuring the same thing. For example, if one measure is of reading ability and another is of mathematical ability, the low correlation would discredit the possibility that the best readers tend to be the best at math. Such a finding might or might not be surprising.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 3: Finding Relationships among Variables

C.6.

In the final round of most professional golf tournaments, two players play together in a “twosome.” It often appears that good play or bad play is contagious. That is, if one player in the twosome plays well, the other plays well, and if one player plays badly, the other player plays badly. What data would you collect, and how would you use the data, to check whether this “contagious” property is true?

Answer: You could collect the two 18-hole scores for both players in many final-round twosomes. There would be two columns of scores: Player 1 Score and Player 2 score.

Then, you would calculate the correlation between the two columns. If the contagious property is true, the correlation should be reasonably large and positive.

C.7.

Suppose you have a large dataset for some sport. Each row might correspond to a particular team (as in the file P03_57.xlsx on football outcomes, for example) or it might even correspond to a given play. Each row contains one or more measures of success as well as many pieces of data that could be drivers of success. How might you find the most important drivers of success if the success measure is categorical (such as Win or Lose)?

How might you find the most important drivers of success if the success measure is numerical and basically continuous (such as Points Scored in basketball)?

Answer: This is another case of Example 3.5 (or #4 above), where you want to see which variables drive Win/Lose outcomes. If one of these drivers is continuous, you would group on it in the pivot table.

C.8.

If two variables are highly correlated, does this imply that changes in one cause changes in the other? If not, give at least one example from the real world that illustrates what else could cause a high correlation.

Answer: Causality is a common misconception in correlation studies. For example, if the daily returns of stock X and stock Y are positively correlated, as they probably would be if X and Y are in the same industry, it is unlikely that high stock X returns are causing high stock Y returns. It is much more likely that the movements in the overall market (or in that industry) are causing both stocks’ movements. (Do a Web search for “spurious correlation” to learn more.) And even in cases where X and Y are causally related, it isn’t always clear which is causing which.

C.9.

Suppose there are two commodities, A and B, with strongly negatively correlated daily returns, such as a stock and gold. Is it possible to find another commodity with daily returns that are strongly negatively correlated with both A and B? Answer: This is very likely to be impossible. We are told that A and B vary inversely with one another. Suppose C also varies inversely with A, so C decreases when A increases. But B also decreases when A increases. So it’s hard to see how C could be inversely related to B, i.e., how C could increase when B decreases. Actually, it can be shown mathematically that some correlation matrices, such as the one below, can never be valid, i.e., no set of observations could ever result in them.

A

B

A

1

B

-0.8

1

C

-0.7

-0.9

C

1

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 3: Finding Relationships among Variables

C.10. In checking whether several times series, such as monthly exchange rates of various currencies, move together, why do most analysts look at correlations between their differences rather than correlations between the original series?

Answer: This is a difficult question. Probably the best answer is that it makes more sense economically to check whether changes in the rates of the currencies are correlated. For example, does an increase in the dollar tend to go along with a decrease in the Euro? You could instead check the correlations between the original series, but such correlations are probably not as meaningful. This would check, say, whether high values of the dollar throughout history have been associated with low values of the Euro. But if this turns out to be the case, it’s probably because the changes are correlated.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 4: Business Intelligence (BI) Tools for Data Analysis: Power Query

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 4: BUSINESS INTELLIGENCE (BI) TOOLS FOR DATA ANALYSIS: POWER QUERY

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

What does the term “self-service BI” mean, and why is it important in today’s business world?

Answer: Self-service BI refers to powerful but user-friendly tools for data analysis that allow employees to perform their own data analysis without relying on an IT department for analyses that are often late and out-of-date.

C.2.

Explain how primary and foreign key fields work and why they are needed in a relational database.

Answer: A primary key is a unique identifier in a table. For example, in a Customers table, no two customers have the same primary key value. A foreign key is a nonunique field in another table that permits lookups. For example, a foreign CustomerID field in an Orders table lets you look up information in the Customers table about the customer who placed an order. Without these keys, it is impossible to relate tables.

C.3.

Suppose your company takes orders for its products, which are supplied to your company by vendors. You want to create a relational database of this information.

Discuss the relationship between orders and products and between products and vendors. What tables would you create for this database, and what would you include in these tables?

Answer: The relationship between Products and Orders is almost surely many-to-many; any product can be part of many orders, and any order can include many products. The relationship between Vendors and Products could be many-to-many (if products are multi-sourced) or one-to-many (if products are single-sourced). A bridge table is required for linking Orders and Products. It will have the OrderID and ProductID foreign keys, plus possibly other fields. A similar bridge table between Vendors and Products is necessary only if the relationship is many-to-many.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 4: Business Intelligence (BI) Tools for Data Analysis: Power Query

C.4.

A bridge table is required for a many-to-many relationship, like between orders and products. Explain what fields a bridge table must include at the very least. Give examples of other fields it might also include.

Answer: A bridge table between Orders and Products, for example, must have foreign key fields relating to the Orders table and the Products table. It might also have other information not included in the Orders or Products tables, such as the discount on any particular order for any particular product.

C.5.

What is the role of a Data Model?

Answer: Before Excel Data Models, relational databases were best stored in an RDBMS such as Access or Microsoft SQL Server. That’s what they’re good at. However, related tables can now be stored directly in Excel with a Data Model. This means you can now store the data in Excel that you need to analyze in Excel. (A Data Model can also be stored in Power BI Desktop.)

C.6.

Can you see data in a Data Model if Power Pivot is not loaded? Can you use data in an Excel Data Model in a pivot table if Power Pivot is not loaded?

Answer: Without Power Pivot loaded, you can’t open the Power Pivot window, which allows you to see (and manage) the data in a Data Model. However, the data in a Data Model is still there, even if Power Pivot isn’t loaded, and you can build pivot tables from this data.

C.7.

When you want to filter data for import into Excel, you could filter in the Power Query Editor, or you could import all the data and then filter inside Excel. Which approach would you recommend? Why? Answer: The primary issue is size versus flexibility. If you know exactly which data you want to filter out, then creating the filter in the Query Editor will reduce the size of the Excel workbook with the imported data. However, if you might want to filter in alternative ways, importing all the data into Excel and then filtering provides more flexibility. The mechanics of both approaches are essentially equivalent.

C.8.

You have a multi-table sales database with a foreign ProductID key in a Sales table and a primary ProductID key in a Products table. You notice that one ProductID value in the Sales table has no matching ProductID value in the Products table. Which of the following can you conclude? (1) The ProductID value in the Sales table is necessarily wrong; (2) something is necessarily wrong in the Products table; (3) there is something necessarily wrong in both tables; or (4) there is not necessarily anything wrong in either table.

Answer: This seems to indicate that you sold a product that isn’t in your product assortment. The ProductID in the Sales table is probably wrong. There are other possibilities, such as someone forgetting to enter a product in the Products table, but this is the most likely possibility.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 4: Business Intelligence (BI) Tools for Data Analysis: Power Query

C.9.

In the Power Query Editor, there are two items on the Home ribbon, Merge Queries and Append Queries. Explain the difference between them, and give an example of how each could be used.

Answer: You use Append when you want to stack similarly structured data “on top of” one another, such as sales for different years. You use Merge when you want to combine two tables, such as a facts and a lookup table on key fields.

C.10. In the Power Query Editor, the are three items on the Add Column ribbon, Column From Examples, Custom Column, and Conditional Column. Give an example of how each could be used.

Answer: Column From Examples is useful for picking up patterns such as merging First Name and Last Name fields. You type a full name and hope it picks up the pattern.

Custom Column is more flexible. It lets you define exactly what you want in a column, even a constant value. Conditional Column is like an IF formula in Excel. You define the new column according to one or more conditions.

C.11.

List the advantages of a Dates table in a Data Model, assuming that there is already a Date column in the facts table.

Answer: A Dates table lets you show date in various ways, such as Year, Month Name, Weekday, and others that might be useful in a pivot table.

C.12. Suppose you import data from some source into a Data Model and create a pivot table from it. You notice that an item in the pivot table is spelled wrong. For example, the City field might be in the Rows area of the pivot table, and you see two rows: Pittsburgh and Pittsburg. Assuming you can’t go back to the source to change the spelling, how can you fix it for your pivot table?

Answer: This is the beauty of the Power Query Editor. You can go into it, change the data as you need (which does not change the data in the original source), and load it back into Excel.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 5: Business Intelligence (BI) Tools for Data Analysis: Power Pivot

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 5: BUSINESS INTELLIGENCE (BI) TOOLS FOR DATA ANALYSIS: POWER PIVOT

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

If you examine one of the pivot table example files from Chapter 3, can you observe the data in the Power Pivot window?

Answer: No, the data in Chapter 3 aren’t stored in Data Models, so they can’t be observed in the Power Pivot window. Only data in a Data Model can.

C.2.

In terms of the material in this chapter and the previous chapter, what is common between Excel and Power BI Desktop? What is different?

Answer: Power Query and the measures from Power Pivot are the same in Excel and Power BI Desktop. The difference is the user interface. The advantage of Power BI Desktop is that it enables you to create great visualizations more quickly and easily.

C.3.

What is the difference between an implicit measure and a measure? How is each created?

Answer: An implicit measure occurs when you drag a numeric variable to the Values area of a pivot table and then choose a summarization—sum, average, or whatever. For an explicit measure, the summarization type is part of the definition of the measure.

C.4.

What is the purpose of a hierarchy in a Data Model, and how do you create one?

Answer: A hierarchy such as continent, country, state/province, and city is useful for drilling down to finer and finer detail in a pivot table

C.5.

What is the purpose of the RELATED function in DAX? Answer: The RELATED function lets you create a calculated column by pulling in data from a related table. It can be used in the same way in a measure.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 5: Business Intelligence (BI) Tools for Data Analysis: Power Pivot

C.6.

What is the difference between a calculated column and a measure in DAX? How are they evaluated when dragged to the Values area of a pivot table?

Answer: A calculated column is a new column in a Data Model table. It is calculated row by row through a DAX formula. A measure is a summarization created by a DAX formula for dragging into the Values area of a pivot table. A measure isn’t calculated row by row. It is calculated only as needed for the filter(s) in place in a pivot table.

C.7.

What is the role of row context when defining a DAX calculated column?

Answer: Row context is what allows a DAX formula to use the data in that row when calculating any particular value in a calculated column or measure.

C.8.

Measures can be defined in terms of other measures. What are the advantages of doing this?

Answer: If a measure is defined in terms of one or other measures and those other measures are then changed, the measure updates automatically.

C.9.

What is the difference between removing a column in the Power Query Editor and hiding a column from client tools in the Power Pivot window? When might you remove a column? When might you hide a column? What kinds of columns would you never remove but would probably hide?

Answer: When you hide a column from client tools, it doesn’t show up in the PivotTable Fields pane. This is done so a user doesn’t mistakenly drag such a column to a pivot table. It also creates less clutter in the PivotTable Fields pane. You typically hide key fields and numeric columns to prevent them from being used as implicit measures. You don’t hide fields that might be useful as dimensions in a pivot table.

C.10. We suggested that the only fields that should appear under a facts table in the PivotTable Fields pane should be measures. Why is this a good idea, and how do you accomplish it?

Answer: This is to prevent implicit measures from being created in a pivot table.

Creating measures that define exactly what summarizations you want in pivot tables is simply good practice.

C.11.

How can you create a hierarchy based on fields in different tables of a Data Model?

Answer: You should create calculated columns with the RELATED function to bring all columns to be used in the hierarchy into a single table.

C.12. DAX has functions such as SUM and SUMX or AVERAGE and AVERAGEX. What is the difference between the non-X and X versions?

Answer: The non-X versions take only a single argument, a column from a table. The X versions allow more complex expressions, so they are more flexible.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 5: Business Intelligence (BI) Tools for Data Analysis: Power Pivot

C.13. How does the combination of the CALCULATE and ALL functions alter filter context?

Why might you want to do this?

Answer: The CALCULATE and ALL combination remove all filter context, that is, all filters are ignored. This is useful for getting the denominator for a percent of total summarization.

C.14. How does the combination of the CALCULATE and ALLEXCEPT functions alter filter context? Why might you want to do this?

Answer: The CALCULATE and ALLEXCEPT combination removes all filters except those in the ALLEXCEPT part. This is useful for getting the denominator for a percent of total summarization broken down by continent, for example.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 6: PROBABILITY AND PROBABILITY DISTRIBUTIONS

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Suppose you want to find the probability that event A or event B will occur. If these two events are not mutually exclusive, explain how you would proceed.

Answer: If you add the individual probabilities of A and B, you double-count their intersection, the event where A and B both occur. You should subtract the probability of their intersection from the sum of their probabilities to get the probability that at least one of them occurs.

C.2.

“If two events are mutually exclusive, they must not be independent events.” Is this statement true or false? Explain your choice.

Answer: When events are mutually exclusive, this is an extreme form of nonindependence. If you are told that A occurs, the probability that B occurs is 0, but if you are told that A doesn’t occur, the probability that B occurs becomes positive.

C.3.

Is the number of passengers who show up for a particular commercial airline flight a discrete or a continuous random variable? Is the time between flight arrivals at a major airport a discrete or a continuous random variable? Explain your answers.

Answer: The number of people who show up is discrete because it is a count. However, there might be such a large number of possibilities that its distribution is approximated by a continuous distribution. In contrast, the time between flight arrivals is a continuous measurement, so it would probably be modeled as a continuous random variable. However, as an approximation, it might be discretized, say, into multiples of 5 minutes.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

C.4.

Suppose officials in the federal government are trying to determine the likelihood of a major smallpox epidemic in the United States within the next 12 months. Is this an example of an objective probability or a subjective probability? How might the officials assess this probability?

Answer: This is partly objective and partly subjective. The objective part is that similar epidemics have been observed in the past, so some historical data are available. The subjective part is that each epidemic is somewhat unique. In any case, there would almost surely be some subjectivity in the officials’ assessment.

C.5.

Consider the statement, “When there are a finite number of outcomes, then all probability is just a matter of counting. Specifically, if n of the outcomes are favorable to some event E, and there are n outcomes total, then the probability of E is n/N.” Is this statement always true? Is it always false?

Answer: This is true only if the N possible outcomes are equally likely. For example, you might be able to count that there are three weather possibilities for tomorrow: sunny, partly cloudy, and rainy. But there is probably no reason to believe that they are equally likely. These counting rules for probability usually arise in games of chance: dealing cards, throwing dice, and so on.

C.6.

If there is uncertainty about some monetary outcome and you are concerned about return and risk, then all you need to see are the mean and standard deviation. The entire distribution provides no extra useful information. Do you agree or disagree?

Provide an example to back up your argument.

Answer: The mean and standard deviation are very important measures of a probability distribution, but the entire shape of the distribution, especially the tails, can be relevant in decision making. This is especially true when you are risk averse and large losses (or gains) are possible. The last section of Chapter 6 discusses this in more detail.

C.7.

Choose at least one uncertain quantity of interest to you. For example, you might choose the highest price of gas between now and the end of the year, the highest point the Dow Jones Industrial Average will reach between now and the end of the year, the number of majors Tiger Woods will win in his career, and so on. Using all of the information and insight you have, assess the probability distribution of this uncertain quantity. Is there one “right answer?”

Answer: As of this writing, Tiger Woods has stalled at 14 major tournament wins, and there is a question whether his health will allow him to ever play tournament golf again. So, rather than ask whether he will surpass Jack Nicklaus’s record of 18 majors, it’s probably more relevant to ask whether Tiger will ever win another major. I’d put this probability at about 20%, but talk about subjective!

C.8.

Historically, the most popular measure of variability has been the standard deviation, the square root of the weighted sum of squared deviations from the mean, weighted by their probabilities. Suppose analysts had always used an alternative measure of variability, the weighted sum of the absolute deviations from the mean, again weighted by their probabilities. Do you think this would have made a big difference in the theory and practice of probability and statistics?

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

Answer: Much of today’s statistical analysis is based on statistical theory developed many years ago. The early researchers decided to use standard deviation, with its squared deviations from the mean, as their measure of variability, and almost everything since then has been based on this assumption. It’s hard to say whether this decision was inevitable, or how far statistical theory could have progressed if a weighted sum of absolute deviations from the mean had been used instead. Your statistics textbooks might look considerably different if this alternative measure had been adopted.

C.9.

Suppose a person flips a coin, but before you can see the result, the person puts her hand over the coin. At this point, does it make sense to talk about the probability that the result is heads? Is this any different from the probability of heads before the coin was flipped?

Answer: The best way to answer this is that “uncertainty” is always relative to your current knowledge. So to you, the observer, the result of the coin that has already been flipped is still as uncertain as before it was flipped, and the rules of probability still apply in exactly the same way.

C.10. Consider an event that will either occur or not. For example, the event might be that California will experience a major earthquake in the next five years. You let p be the probability that the event will occur. Does it make any sense to have a probability distribution of p? Why or why not? If so, what might this distribution look like? How would you interpret it?

Answer: A distribution on p makes sense. For example, it might be based on interviews with many “experts” with different opinions and different sets of knowledge about California earthquakes. Each of them might give you their estimate of p, and then you could aggregate these into a distribution on p.

C.11.

Suppose a couple is planning to have two children. Let B1 be the event that the first child is a boy, and let B2 be the event that the second child is a boy. You and your friend get into an argument about whether B1 and B2 are independent events. You think they are independent, and your friend thinks they aren’t. Which of you is correct? How could you settle the argument?

Answer: You and your friend could argue all day about this, even using mathematical arguments, but you wouldn’t get anywhere without data. The only way to resolve the independence issue is to gather data on many families with at least two children. You would divide them into two groups, one where the first child is a boy and the other where the first child is a girl. Then, you would check whether the proportion in the first group that has a boy for the second child is the same as the similar proportion in the second group. If these two proportions are nearly the same, you can conclude independence.

C.12. For each of the following uncertain quantities, discuss whether it is reasonable to assume that the probability distribution of the quantity is normal. If the answer isn’t obvious, discuss how you could discover whether a normal distribution is reasonable. a. The change in the Dow Jones Industrial Average between now and a year from now b. The length of time (in hours) a battery that is in continuous use lasts c. The time between two successive arrivals to a bank

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

The time it takes a bank teller to service a random customer The length (in yards) of a typical drive on a par 5 by Phil Michelson The amount of snowfall (in inches) in a typical winter in Minneapolis The average height (in inches) of all boys in a randomly selected seventh-grade middle school class h. Your bonus from finishing a project, where your bonus is $1000 per day under the deadline if the project is completed before the deadline, your bonus is $500 if the project is completed right on the deadline, and your bonus is $0 if the project is completed after the deadline i. Your gain on a call option on a stock, where you gain nothing if the price of the stock a month from now is less than or equal to $50 and you gain (P – 50) dollars if the price P a month from now is greater than $50

d. e. f. g.

Answer: Possible answers follow. a. First, don’t confuse the question of normality with the question of whether the Dow will go up or down. The distribution of the change could be normal even if you expect the Dow to increase. Then, its mean would be positive, while a belief in up being as likely as down would imply a mean of 0. In either case, an assumption of normality of the change is certainly reasonable. b. This distribution is probably not normal. It is more likely to be “bathtub-shaped,” where the battery has some chance of failing early (infant mortality) and a decent chance of lasting a long time. Even without infant mortality, the distribution is probably skewed to the right. c. This is more likely to be exponentially distributed than normally distributed, where short times between arrivals are most likely, but every now and then, there is a long time between arrivals. Many actual data sets have confirmed this behavior of customer arrivals. d. This is probably more likely to be skewed to the right than symmetric. Relatively short times are the norm, but every once in a while, a customer takes forever to be served. (You’ve undoubtedly been behind such customers.) e. Data would be needed to confirm this, but a normal distribution (mean about 290300 yards, relatively small standard deviation) is probably reasonable. f. It’s hard to know without examining historical data. The distribution could be normal, but it’s probably skewed to the right due to really bad winters. g. This is the classic case of when a normal distribution almost surely applies. h. This can’t be normal because it’s discrete with only three possible values. i. The price in one month might be normally distributed, but your gain isn’t. Your gain is partly discrete, with positive probability of no gain, and partly continuous, when you are “in the money.” This type of “mixture” distribution can’t be normal.

C.13. For each of the following uncertain quantities, discuss whether it is reasonable to assume that the probability distribution of the quantity is binomial. If you think it is, what are the parameters n and p? If you think it isn’t, explain your reasoning. a. The number of wins the Boston Red Sox baseball team has next year in its 81 home games b. The number of free throws Kobe Bryant misses in his next 250 attempts c. The number of free throws it takes Kobe Bryant to achieve 100 successes d. The number out of 1000 randomly selected customers in a supermarket who have a bill of at least $150 © 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

e. The number of trading days in a typical year where Microsoft’s stock price increases f. The number of spades you get in a 13-card hand from a well-shuffled 52-card deck g. The number of adjacent 15-minute segments during a typical Friday where at least 10 customers enter a McDonald’s restaurant h. The number of pages in a 500-page book with at least one misprint on the page Answer: Possible answers follow. a. If Boston tends to win, say, 50% of its home games, then a binomial distribution with parameters 81 and 0.5 is probably a good approximation. However, the probability of winning certainly varies, at least to some extent, on its opponent and possibly other factors. b. A binomial distribution is probably reasonable here, with a very low value of p. Kobe seems resistant to pressure, so his probability of missing doesn’t seem to vary much according to game conditions. c. If Kobe’s successes are binomially distributed (using the argument in part b), then the number of trials required to make at least 100 successes has a non-binomial distribution. It is called a negative binomial distribution, and it’s skewed to the right.

Note that it has to be at least 100. This is essentially the opposite of the binomial situation. With a binomial random variable, the number of trials is given, and the uncertain quantity is the number of successes. With a negative binomial random variable, the number of successes is given, and the number of trials necessary to achieve them is uncertain. d. This number is probably approximately binomially distributed, with p equal to the probability that a typical customer spends at least $150. It’s only approximate because different customers will have different values of p. e. This is approximately binomially distributed if you believe the probability of a price increase is reasonably constant from day to day. f. This is sampling without replacement (you can’t get the same card twice), so the binomial distribution is, at best, an approximation. The correct distribution is hypergeometric, as discussed in Section 5-4b. g. This is not binomially distributed over an entire day because the probability p of at least 10 customers in a 15-minute interval will vary, probably considerably, over the day. It’s hard to say how well a binomial distribution with an “average” value of p will approximate the true distribution. h. A binomial distribution should work well here. However, this is a case with a lot of trials, each with a small value of p. In this case, a Poisson distribution with parameter np should also work well. (See the next answer.)

C.14. One disadvantage of a normal distribution is that there is always some probability that a quantity is negative, even when this makes no sense for the uncertain quantity. For example, the time a light bulb lasts cannot be negative. In any particular situation, how would you decide whether you could ignore this disadvantage for all practical purposes?

Answer: One quick check is to see whether 0 is more than 3 standard deviations to the left of the mean. If it is, then the probability of negative values is practically negligible, and you don’t have to worry. But if it is, you might decide instead to use a non-normal distribution that doesn’t allow negative values. (There are plenty to choose from.)

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 6: Probability and Probability Distributions

C.15. For real applications, the normal distribution has two potential drawbacks: (1) it can be negative and (2) it isn’t symmetric. Choose some continuous random numeric outcomes of interest to you. Are either potential drawbacks really drawbacks for your random outcomes? If so, which is the more serious drawback?

Answer: Here’s one example. Suppose you gather data on the amounts that customers spend for online purchases, and the mean and standard deviation are $100 and $75.

Clearly, a normal distribution doesn’t apply because it would allow nonsensical negative values with fairly high probability. So the “negative” issue is the main problem here. As another example, suppose you gather data on household incomes in some affluent section of town, and the mean and standard deviation are $250,000 and $75,000. Then, a normal distribution with these parameters has a very low probability of being negative, but it still might not be applicable if the histogram of incomes is skewed to the right. In this case, skewness is the problem.

C.16. Many basketball players and fans believe strongly in the “hot hand.” That is, they believe that players tend to shoot in streaks, either makes or misses. If this is the case, why does the binomial distribution not apply, at least not exactly, to the number of makes in a given number of shots? Which assumption of the binomial model is violated, the independence of successive shots or the constant probability of success on each shot? Or can you tell?

Answer: This is very difficult to untangle, given data on successive makes and misses.

Either a lack of independence or a changing value of p could lead to more streakiness than would be expected under a binomial model. You could argue that, for whatever reason, the player’s probability of success, p, varies over time. Or, you could argue that when a player has a success, he is more likely to have a success on the next shot, feeling that he’s “hot.” Both these assumptions could lead to the same data, so it is virtually impossible to say which of them causes the behavior.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 7: Decision Making under Uncertainty

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 7: DECISION MAKING UNDER U NCERTAINTY

TABLE OF CONTENTS Answers to Conceptual Questions................................................................. 1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Your company needs to make an important decision that involves large monetary consequences. You have listed all of the possible outcomes and the monetary payoffs and costs from all outcomes and all potential decisions. You want to use the EMV criterion, but you realize that this requires probabilities, and you see no way to find the required probabilities. What can you do?

Answer: The point is that you don’t have the luxury of saying, “There is no way to find the probabilities.” You must estimate them, even if your estimates are highly subjective.

After all, you are required to make a decision. Then, you can always run a sensitivity analysis on the probabilities to see how sensitive the results are to them.

C.2.

If your company makes a particular decision in the face of uncertainty, you estimate that it will either gain $10,000, gain $1000, or lose $5000, with probabilities 0.40, 0.30, and 0.30, respectively. You (correctly) calculate the EMV as $2800. However, you distrust the use of this EMV for decision-making purposes. After all, you reason that you will never receive $2800; you will receive $10,000, $1000, or lose $5000. Discuss this reasoning.

Answer: As discussed in Section 6-6, this largely depends on your wealth and your feelings about risk. If these dollar amounts won’t kill you (because you’re wealthy), using EMV to make your decision makes sense. If these dollar amounts are relatively large for you and you are risk averse, then you should use another decision -making criterion, probably expected utility.

C.3.

In the previous question, suppose you have the option of receiving a check for $2700 instead of making the risky decision described. Would you make the risky decision, where you could lose $5000, or would you take the sure $2700? What would influence your decision?

Answer: There’s no correct answer to this question. It depends on the person. However, the fact that many very rational people would take the $2700 instead of the gamble with EMV $2800 implies that even rational people don’t always act according to EMV.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 7: Decision Making under Uncertainty

C.4.

In a classic oil-drilling example, you are trying to decide whether to drill for oil on a field that might or might not contain any oil. Before making this decision, you have the option of hiring a geologist to perform some seismic tests and then predict whether there is any oil or not. You assess that if there is actually oil, the geologist will predict there is oil with probability 0.85. You also assess that if there is no oil, the geologist will predict there is no oil with probability 0.90. Why will these two probabilities not appear on the decision tree? Which probabilities will be on the decision tree?

Answer: These probabilities are of the form P(geologist’s prediction, given amount of oil). However, the “given” part won’t be observed until after the geologist’s prediction (i.e., it will be further to the right in the tree). The probabilities in the tree will be backwards of this, of the form P(amount of oil, given geologist’s prediction), and must be found with Bayes’ rule.

C.5.

Your company has signed a contract with a good customer to ship the customer an order no later than 20 days from now. The contract indicates that the customer will accept the order even if it is late, but instead of paying the full price of $10,000, it will be allowed to pay 10% less, $9000, due to lateness. You estimate that it will take anywhere from 17 to 22 days to ship the order, and each of these is equally likely. You believe you are in good shape, reasoning that the expected days to ship is the average of 17 through 22, or 19.5 days. Because this is less than 20, you will get your full $10,000. What is wrong with your reasoning?

Answer: Imagine a tree with 6 chance branches corresponding to the 17 to 22 possibilities. On the first three of these, you get the full $10K, but on the last three, you get only $9K. Therefore, your expected revenue (because of the equally likely possibilities) is only $9.5K. Essentially, you can’t treat your gain as a sure thing based on the average of the days to ship. It must be a weighted average of the gains you can make. This is a very subtle distinction. It can be stated abstractly as follows: The expected value of a function of a random variable is generally not equal to the function evaluated at the expected value of the random variable. (The same idea is discussed in Chapter 15 as the “flaw of averages.”)

C.6.

You must make one of two decisions, each with possible gains and possible losses. One of these decisions is much riskier than the other, having much larger possible gains but also much larger possible losses, and it has a larger EMV than the safer decision.

Because you are risk averse and the monetary values are large relative to your wealth, you base your decision on expected utility, and it indicates that you should make the safer decision. It also indicates that the certainty equivalent for the risky decision is $210,000, while its EMV is $540,000. What do these two numbers mean? What do you know about the certainty equivalent of the safer decision?

Answer: The risky decision’s EMV of $540K has the usual interpretation: a weighted average, weighted by the probabilities, of the possible gains and losses. Its certainty equivalent of $210K means that you would trade this risky gamble for a sure $210K gain.

That is, you are willing to give up $330K in expected value to avoid the gamble.

Evidently, the safer decision has a larger certainty equivalent because it is favored by the expected utility criterion.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 7: Decision Making under Uncertainty

C.7.

A potentially huge hurricane is forming in the Caribbean, and there is some chance it might make a direct hit on Hilton Head Island, South Carolina, where you are in charge of emergency preparedness. You have made plans for evacuating everyone from the island, but such an evacuation is obviously costly and upsetting for all involved, so the decision to evacuate shouldn’t be made lightly. Discuss how you would make such a decision. Is EMV a relevant concept in this situation? How would you evaluate the consequences of uncertain outcomes?

Answer: As the “M” in EMV indicates, decision-making with the EMV criterion is based on monetary consequences. The decision to evacuate a whole island certainly involves monetary consequences, but it also involves nonmonetary human consequences, particularly what would happen if a giant hurricane hit, and people were not evacuated.

To say the least, these decisions involve very difficult trade-offs. How much is a human life worth, compared to the dollars (and other consequences) involved in evacuating?

Still, a decision tree, even just a skeleton of a tree, might help to better understand the problem.

C.8.

It seems obvious that if you can purchase information before making an ultimate decision, this information should generally be worth something, but explain exactly why (and when) it is sometimes worth nothing.

Answer: Imagine Congress spends lots of money on a feasibility study, knowing ahead of time that they are going to vote in favor of a bill regardless of the results of the study. (Far-fetched? Hardly.) Then, the expected value of the information from the study is 0 because the results of the study won’t have any effect on the ultimate decision.

C.9.

Insurance companies wouldn’t exist unless customers were willing to pay the price of the insurance and the insurance companies were making a profit. So explain how insurance is a win-win proposition for customers and the company.

Answer: This is straightforward. Insurance companies can behave according to EMV because they insure so many people—they can afford to win some and lose some. They figure the EMV from lots of historical data and charge a bit more to make a profit.

Customers, on the other hand, are risk averse, so they are willing to pay the extra premium (above their EMV of loss) to avoid big losses. The result is win-win (unless the insurance companies charge exorbitant rates or don’t pay legitimate claims).

C.10.

You often hear about the trade-off between risk and reward. Is this trade-off part of decision making under uncertainty when the decision maker uses the EMV criterion?

For example, how does this work in investment decisions?

Answer: EMV essentially combines risk and reward into a single number for any decision, and the decision with the largest EMV is chosen. However, in other contexts, particularly investing, the mean return of each investment portfolio (the reward) and standard deviation of each investment portfolio (the risk) are both calculated, and an efficient frontier of portfolios is created. The investor can then choose from the portfolios on this efficient frontier based on their feelings toward risk. This doesn’t exactly match the procedure discussed in this chapter, but it is discussed in the last example in Chapter 14.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 7: Decision Making under Uncertainty

C.11.

Can you ever use the material in this chapter to help you make your own real -life decisions? Consider the following. You are about to take an important and difficult exam in one of your MBA courses, and you see an opportunity to cheat. Obviously, from an ethical point of view, you shouldn’t cheat, but from a purely monetary point of view, could it also be the wrong decision? To model this, consider the long-term monetary consequences of all possible outcomes.

Answer: Several years ago, I developed such a model in an MBA course (right before the midterm), and it was difficult to tweak the numbers so cheating would make financial sense, let alone ethical sense. You might try developing your own version. (My version, left as is from an older version of Excel, is available. The file is titled Albright 7e_Ch 06_Cheating in the Core.xlsx.)

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 8: STATISTICAL INFERENCE

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

A large corporation has 4520 male and 567 female employees. The organization’s equal employment opportunity officer wants to poll the opinions of a random sample of employees. To give adequate attention to the opinions of female employees, why might stratified sampling be preferable to simple random sampling?

Answer: A simple random sample should draw a representative sample from both genders, so a simple random sample could be used. But it might be safer to use a stratified sample, stratified by gender. Then, you can be sure that adequate numbers of both genders will be sampled.

C.2.

Provide an example of when you might want to take a stratified random sample instead of a simple random sample, and explain what the advantages of a stratified sample might be.

Answer: Suppose an agency is surveying opinions of government responsiveness in a large, cosmopolitan city. Then, it might make sense to stratify by any number of factors, such as income level, area of the city, or ethnicity. This guarantees representative samples from all of the ethnic groups, for example, which might not happen with a simple random sample.

C.3.

Provide an example of when you might want to take a cluster random sample instead of a simple random sample, and explain what the advantages of a cluster sample might be. Also, explain how you would choose the cluster sample.

Answer: Suppose the state government in a rural state like Iowa wants to survey the opinions of residents in small rural towns. Then, it might make sense to choose a representative small rural town at random—the cluster—and then sample heavily from this one town. The thinking is that this will be much more convenient, and this town is probably representative of similar small rural towns.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

C.4.

When pollsters take a random sample of about 1000 people to estimate the mean of some quantity over a population of millions of people, how is it possible for them to estimate the accuracy of the sample mean?

Answer: The key is the standard error of the sample mean, which contains the square root of the sample size in the denominator. Even with a sample size of only about 1000, this standard becomes fairly small. So, when you go out two standard errors on either side of the sample mean as a measure of the accuracy of the estimate, the interval is fairly narrow (i.e., the accuracy of the estimate is quite good).

C.5.

Suppose you want to estimate the population mean of some quantity when the population consists of millions of members (such as the population of all U.S. households). How is it possible that you can obtain a fairly accurate estimate using the sample mean of only about 1000 randomly selected members?

Answer: This is basically the same as the previous question, just stated slightly differently.

C.6.

What is the difference between a standard deviation and a standard error? Be precise.

Answer: For example, consider the distribution of salaries at some large company. The standard deviation of this population measures how much salaries vary across the company. The standard error of the sample mean, on the other hand, measures how much many sample means would vary. Because of the averaging effect (lows tend to cancel highs), these sample means will tend to vary much less than the individual salaries. This explains intuitively why the standard error of the mean is considerably smaller than the standard deviation of the population. It also explains why the standard error of the mean decreases as the sample size increases—the averaging effect increases.

C.7.

Explain as precisely as possible what it means that the sample mean is an unbiased estimate of the population mean [as indicated in Equation (8.1)].

Answer: This unbiased property means that if you calculate the sample mean from many different random samples, all taken from the same population, some of the sample means will be lower than the population mean and some will be higher, but the average of them will be right on target (assuming sufficiently many sample means).

C.8.

In the wheel of fortune simulation with, say, three spins, many people mistakenly believe that the distribution of the average is the flat graph in Figure 8.2; that is, they believe the average of three spins is uniformly distributed between $0 and $1000.

Explain intuitively why they are wrong.

Answer: The intuition is again the averaging effect, which increases with more spins.

Imagine how unlikely it is to get an average from 3 spins of, say, $300. You would need 3 unlikely spins. Averages closer to $500 are more likely because they are the result of lows cancelling highs.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

C.9.

Explain as precisely as possible what the central limit theorem says about averages.

Answer: The central limit theorem states that the distribution of averages (or sums) from any population distribution will be approximately normally distributed if the sample size is sufficiently large. This theorem is the main reason why the normal distribution plays such a key role in statistical theory and practice.

C.10. Many people seem to believe that the central limit theorem “kicks in” only when n is at least 30. Why is this not necessarily true? When is such a large n necessary?

Answer: As you saw in the wheel of fortune example, the distribution of the sample mean is approximately normal even for small sample sizes such as n=6. It all depends on how non-normal the population is. In the wheel of fortune example, the central limit “kicked in” for small sample sizes because the original distribution, uniform from 0 to 100, is not terribly non-normal—at least it’s symmetric, if not bell-shaped.

C.11.

Suppose you are a pollster and are planning to take a sample that is very small relative to the population. In terms of estimating a population mean, can you say that a sample of size 9n is about 3 times as accurate as a sample of size n? Why or why not? Does the answer depend on the population size? For example, would it matter if the population size were 50 million instead of 10 million?

Answer: This is a true statement, and it is virtually independent of the population size unless the sample size is a large percentage of the population size. The reason is the “square root of sample size” in the denominator of the standard error of the mean. This is the square root of 9n, or 3 times the square root of n, for a sample size of 9n. Hence, a confidence interval for the mean will be narrower by a factor of 3.

C.12. You saw in Equation (8.1) that the sample mean is an unbiased estimate of the population mean. However, some estimates of population parameters are biased. In such cases, there are two sources of error in estimating the population parameter: the bias and the standard error. To understand these, imagine a rifleman shooting at a bull’s-eye. The rifleman could be aiming incorrectly and/or the shots could vary wildly from shot to shot. If the rifleman is aiming incorrectly, but the shots are very consistent, what can you say about the bias and standard error? Answer the same question if the rifleman is aiming correctly at the bull’s-eye but is very inconsistent.

Can you say which of these two situations is worse?

Answer: If he is aiming wrong but his shots are consistent, this is an example of bias but a small standard error. If he is aiming correctly but is very inconsistent, this is an example of no bias but a large standard error. Of course, both are bad, just in different ways. Sampling procedures should strive to reduce both bias and standard error, that is, bad aims and inconsistency.

C.13. Under what conditions, if any, is it not correct to assume that the sampling distribution of the sample mean is approximately normally distributed?

Answer: The only problem is with a sample size that is not large enough to overcome a non-normal population distribution. If the sample size is reasonably large and/or the population distribution isn’t too non-normal, the confidence interval formulas in this chapter should apply quite well.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

C.14. When, if ever, is it appropriate to use the standard normal distribution as a substitute for the t distribution with n – 1 degrees of freedom in estimating a population mean?

Answer: The whole reason for using the t distribution with n-1 degrees of freedom rather than the standard normal distribution for the sampling distribution of the sample mean is that the population standard deviation is unknown. If it were known and could be used in the standard error of the sample mean (sigma over the square root of n), there would be no need for the t distribution.

C.15. “Assuming that all else remains constant, the length of a confidence interval for a population mean increases when the confidence level and sample size increase simultaneously.” Is this statement true or false? Explain your choice.

Answer: When the confidence level increases, say, from 90% to 95% or from 95% to 99%, the confidence interval length will surely increase. But when the sample size increases, the standard error of the mean will usually decrease, implying a narrower confidence interval. So it’s not clear what will happen to the confidence interval length when the confidence level and the sample size both increase.

C.16. Assuming all else remains constant, what happens to the length of a 95% confidence interval for a population parameter when the sample size is reduced by half? You can assume that the resulting sample size is still quite large. Justify your answer.

Answer: The width of the interval decreases by the square root of 2. This is due to the square root of n in the formula for the standard error of the mean.

C.17. Suppose you have a list of salaries of all professional athletes in a given sport in a given year. For example, you might have the salaries of all Major League Baseball players in 2022. Does it make sense to find a 95% confidence interval for the mean salary? If so, what is the relevant population?

Answer: If you are interested only in 2010 Major League Baseball players and you have all of their salaries, there is no need for statistical inference; you know the population mean.

Actually, it’s not clear what other population might be relevant here. The population of all future Major League Baseball players? If so, their salaries are likely to be inflated from those in 2010, so inferring future salaries from those in 2010 might not be such a good idea.

C.18. Researchers often create multiple 95% confidence intervals based on a given data set.

For example, if the variable of interest is home price and there are five neighborhoods in the population, they might create 10 confidence intervals, one for each difference between mean home prices for a given pair of neighborhoods. (There are 10 pairs.) Can they then conclude that there is 95% confidence that all 10 of their confidence intervals will include the corresponding population mean differences? Why or why not?

Answer: A “95% confidence interval” is constructed so any one of them has a 95% chance of including the true population parameter. But, if you calculate many such confidence intervals, each for a different population parameter, the probability that all of them contain their respective population parameters is considerably less than 95%.

An analogy is with a really good free-throw shooter in basketball who makes 95% of his free throws. In this case, the probability that he makes a given free throw is 95%, but if he shoots many free throws, the probability that he makes all of them is clearly less than 95%.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

C.19. Based on a given random sample, suppose you calculate a 95% confidence interval for the following difference: the mean test score for students under 25 years old minus the mean test score for students at least 25 years old, and the confidence interval extends from –14.3 to 1.2. How would you interpret these results? Would you claim that older students, on average, score higher on this test? Would you claim that, on average, it is possible that the younger students score higher on this test?

Answer: In this case, the confidence interval for the “younger minus older” mean difference extends from a negative number, -14.3, to a positive number, 1.2. If the younger students scored, on average, exactly the same as the older students, the true mean difference would be 0. We can’t totally rule out this possibility because 0 is inside the calculated confidence interval. However, because the interval is much more negative than positive, there is more evidence that the true difference is negative, implying that younger students score lower.

C.20. Suppose you are testing the null hypothesis that a mean equals 75 versus a two-tailed alternative. If the true (but unknown) mean is 80, what kind of error might you make?

When will you not make this error?

Answer: Because the null hypothesis is not true (the true mean is 80, not 75), the only type of error you can make is a type II error, the error of not rejecting the null hypothesis. You won’t make this error if you do indeed reject the null hypothesis.

C.21. Give an example of when a one-sided test on a population mean would make more sense than a two-tailed test. Give an example of the opposite. In general, why do we say that there is no statistical way to decide whether a test should be run as a onetailed test or a two-tailed test?

Answer: An example of when a one-tailed hypothesis is more natural is when a pharmaceutical company develops a drug that they believe is more likely to be more effective on a given illness than the best existing drug. Then, the alternative (research) hypothesis is that the new drug is, on average, better than the existing drug, and only evidence in this direction will be sufficient to reject the null hypothesis. An example of when a two-tailed hypothesis is more natural is when two new drugs are introduced and there is no a priori reason to believe that either is better than the other. Then, strong enough evidence in either direction (drug 2 is better than drug 1, or the opposite) will lead to the rejection of the null hypothesis that they are equally effective on average. Whether you run the test with a one-tailed or a two-tailed alternative depends completely on what you’re trying to prove.

C.22. For any given hypothesis test, that is, for any specification of the null and alternative hypotheses, explain why you could make only a type I error or a type II error, but not both. When would you make a type I error? When would you make a type II error?

Answer as generally as possible.

Answer: Either the null hypothesis is true, or the alternative hypothesis is true; they can’t both be true (and they can’t both be false). If the null is true, you can only make a type I error by incorrectly rejecting the null. If the alternative is true, you can only make a type II error by incorrectly not rejecting the null.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 8: Statistical Inference

C.23. Suppose you wish to test a manager’s claim that the proportion of defective items generated by a particular production process has decreased from its longrun historical value of 0.30. To carry out this test, you obtain a random sample of 300 items produced through this process. The test indicates a p-value of 0.01. What exactly is this p-value telling you? At what levels of significance can you reject the null hypothesis?

Answer: This small p-value indicates that such an extreme sample would occur only 1 time out of 100 in repeated samples if the null hypothesis is true, i.e., if the current proportion of defectives is still 0.30. Because of this p-value, the null can be rejected at the 1% significance level or any lower level, like 0.5%.

C.24. Suppose that a 99% confidence interval for the proportion p of all Lakeside residents whose annual income exceeds 80,000 extends from 0.10 to 0.18. The confidence interval is based on a random sample of 150 Lakeside residents. Using this information and a 1% significance level, you wish to test H0: p = 0.08 versus Ha: p ≠ 0.08. Based on the given information, are you able to reject the null hypothesis? Why or why not?

Answer: Yes, you can reject the null hypothesis versus a two-tailed alternative at the 1% significance level because the null value, 0.08, is not inside the 99% confidence interval.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 9: Regression Analysis: Estimating Relationships

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 9: REGRESSION ANALYSIS: ESTIMATING RELATIONSHIPS

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Consider the relationship between yearly wine consumption (liters of alcohol from drinking wine, per person) and yearly deaths from heart disease (deaths per 100,000 people) in 19 developed countries. Suppose that you read a newspaper article in which the reporter states the following: Researchers find that the correlation between yearly wine consumption and yearly deaths from heart disease is –0.84. Thus, it is reasonable to conclude that increased consumption of alcohol from wine causes fewer deaths from heart disease in industrialized societies.

Comment on the reporter’s interpretation of the correlation in this situation.

Answer: This is a case where it is dangerous to infer causality from correlation. Yes, moderately increased wine consumption could have some beneficial effect on preventing deaths from heart disease, but there are undoubtedly other factors involved.

In fact, it is very conceivable that some other factor, possibly a cultural factor, drives both increased wine consumption and fewer deaths from heart disease. At the very least, you shouldn’t infer that drinking more wine is a magic bullet that will prevent death from heart disease!

C.2.

“It is generally appropriate to delete all outliers in a dataset that are apparent in a scatterplot.” Do you agree with this statement? Explain.

Answer: Hopefully, the chapter drove the point home that this is usually a bad practice.

You should have a good reason for deleting outliers. You shouldn’t delete them just to make an analysis come out better.

C.3.

How would you interpret the relationship between two numeric variables when the estimated least squares regression line for them is essentially horizontal (slope 0)?

Answer: This flat scatter has a simple interpretation: Y is not related to X in the sense that your prediction of Y will be the same (the height of the flat scatter) regardless of the value of X.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 9: Regression Analysis: Estimating Relationships

C.4.

Suppose you generate a scatterplot of residuals versus fitted values of the dependent variable for an estimated regression equation. Furthermore, you find the correlation between the residuals and fitted values to be 0.829. Does this provide a good indication that the estimated regression equation is satisfactory? Explain why or why not.

Answer: This high correlation between fitted values and residuals means that you have missed something. That is, it means that the regression equation could be improved upon by adding another term, transforming an existing term, or making some other modification. In a “good” regression, the scatterplot of residuals versus fitted values should be a shapeless swarm with a very small correlation.

C.5.

Suppose you have generated three alternative multiple regression equations to explain the variation in a particular dependent variable. The regression output for each equation can be summarized as follows: Equation 1

Equation 2

Equation 3

4

6

9

R2

0.76

0.77

0.79

Adjusted R2

0.75

0.74

0.73

No. of Xs

Which of these equations would you select as “best”? Explain your choice.

Answer: The key here is that the adjusted R-square has decreased as more terms have been added. This implies that the extra terms are “not worth their weight” and should probably be omitted. Based on parsimony, most analysts would consider Equation 1 the best. However, keep in mind that such choices are “at the margin.” Probably all three equations provide very similar predictions.

C.6.

Suppose you want to investigate the relationship between a dependent variable Y and two potential explanatory variables X1 and X2. Is the R2 value for the equation with both X variables included necessarily at least as large as the R2 value from each equation with only a single X? Explain why or why not. Could the R2 value for the equation with both X variables included be larger than the sum of the R2 values from the separate equations, each with only a single X included? Is there any intuitive explanation for this?

Answer: The answer to the first question is Yes because it is a mathematical fact that R-square can only increase when an extra variable is added to the equation. The answer to the second question is also Yes, even though the reason is not at all intuitive.

This is exactly what happened in the Bendrix Overhead example. Without seeing this in a numeric example, you would probably conclude, intuitively, that it couldn’t happen.

C.7.

Suppose you believe that two variables X and Y are related, but you have no idea which way the causality goes. Does X cause Y or vice versa (or maybe even neither)? Can you tell by regressing Y on X and then regressing X on Y? Explain. Also, provide at least one real example where the direction of causality would be ambiguous.

Answer: Causality can’t be seen from a regression output. Such an output only indicates whether the variables are related. But for all you know, X could be driving Y, Y

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 9: Regression Analysis: Estimating Relationships

could be driving X, or some other variable could be driving both X and Y, and the regression outputs would be identical. For examples, look at macroeconomic variables.

There are plenty of relationships, but it’s not usually clear what is causing what.

C.8.

Suppose you have two columns of monthly data, one on advertising expenditures and one on sales. If you use this dataset, as is, to regress sales on advertising, will it adequately capture the behavior that advertising in one month doesn’t really affect sales in that month but only in future months? What should you do, in terms of regression, to capture this timing effect?

Answer: You need to lag advertising by “pushing its column down” one or more rows.

Then, for example, Sales in May will be in the same row as Advertising in April, Advertising in March, and maybe others. (You get to choose the number of lagged columns.) Then, you can regress the Sales variable against the lagged Advertising variables.

C.9.

Suppose you want to predict reading speed using, among other variables, the device the person is reading from. This device could be a regular book, an iPad, a Kindle, or others.

Therefore, you create dummy variables for the device. How, exactly, would you do it? If you use regular book as the reference category and another analyst uses Kindle as the reference category, will you get the same regression results? Explain.

Answer: Creating the dummy variables is easy enough, either with StatTools or with simple IF formulas. You can decide which category to use as the reference category (by ignoring its dummy). However, the regression results will be equivalent, regardless of the category chosen as the reference category. You should interpret each equation’s coefficients relative to the chosen reference category.

C.10. Explain the benefits of using natural logarithms of variables, either of Y or of the Xs, as opposed to other possible nonlinear functions, when scatterplots (or possibly economic considerations) indicate that nonlinearities should be taken into account. Explain exactly how you interpret regression coefficients if logs are taken only of Y, only of the Xs, or of both Y and the Xs. Answer: As explained in Section 10-6c, the interpretation of regression results is simplified when using log transformations as opposed to other nonlinear transformations. (Besides, they often provide good fits.) The interpretation is always something like: if X increases by a given (unit or percentage), Y will change by a given (unit or percentage), where the correct words in parentheses depend on whether X, Y, or both are log-transformed.

C.11.

The number of cars per 1000 people is known for virtually every country in the world.

For many countries, however, per capita income is not known. How might you estimate per capita income for countries where it is unknown?

Answer: You could try regressing Per Capita Income (Y) on Number of Cars Per 1000 People (X) for countries where both are known. Assuming the fit is reasonably good, you could then use this equation to predict Y from X for countries where X is known but Y isn’t.

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3

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 10: Regression Analysis: Statistical Inference

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 10: REGRESSION ANALYSIS: STATISTICAL INFERENCE

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Suppose a regression output produces the following 99% confidence interval for one of the regression coefficients: [–32.47, –16.88]. Given this information, should an analyst reject the null hypothesis that this population regression coefficient is equal to zero?

Explain your answer.

Answer: Yes, this null hypothesis should be rejected at the 1% significance level. The reason is that the 99% confidence for this coefficient is all negative, i.e., it doesn’t include 0.

C.2.

Explain why it is not possible to estimate a linear regression model that contains all dummy variables associated with a particular categorical explanatory variable.

Answer: The reason is technical, not intuitive. Obviously, there is some redundancy in including all of the dummies for a given categorical variable. For example, if the categories are Male and Female, the dummy for Male is redundant, given the dummy for Female. This redundancy creates a perfect linear relationship between the dummies, and least squares calculations fail when there is a perfect linear relationship between explanatory variables. It’s a bit like dividing by 0—you aren’t allowed to do it.

C.3.

Suppose you have a dataset that includes all of the professional athletes in a given sport over a given period of time, such as all NFL football players during the 2017–2019 seasons, and you use regression to estimate a variable of interest. Are the inferences discussed in this chapter relevant? Recall that we have been assuming that the data represent a random sample of some larger population. In this sports example, what is the larger population—or is there one?

Answer: The key question is whether a larger population makes sense. It might in this case, if you think you can project properties of these seasons to future seasons. Then, the observed data would essentially be interpreted as a random sample from the population of all relevant seasons, present and future. For this to make sense, you would have to assume that nothing is changing much over time.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 10: Regression Analysis: Statistical Inference

C.4.

Distinguish between the test of significance of an individual regression coefficient and the ANOVA test. When, if ever, are these two statistical tests essentially equivalent?

Answer: Each individual t test is for a single X coefficient. The ANOVA test is for the whole equation. It is very possible, for example, that an equation with several Xs has significant explanatory power as a whole—a very low p-value in the ANOVA table—even though one or more of the Xs has a large p-value and hence could be omitted from the equation.

C.5.

Which of these intervals based on the same estimated regression equation with fixed values of the explanatory variables would be wider: (1) a 95% prediction interval for an individual value of Y or (2) a 95% confidence interval for the mean value of Y? Explain your answer. How do you interpret the wider of these two intervals in words?

Answer: The first interval would be wider. It is harder to accurately predict the value of Y for a single individual than to accurately predict the mean of Y for a group of individuals (all with the same X values). The intuitive reason is the averaging effect.

C.6.

Regression outputs from virtually all statistical packages look the same. In particular, the section on coefficients lists the coefficients, their standard errors, their t-values, their p-values, and (possibly) 95% confidence intervals for them. Explain how all of these are related.

Answer: To get any of these t-values, you divide the estimated coefficient by its corresponding standard error. Then, you get the p-value by seeing how much area is beyond the t-value (in both tails) in a t distribution with the appropriate degrees of freedom.

C.7.

If you are building a regression equation in a forward stepwise manner, that is, by adding one variable at a time, explain why it is useful to monitor the adjusted R2 and the standard error of estimate. Why is it not as useful to monitor R2?

Answer: Monitoring R-squares isn’t much help because R-square can only increase when more Xs are added. However, adjusted R-square is worth monitoring because it can decrease when an X is added that doesn’t contribute much to the fit. In fact, this is the only purpose of adjusted R-square.

C.8.

You run a regression with two explanatory variables and notice that the p-value in the ANOVA table is extremely small, but the p-values of both explanatory variables are larger than 0.10. What is the probable reason? Can you conclude that neither explanatory variable does a good job in predicting the dependent variable?

Answer: This is due to high multicollinearity between the two Xs, i.e., they are highly correlated, and both are related to Y. In this case, you can do just as well with only one of the Xs—either one. This is the essence of the height versus foot length example.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 10: Regression Analysis: Statistical Inference

C.9.

Why are outliers sometimes called influential observations? What could happen to the slope of a regression of Y versus a single X when an outlier is included versus when it is not included? Will this necessarily happen when a point is an outlier? Answer by giving a couple of examples.

Answer: Figure 11.12 says it all. One point, like an executive’s extremely large salary, can change the slope of a regression line considerably. However, an outlier won’t necessarily have this effect. For example, the slope of the regression for the points in Figure 11.13 will be similar with or without the obvious outlier.

C.10. The Durbin–Watson test is for detecting lag 1 autocorrelation in the residuals. Which values of DW signal positive autocorrelation? If you observe such a DW value but ignore it, what might go wrong with predictions based on the regression equation? Specifically, if the data are time series data, and your goal is to predict the next six months, what might go wrong with the predictions?

Answer: DW values less than 2 are a sign of positive autocorrelation of the residuals.

Although this is a somewhat advanced topic, and DW values are usually reported only in time series studies, the problem is that when residuals are positively autocorrelated (the usual case), an overprediction in March, say, will likely lead to an overprediction in April. Similarly, an underprediction in June will likely lead to an underprediction in July.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 11: Time Series Analysis and Forecasting

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 11: TIME SERIES ANALYSIS AND FORECASTING

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

“A truly random series will likely have a very small number of runs.” Is this statement true or false? Explain your choice.

Answer: The true statement is that a random series will have a predictable number of runs. This number is not necessarily as small as you might imagine, but (within limits), it is predictable. This is the basis of the runs test. If the observed number of runs is either considerably smaller or larger than the number expected, this is evidence of non-randomness. It is more common to see too few runs, based on the “meandering” pattern of many time series (high for a while, then low for a while, etc.).

C.2.

Distinguish between a correlation and an autocorrelation. How are these measures similar? How are they different?

Answer: These measures are similar intuitively. For example, positive values indicate that highs tend to go with highs, and lows tend to go with lows. However, a correlation is between two different variables, like monthly sales of one product and monthly sales of another product. An autocorrelation, by contrast, measures whether nearby values of a single variable are correlated. For example, it might measure whether sales of a given product in successive months are correlated.

C.3.

Under what conditions would you prefer a simple exponential smoothing model to the moving averages method for forecasting a time series?

Answer: Some companies prefer simple exponential smoothing to moving averages because the latter requires more information to be stored in computer memory, especially when it is applied to many (maybe thousands) of time series variables such as demands for SKUs. However, with today’s cheap memory, this is hardly an issue.

Aside from this, they tend to be equally simple, easy to understand, and accurate.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 11: Time Series Analysis and Forecasting

C.4.

Is it more appropriate to use an additive or a multiplicative model to forecast seasonal data? Summarize the difference(s) between these two types of seasonal models.

Answer: Both can be useful, and it’s hard to say that one is better than the other. The additive model is relevant if you believe each season’s typical value is a certain amount above or below the overall average. The multiplicative model is relevant if you believe each season’s typical value is a certain percentage above or below the overall average.

C.5.

Suppose that monthly data on some time series variable exhibit a clear upward trend but no seasonality. You decide to use moving averages with any appropriate span. Will there tend to be a systematic bias in your forecasts? Explain why or why not.

Answer: The forecasts will tend to stay too low. After all, you’re basing each forecast on an average of past values, which, because of the upward trend, are typically lower than the amount you’re trying to forecast. In this case, a method like Holt’s exponential smoothing method is better because it captures the upward trend.

C.6.

Suppose that monthly data on some time series variable exhibit obvious seasonality.

Can you use moving averages, with any appropriate span, to track the seasonality well?

Explain why or why not.

Answer: Moving averages are not relevant when seasonality is present. The reason is that they average the seasonal highs and lows, essentially removing them. Therefore, the forecast of the next season’s value won’t include the seasonal component, or at least not the correct one.

C.7.

Suppose that quarterly data on some time series variable exhibit obvious seasonality, although the seasonal pattern varies somewhat from year to year. Which method will work best: Winters’ method or regression with dummy variables for quarters (and possibly a time variable for trend)? Why? Answer: Winters’ method should work better because its seasonal smoothing constant lets the method learn from year to year about possible changes in the seasonal pattern. In contrast, regression will find coefficients of the seasonal dummy variables once and for all. That is, it will estimate only one season pattern, even if the actual seasonal pattern varies through time.

C.8.

Most companies that use (any version of) exponential smoothing use fairly small smoothing constants such as 0.1 or 0.2. Why don’t they tend to use larger values?

Answer: Larger smoothing constants track the ups and downs (the zigzags) more closely. This would be a good feature if you thought these ups and downs were predictable, but they are often unpredictable noise that is not worth tracking. Lower smoothing constants like 0.1 or 0.2 are better at tracking any underlying trends without tracking the unpredictable noise.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 12: Introduction to Optimization Modeling

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 12: INTRODUCTION TO OPTIMIZATION MODELING

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Suppose you use Solver to find the optimal solution to a maximization model. Then, you remember that you omitted an important constraint. After adding the constraint and running Solver again, is the optimal value of the objective guaranteed to decrease?

Why or why not?

Answer: It all depends on whether the previous optimal solution happens to satisfy the new constraint. If it doesn’t, then the new optimal objective value will almost surely be lower. But if it does, then the previous optimal solution is still optimal.

C.2.

Consider an optimization model with a number of resource constraints. Each indicates that the amount of the resource used cannot exceed the amount available. Why is the shadow price of such a resource constraint always zero when the amount used in the optimal solution is less than the amount available?

Answer: The shadow price of a resource constraint represents the most you would pay for an extra unit of that resource. But if you already have more than you need—the optimal solution doesn’t use all of the available resource—it makes no sense to pay for more. Hence, the shadow price is 0.

C.3.

If you add a constraint to an optimization model, and the previously optimal solution satisfies the new constraint, will this solution still be optimal with the new constraint added? Why or why not?

Answer: Yes, it will still be optimal. The reason is that the feasible region has gotten smaller, so there are no new feasible solutions. If the current solution was optimal with the previous larger feasible region, it will still be the best among the now smaller feasible region. The key, of course, is that the current solution satisfies the new constraint.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 12: Introduction to Optimization Modeling

C.4.

Why is it generally necessary to add nonnegativity constraints to an optimization model? Wouldn’t Solver automatically choose nonnegative values for the decision variable cells?

Answer: The nonnegativity constraints are required for obvious physical reasons. For example, you can’t produce a negative amount of some product. But an optimization algorithm such as Solver’s has no way of knowing this mathematically. You must tell it explicitly that negative values aren’t allowed.

C.5.

Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit. What does the additive assumption imply about the profit objective? What does the proportionality assumption imply about the profit objective? Be as specific as possible. Can you think of any reasonable profit functions that would not be linear in the amounts of the products produced?

Answer: For the objective, additivity means that the contributions from the various products should be added to get the overall objective value. Proportionality means that if you produce, say, twice as much of some product, the corresponding term in the objective should be doubled. Probably the simplest example of nonlinearity is with quantity discounts. Then, proportionality is violated.

C.6.

In a typical product mix model where a company must decide how much of each product to produce to maximize profit, discuss possible situations where there might not be any feasible solutions. Could these be realistic? If you had such a situation in your company, how might you proceed?

Answer: This could easily happen. Various departmental units of a company might add their own constraints, and this could lead overall to no feasible solutions at all. The only way out of this dilemma is to ask for compromise from the various departmental units. They need to relax their constraints to some extent (not necessarily eliminate them) so that something is possible.

C.7.

In a typical product mix model where a company must decide how much of each product to produce to maximize profit, there are sometimes customer demands for the products. We used upper-bound constraints for these: Don’t produce more than you can sell. Would it be realistic to have lower-bound constraints instead: produce at least as much as is demanded? Would it be realistic to have both (where the upper bounds are greater than the lower bounds)? Would it be realistic to have equality constraints: produce exactly what is demanded?

Answer: Lower bounds and upper bounds could easily make sense. The lower bounds might arise from promises made to some customers that must be kept. So, for example, you could have a lower bound of 100 for demand that must be met and an upper bound of 600 to limit production to what you can sell. As for equality constraints, these make intuitive sense (e.g., you know your customers want exactly 500 units total), but they are typically too constraining. That is, a lot of equality constraints of this type might easily lead to no feasible solutions. Inequality constraints provide more flexibility.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 12: Introduction to Optimization Modeling

C.8.

In a typical production scheduling model like Pigskin’s, if there are no production capacity constraints—the company can produce as much as it needs in any time period—but there are storage capacity constraints and demand must be met on time, is it possible that there will be no feasible solutions? Why or why not?

Answer: In the absence of production capacity constraints, there is always at least one feasible solution: produce exactly to demand each period. This might not be a good solution, but it’s certainly feasible.

C.9.

In a production scheduling problem like Pigskin’s, suppose the company must produce several products to meet customer demands. Would it suffice to solve a separate model for each product, as we did for Pigskin, or would one big model for all products be necessary? If the latter, discuss what this big model might look like.

Answer: A separate model for each product is appropriate if the products are competing for resources, but this is unlikely. They are probably competing for production capacity and storage capacity. In this case, one big model of all products is necessary. Each period, the total amount produced of all products must be less than or equal to production capacity, and the total ending inventory of all products must be less than or equal to storage capacity. Nevertheless, this big model will look a lot like the single-product model in the chapter.

C.10. In any optimization model such as those in this chapter, we say that the model is unbounded (and Solver will indicate as such) if there is no limit to the value of the objective. For example, if the objective is profit, then for any dollar value, no matter how large, there is a feasible solution with profit at least this large. In the real world, why are there never any unbounded models? If you run Solver on a model and get an “unbounded” message, what should you do?

Answer: Except possibly in a few financial situations where short-term arbitrage opportunities are available (and even these will be very short term), the presence of an unbounded solution would indicate the availability of a “get infinitely rich” opportunity, and such opportunities just don’t exist. If you encounter such an unbounded solution, you should check your constraints carefully. You probably either omitted a constraint or the direction of the inequality is wrong.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 13: Optimization Models

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 13: OPTIMIZATION MODELS

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

The employee scheduling model in this chapter was purposely made small (only seven decision variable cells). What would make a similar problem for a company like McDonald’s much harder? What types of constraints would be required? How many decision variable cells (approximately) might there be?

Answer: Instead of 7 staffing levels, one per day of the week, McDonald’s might have one for each 15-minute period of each day of the week. This by itself makes the model much larger and more difficult to solve—there are many more changing cells. Besides this, the company probably has different categories of workers (those in the kitchen, those behind the counter), different tasks these workers can be assigned to, and different types of allowable “shifts.” The same basic approach of our example can still be used, but the model complexity will grow significantly.

C.2.

Explain why it is problematic to include a constraint such as the following in an LP model for a blending problem:

Total octane in gasoline 1 blend ≥ 10 Barrels of gasoline 1 blended daily Answer: There are two problems with this approach: (1) It can lead to division by 0, and (2) it is nonlinear because of changing cells in both the numerator and denominator.

Clearing fractions (multiplying through by the denominator) fixes both of these problems.

C.3.

“It is essential to constrain all shipments in a transportation problem to have integer values to ensure that the optimal LP solution consists entirely of integer-valued shipments.” Is this statement true or false? Why? Answer: Fortunately, it is often not necessary to impose explicit integer constraints on logistics models. This is not at all intuitive, but it is a result of a special mathematical structure of such models. This structure guarantees integer solutions. It is a fortunate property because linear models without explicit integer constraints are much easier to

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 13: Optimization Models

solve than models with integer constraints, and it is also fortunate because real logistics models tend to be very large.

C.4.

What is the relationship between transportation models and more general logistics models? Explain how these two types of linear optimization models are similar and how they are different.

Answer: The “transportation” model is a simple special case of general logistics models. They all involve shipping goods through a network to minimize some cost or maximize some profit. In a transportation model, all shipments are made from a set of suppliers to a set of demanders. This leads to a simple network with all arrows pointing from left to right. A more general logistics model can have a network of any shape with arrows pointing in any direction. The key to solving these general models is imposing the correct node balance constraints.

C.5.

Unlike the small logistics models presented here, realworld logistics problems can be huge. Imagine the global problem a company like FedEx faces each day. Describe as well as you can the types of decisions and constraints it has. How large (number of decision variables, number of constraints) might such a problem be?

Answer: The size of the models FedEx (or Hertz, etc.) faces is mind-boggling. Every decision they have to make (for a one-day period, say) corresponds to a separate changing cell: which trucks (or planes) to send where, what to pack on them, when to send them, and so on. Then, of course, there are all sorts of constraints. The number of decision variables can easily be in the hundreds of thousands (or more), and the number of constraints can easily be in the thousands (or more). But companies like FedEx do develop and solve such models on a daily basis. Of course, they do it with powerful, non-spreadsheet software, not with Solver in Excel.

C.6.

Suppose you develop and solve an integer programming model with a costminimization objective. Assume the optimal solution yields an objective cell value of $500,000. Now, consider the same linear optimization model without the integer restrictions. That is, suppose you drop the requirement that the decision variable cells be integer-valued and reoptimize with Solver. How does the optimal objective cell value for this modified model (called the LP relaxation of the IP model) compare to the original total cost value of $500,000? Explain your answer.

Answer: By dropping the integer constraints, you have made the feasible region much larger, and with many more feasible (non-integer) solutions, the optimal objective value can only get better, i.e., a lower cost than $500K.

C.7.

The portfolio optimization model presented here is the standard model: minimize the variance (or standard deviation) of the portfolio, as a measure of risk, for a given required level of expected return. In general, the goal is to keep risk low and expected return high. Can you think of other ways to model the problem to achieve these basic goals? Is high variability all bad risk?

Answer: The possible drawback to our portfolio model is that “risk” is measured by the standard deviation of the portfolio. However, this risk includes both downside risk (bad) and upside risk (good), so it would be better to have a measure that includes only downside risk. Fortunately, such measures are available and are often used.

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2

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 14: Introduction to Simulation Modeling

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 14: INTRODUCTION TO SIMULATION MODELING

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

You are making several runs of a simulation model, each with a different value of some decision variable (such as the order quantity in the Walton calendar model), to see which decision value achieves the largest mean profit. Is it possible that one value beats another simply by random luck? What can you do to minimize the chance of a “better” value losing out to a “poorer” value?

Answer: Yes, luck can definitely play a role. The best way to prevent one decision value from beating another just by luck is to use the same random numbers in the simulation of each. This is a bit difficult to do if you are simulating with Excel-only tools, but @RISK makes it easy by using its RiskSimtable function. (Actually, you should check one setting.

Go into the @RISK Simulation Settings dialog box, click the Sampling tab, and make sure the Multiple Simulations setting is “All Use Same Seed.” This forces each simulation to use the same random numbers. Fortunately, this is the default setting.)

C.2.

If you want to replicate the results of a simulation model with Excel functions only, not @RISK, you can build a data table and let the column input cell be any blank cell.

Explain why this works.

Answer: It is a trick made available by the way data tables work in Excel. They take each value in the left column of the data table, one at a time, plug it into the column input cell, and then recalculate. The plugging in part is completely irrelevant. The important part is the recalculation. This leads to new random numbers for each row of the data table—exactly what you want.

C.3.

Suppose you simulate a gambling situation where you place many bets. On each bet, the distribution of your net winnings (loss if negative) is highly skewed to the left because there are some possibilities of really large losses but not much upside potential. Your only simulation output is the average of the results of all the bets. If you run @RISK with many iterations and look at the resulting histogram of this output, what will it look like? Why? Answer: The “population distribution” you are sampling (i.e., simulating) from is heavily skewed to the left. But this is for an individual bet. Because the simulation output is

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 14: Introduction to Simulation Modeling

the average of many such bets, the central limit theorem implies that the resulting histogram of the average will be approximately normal, i.e., bell-shaped and not skewed (at least, not very much, depending on the number of terms in the average).

C.4.

You plan to simulate a portfolio of investments over a multiyear period, so for each investment (which could be a particular stock or bond, for example), you need to simulate the change in its value for each of the years. How would you simulate these changes in a realistic way? Would you base it on historical data? What about correlations? Do you think the changes for different investments in a particular year would be correlated? Do you think changes for a particular investment in different years would be correlated? Do you think correlations would play a significant role in your simulation in terms of realism?

Answer: There are different ways to do this depending on how you think the world works. One popular way is to model the changes of the different investments in a given year with correlated lognormal (or possibly normal) distributions but to let changes in different years be uncorrelated—correlation within years, no correlations across years. @RISK lets you do this with a correlation matrix for a given year and different years using a separate instance of this correlation matrix. You can look this up in @RISK help, but the point is that it’s possible and is often realistic.

C.5.

Big Hit Video must determine how many copies of a new video to purchase. Assume that the company’s goal is to purchase a number of copies that maximizes its expected profit from the video during the next year. Describe how you would use simulation to shed light on this problem. Assume that each time a video is rented, it is rented for one day.

Answer: This is very similar to the Walton bookstore problem. Assume initially that the daily demand is normally distributed. Then, data on past video sales can be used to fit the mean and standard deviation of daily demand. As time goes on, the mean will probably decrease. Then, you can try different order quantities to see which order quantity maximizes expected profit. Note that the number of daily rentals is the smaller of a day's simulated demand and the number of videos ordered. The daily demand for videos is probably Poisson distributed. (This has been found in empirical studies.)

C.6.

Many people who are involved in small auto accidents do not file a claim because they are afraid their insurance premiums will be raised. Suppose your insurance company has three rates. If you file a claim, you are moved to the next higher rate. How might you use simulation to determine whether you should file a claim?

Answer: You need to fit some distribution to the size of claims. Lognormal (a distribution with positive values that is skewed to the right) will probably work here.

Policies such as file if claim >$500, file if claim >$1000, etc. could then be evaluated against the goal of minimizing average cost per year. You must also model the frequency of accidents. (The time between accidents is probably exponentially distributed, as has been found in empirical studies.)

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 14: Introduction to Simulation Modeling

C.7.

A building contains 1000 lightbulbs. Each bulb lasts at most five months. The company maintaining the building is trying to decide whether it is worthwhile to practice a “group replacement” policy. Under a group replacement policy, all bulbs are replaced every T months (where T is to be determined). Also, bulbs are replaced when they burn out.

Assume that it costs $0.05 to replace each bulb during a group replacement and $0.20 to replace each burned-out bulb if it is replaced individually. How would you use simulation to determine whether a group replacement policy is worthwhile?

Answer: You need to estimate the probability that a bulb will fail during the current month given its age. For example, 10% might fail during their first month, 20% during their second month, and so on. Then, each month, you need to keep track of how many bulbs of each age are in the building. Remember that a planned replacement sets all bulbs back to being new. Then, it is straightforward to keep track of costs.

C.8.

Why is the RiskCorrmat function necessary? How does @RISK generate random inputs by default, that is, when RiskCorrmat is not used?

Answer: The RiskCorrmat function is @RISK’s way of specifying that you want uncertain input variables to be correlated. Without this function, each input’s value on each iteration is an independent draw from that input’s distribution, regardless of the values of the other inputs on that iteration. With RiskCorrmat, the inputs’ values for all iterations are first generated in this way (you can think of them as stored in long columns, one column per input variable), and then the numbers in these columns are somehow “shuffled” so the columns are correlated as desired. @RISK takes care of the shuffling details. Your job is to recognize when correlations are necessary and then to build them in explicitly with RiskCorrmat.

C.9.

Consider the claim that normally distributed inputs in a simulation model are bound to lead to normally distributed outputs. Do you agree or disagree with this claim? Defend your answer.

Answer: This is often true, but there are a lot of cases where it isn’t true. For example, suppose the amount of time it takes to complete a project is normally distributed with mean 30 days, and you get a bonus if you finish the project in 28 days or fewer. If your bonus is an output, it certainly won’t be normally distributed. It will be discrete with possible values 0 and the size of the bonus.

C.10. It is very possible that when you use a correlation matrix as input to the RiskCorrmat function in an @ RISK model, the program will inform you that this is an invalid correlation matrix. Provide an example of an obviously invalid correlation matrix involving at least three variables, and explain why it is invalid.

Answer: A simple example is when all three of the correlations (between 1 and 2, between 1 and 3, and between 2 and 3) are all large negative, such as -0.9. You can try this in @RISK, and it will inform you that this correlation matrix is invalid. The intuition is that if 1 and 2 go in very opposite directions, and 1 and 3 go in very opposite directions, then it is impossible for 2 and 3 to also go in very opposite directions.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 14: Introduction to Simulation Modeling

C.11.

When you use a RiskSimtable function for a decision variable, such as the order quantity in the Walton model, explain how this provides a “fair” comparison across the different values tested.

Answer: It is fair because each of the simulations is exposed to the same random numbers. See more about this in the answer to #1 above.

C.12. Consider a situation where there is a cost that is either incurred or not. It is incurred only if the value of some random input is less than a specified cutoff value. Why might a simulation of this situation give a very different average value of the cost incurred than a deterministic model that treats the random input as fixed at its mean? What does this have to do with the “flaw of averages”?

Answer: In a deterministic model, the cost would be incurred based on the average value of the input. Let’s say this average value is below the cutoff. Then, the cost would never be incurred in the deterministic model. In contrast, it would sometimes be incurred and sometime not be incurred in a simulation model. Hence, the average of these, over all iterations, would be positive—very different from the 0 in the deterministic model. (The situation is just as bad if the average value of the input is above the cutoff, so the cost is always incurred in the deterministic model.) This example is the essence of the flaw of averages.

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4

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 15: Simulation Models

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 15: SIMULATION MODELS

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

We have separated the examples in this chapter into operations, finance, marketing, and games of change categories. List at least one other problem in each of these categories that could be attacked with simulation. For each, identify the random inputs, possible probability distributions for them, and any outputs of interest.

Answer: In operations, a hot topic these days is supply chain operations. For example, an interesting supply chain simulation could model the effect of random interruptions in the supply chain (hurricanes, worker stoppages, etc.). In finance, the sky is the limit.

For example, you could project uncertain values in future financial statements. In marketing, you could simulate sales of a new drug over its (possibly random) lifetime, using market size, market share, and possibly price as uncertain driving factors. In sports, the sky is again the limit. For example, you could simulate the final seconds of a basketball game where one team is ahead by 3 points and the other team has the ball.

The age-old question is then whether the team that’s ahead should foul. (This simulation is actually not as simple as it sounds!)

C.2.

Suppose you are an HR (human resources) manager at a big university, and you sense that the university is becoming too top-heavy with full professors. That is, there do not seem to be as many younger professors at the assistant and associate levels as there ought to be. How could you study this problem with a simulation model, using current and/or proposed promotions, hiring, firing, and retirement policies?

Answer: This type of simulation has been developed. Typically, the current hiring/promotion/retirement policy leads to input probabilities, such as the probability that an associate professor will be promoted to full rank in a given year. Then, starting with an initial number of professors in the various ranks in year 1, you can simulate where they move in succeeding years. The outputs can be the numbers in the various ranks in some future year, say, 10 years from now. If these numbers aren’t the profile desired by the university, the simulation can be run again (or as many times as you like) with different input probabilities.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 15: Simulation Models

C.3.

You are an avid basketball fan, and you would like to build a simulation model of an entire game so you could compare two different strategies, such as manto- man versus zone defense. Is this possible? What might make this simulation model difficult to build?

Answer: This would not be the simple type of spreadsheet simulation illustrated in this chapter. It would instead be a “discrete event” simulation, where various events (a shot is made, a player is fouled, a turnover occurs, etc.) are made at random points in time.

Usually, a computer program has to be written to “play out” these events throughout the game, and aggregate statistics are tallied throughout the game. It can require tricky programming, but it can certainly be done.

C.4.

Suppose you are a financial analyst, and your company runs many simulation models to estimate the profitability of its projects. If you had to choose just two measures of the distribution of any important output, such as net profit, to report, which two would you choose? Why? What information would be missing if you reported only these two measures? How could they be misleading?

Answer: The point here is that there is an entire distribution of any such output, and we are limited to reporting only two of its measures. (Granted, this is unrealistic. Why not report all the information available from @RISK?) Anyway, you probably want a measure of central tendency and a measure of variability. Two useful measures for a financial analyst might be the median (half the time, net profit is above it; half the time, it’s below it) and the 5th percentile (this is nearly as bad as net profit will be).

C.5.

Software development is an inherently risky and uncertain process. For example, there are many examples of software that couldn’t be “finished” by the scheduled release date—bugs still remained, and features weren’t ready. How might you simulate the development of a software product? What random inputs would be required? Which outputs would be of interest? Which measures of the probability distributions of these outputs would be most important?

Answer: First, this would be useful only if it were done carefully. It wouldn’t be an easy simulation to develop, and it would require advice from various experts to get realistic input distributions. Generally, this is a case of project analysis, and one of Microsoft’s Office programs, Microsoft Project (which you must buy separately), is devoted entirely to project analysis. However, it doesn’t include simulation possibilities. Recognizing this, Palisade added a Project tool to @RISK. This tool works together with Microsoft Project, allowing you to build randomness into times or costs of a project. (It appears in the @RISK ribbon only if you have Microsoft Project installed on your PC.)

C.6.

Health care is continually in the news. Can (or should) simulation be used to help solve, or at least study, some of the difficult problems associated with health care? Provide at least two examples where simulation might be useful.

Answer: The short answer to this question is an obvious YES! You can undoubtedly list any number of uncertainties in health care systems that affect costs. Unfortunately, the overall system is so complex, and there are so many uncertainties, that it’s hard to know where to start.

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2

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 16: Data Mining: Classification

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 16: DATA MINING: CLASSIFICATION

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

How does the general classification problem discussed in Section 16-2 differ from the general problem analyzed with regression, and why does standard regression not work for the classification problem?

Answer: In classification problems, the dependent variable is discrete, not continuous, with 2 or more possible values, each representing a category. Like regression, we want to use explanatory variables, the Xs, to explain which category each observation is in, but standard regression doesn’t work because of the discrete nature of the dependent variable.

C.2.

What is the main purpose of logistic regression, and how does it differ from the type of regression discussed in Chapters 9 and 10?

Answer: Logistic regression is used to relate a categorical variable, usually a 0/1 variable, to any number of explanatory variables. This is basically the classification problem: to see which variables help explain why some customers buy our product and others don’t, for example. Although this sounds very much like regression from Chapters 10 and 11, the fact that the dependent variable is categorical means that logistic regression is quite different, mathematically, from regular least-squares regression. Formally, the log of the odds ratio (i.e., the logit), not a dependent variable Y, is regressed against a linear function of the explanatory variables.

C.3.

Suppose a term in a logistic regression equation is 0.687*Mall_Trips. Explain exactly what this means.

Answer: First, because this coefficient is positive, the probability that a typical person is a 1 (in this case, a lasagna trier) increases as the number of mall trips increases. More specifically, for each extra mall trip, the log of the odds ratio is expected to increase by an amount 0.687. This isn’t an easy concept, but it is explained in more detail with a numerical calculation in Section 17-3a.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 16: Data Mining: Classification

C.4.

Suppose you are trying to classify a variable where 96% of its observations equal 0, and only 4% equal 1. You run a logistic regression, and the classification table shows that 97% of the classifications are correct. Why might this large percentage still not be cause for celebration?

Answer: This is a common situation in classification studies, where the 1s comprise a very small minority of all observations. You could use the naive rule of classifying everyone as a 0, and you would have 96% accuracy. However, this classification rule provides absolutely no insight; it is worthless from a practical point of view. To be really good, a classification rule should have a much better accuracy rate than 96%.

Granted, 97% is better, but it’s not much better.

C.5.

What are the strengths and drawbacks of neural nets versus classification trees?

Answer: These two algorithms are basically at opposite ends of a spectrum. Neural nets often provide very accurate predictions, but because of all the nonlinear functions involved, they are almost impossible to untangle or understand. On the other hand, classification trees might not provide similar prediction accuracy, but they provide classification rules that are easy to understand and often provide useful insights.

Besides, classification trees generally provide more accurate predictions than you might expect. In any case, you can take your pick of these two extremes.

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2

Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 17: Data Mining: Clustering and Market Basket Analysis

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 17: DATA MINING: CLUSTERING AND MARKET BASKET ANALYSIS

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Clustering algorithms always start with a distance measure. Why it is not always obvious how to develop such a measure?

Answer: The problem is that a given observation, that is, a given row of data, might contain continuous numeric measurements, counts, and even text data like Democrat, Republican, or Independent. The challenge is to compare any two such rows for dissimilarity. It’s a bit like comparing apples and oranges. For example, if two people have the same height, weight, and age, but one is Democrat and the other is Republican, are they more or less dissimilar than two Democrats, where one is young, tall, and thin, and the other is older, shorter, and heavier? Careful weighting is required so no characteristic dominates or is dominated by the others.

C.2.

The typical implementation of K-Means is iterative, as in Example 17.2. Explain what this means.

Answer: The typical K-means algorithm starts with some randomly chosen cluster centers and assigns each member to the nearest cluster center. It then determines new cluster centers and repeats, and it keeps repeating until the new cluster centers are practically the same as the previous ones.

C.3.

Cluster analysis tries to define clusters according to two criteria: the members of a cluster should be close to another and the member of different clusters should be distant from one another. How might you measure these two criteria? Be specific.

Answer: One measure of the closeness of members to their cluster center is the average of their distances from it. One measure of the distinctness of clusters is the average of the distances from the cluster centers to one another. Still, different implementations do it in slightly different ways.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 17: Data Mining: Clustering and Market Basket Analysis

C.4.

Given a dendrogram of customers as in Figure 17.19, how might you assign customers to clusters?

Answer: The lines in a dendrogram indicate which members are paired with one another. If you draw a horizontal line in the graph (say, at about 15 or 20 in Figure 17.19), the number of clusters is the number of vertical lines this line crosses. Then, you can look down to see which members are in each cluster.

C.5.

We said in section 17.3 that when the antecedent X and the consequent Y are positively correlated—that is, they tend to be purchased together—P(X and Y) is greater than the product of P(X) and P(Y). Argue why this is true. (Recall Equation 6.4 from Chapter 6.)

Answer: From Chapter 6, P(X and Y) equals the product of P(X) and P(Y|X), so P(X and Y) is greater than the product of P(X) and P(Y), and P(Y|X) must be greater than P(Y). In other words, Y is more likely, given the knowledge that X has occurred, than without this knowledge. In this sense, X and Y are “correlated.”

C.6.

When examining an association rule such as {hot dogs, buns} → {mustard}, explain exactly what support, confidence, and lift tell you, and explain why they are useful.

Answer: The support is the fraction of all baskets that contain all three items, so a large support indicates that the rule applies fairly often. The confidence is the fraction of all baskets containing hot dogs and buns that also contain mustard, so a large confidence indicates that mustard does tend to be purchased when hot dogs and buns are purchased. A large lift (greater than 1) means that the two sets {hot dogs, buns} and {mustard} are correlated in a probabilistic sense.

C.7.

When, if ever, is support for an association rule 0? When, if ever, is it 1? When, if ever, is the confidence for a rule 0? When, if ever, is it 1?

Answer: Support for a rule is 0 only when the antecedent is in none of the baskets. It is 1 only when the antecedent is in all of the baskets. The confidence is 0 only when the consequent is in none of the baskets that contain the antecedent. It is 1 only when the consequent is in all of the baskets that contain the antecedent.

C.8.

Is there a lower bound for the lift of an association rule? If so, when would it occur? Is there an upper bound? If so, when would it occur?

Answer: The lowest lift can be is 0. This happens when either P(X) or P(Y) or both is 0, that is, when no baskets contain the antecedent or no baskets contain the consequent.

From C.5 above, lift is P(Y|X)/P(Y). In the extreme, imagine that Y is very rare, but Y is in every basket that contains X. Then, lift will be very large, and there is no limit on how large it can be.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 18: Analysis of Variance and Experimental Design

Solution and Answer Guide

BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 18: ANALYSIS OF VARIANCE AND EXPERIMENTAL DESIGN

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

ANOVA is always a test of the equality of means. So why is the method called analysis of variance?

Answer: Figures 18.1 and 18.2 answer this question best. Yes, ANOVA is all about testing for equal means, but the results of the test depend heavily on the variances of the various subpopulations. If these variances are large, as in Figure 18.1, it will be difficult to reject the null hypothesis of equal means because the large variances make it hard to estimate the means accurately. But if the variances are small, as in Figure 18.2, the means can be estimated much more accurately, which makes it much easier to reject the equal-means hypothesis (if it is indeed false).

C.2.

In ANOVA terminology, there are “factors” and “treatment levels.” Give at least two examples of possible factors and treatment levels to illustrate that you understand what these terms mean.

Answer: A factor is essentially a categorical explanatory variable, and treatment levels (or simply levels) are its categories. As an example, Gender might be a factor, with levels Male and Female. Or, Shelf Height might be a factor, with levels Low, Medium, and High.

C.3.

Explain what the key ratio, the F ratio, in a one-way ANOVA table is all about and why it is the basis for a test of equal means.

Answer: The F ratio is MSB/MSW, or “mean square between” divided by “mean square within.” The numerator measures differences across the means of the various treatment levels, while the denominator measures variability within the treatment levels. For example, the numerators in Figures 18.1 and 18.2 are identical because the means in both figures are identical. But the denominator in Figure 18.1 is much larger—more variability in each box plot—than in Figure 18.2. Therefore, it makes intuitive sense that a large F ratio, where MSB is large relative to MSW, leads to rejection of the equal-means hypothesis.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 18: Analysis of Variance and Experimental Design

C.4.

Explain why there are several choices (in StatTools and other packages) for the type of confidence intervals in one-way ANOVA. Specifically, what problem do these variations attempt to solve?

Answer: This has to do with the multiple comparison problem discussed in Section 18-4. If you request only the “standard” confidence intervals at the 95% level, say, each will give you 95% confidence that the true mean difference for this particular pair of means is within the confidence limits. But the probability that all of the many mean differences will be within their respective confidence limits will be less than 95%, maybe considerably less than 95%. The other types of confidence intervals adjust for this (in slightly different ways) by expanding each confidence interval to some extent.

The goal is to have a set of confidence intervals where there is 95% confidence that they all contain their respective mean differences.

C.5.

Explain exactly how regression can be used as an alternative to the “standard” one-way ANOVA methodology. Which p-values from the two methods are guaranteed to be the same?

Answer: In the regression approach, you regress the dependent variable against dummies for the levels of the single factor (leaving one dummy out as usual). Each coefficient is then an estimate of the mean difference between its category and the left-out reference category. The procedures are equivalent in that the F value in the regression ANOVA table is the same as the F value in the one-way ANOVA table.

Therefore, their p-values are also the same, and each indicates whether you can reject the equal-means hypothesis at a given significance level.

C.6.

In academic papers that are based on data analysis, you often see a correlation matrix from the observed data with asterisks indicating the “significant” correlations (those that are significantly different from 0). How does Section 18-4 on the multiple comparison problem relate to this situation? (By the way, academic researchers often offer no evidence that they are even aware of the multiple comparison problem. Maybe they are, but who knows?)

Answer: These papers are really running multiple hypothesis tests, one for each correlation. For example, if the correlation table is 5×5, it is running 10 separate hypothesis tests. By quoting that any of these is significant at the 5% level, the paper is claiming that there is a 5% chance of committing a type I error in that correlation’s test. But then, just as in the multiple comparison problem, the probability that there will be at least one type I error for all of the tests is considerably greater than 5%. We usually have no way of knowing whether the authors adjusted for this, but they should.

C.7.

What is the difference between a full factorial design and an incomplete factorial design? If the former is “better,” why would you ever use the latter?

Answer: Consider a 3-factor design where the 3 factors have 4 levels, 5 levels, and 3 levels, respectively. A full factor design would require some number n of observations for each of the 4×5×3 = 60 combinations of the factors. This is the recommended approach—if it is feasible physically and economically. But sometimes, the number of factors and treatment levels is too large to allow a full-factor design. In this case, only some of the factor level combinations are used. Surprisingly, when this is done

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 18: Analysis of Variance and Experimental Design

carefully (as explained in detail in experimental design textbooks), the results can be nearly as good as if a full-factorial design had been used.

C.8.

A study is performed on a sample of residential homes to discover whether the size of the monthly heating bill depends on the type of heat or the type of home. In particular, three types of heat are examined: electric, natural gas, and oil. Also, all homes are classified into two types: those on a single level and those with at least two stories.

What might an interaction effect look like in this situation? Intuitively, do you think there is any reason to expect an interaction effect?

Answer: There will be an interaction effect only if the effect of heat type on the monthly bill depends in some way on the number of levels of the home. For example, heating bills might be higher, on average, in multi-story homes than in single-story homes (a main effect), but this difference might be even larger when electricity is used than when natural gas or oil are used. This is just one form of interaction; there are several other possibilities. A researcher would probably assume that no interaction effects are present here, but the two-way ANOVA procedure can easily test for them.

C.9.

Chapter 8 discussed paired comparisons as one possibility when analyzing the difference between two means. What does this have to do with blocking, as discussed in Section 18-6b?

Answer: Recall the discussion of paired comparison procedures in Chapters 8 and 9.

The purpose of designing an experiment with paired observations, such as before and after test on a single person, or husbands and wives viewing the same sales presentation, is that it eliminates some sampling error in the estimate of the difference between means. In other words, if pairing in a natural way is possible, it is a good idea because it increases the power of the test. The same goes exactly for blocking. By blocking on an “extraneous” factor (like store in Example 18.5), you are “controlling for” this factor, and you therefore get a more powerful test on the factor of interest (in this case, the type of soap dispenser). So if a “natural” blocking variable can be identified, it is probably worth using in a block design.

C.10. Part of the title of this chapter is “experimental design.” Why is the design of an experiment so important? What is the main disadvantage of an experiment that is not properly designed?

Answer: In general, you can think of hypothesis testing (or any inference method) as trying to detect a signal in the presence of noise. For example, in comparing several population means, the signal consists of the true mean values, and the noise consists of variability within the populations. Experimental design is the study of creating experiments to minimize the noise. This can be done in several ways, such as suitably large sample sizes, randomization, blocking, or “good” incomplete designs. It is a huge topic, and whole (thick) textbooks have been devoted to it.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 19: Statistical Process Control

Solution and Answer Guide BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING, 8E 2025, 9780357984581; CHAPTER 19: STATISTICAL PROCESS CONTROL

TABLE OF CONTENTS Answers to Conceptual Questions .....................................................................................1

ANSWERS TO CONCEPTUAL QUESTIONS Note to Instructors: Student answers will vary. The responses here are intended to provide general guidance in terms of concepts that could be discussed.

C.1.

Why is a process that is “in control” not necessarily a “good” process? What does it really mean for a process to be in control?

Answer: As Deming hammered home in his teaching, a process out of control is unpredictable. So a process in control simply means that it is predictable, subject to common cause variation. However, this doesn’t imply that it’s any good. For example, you might be able to accurately predict that 20 out of every 100 parts will be defective, but this is almost surely an excessively high rate of defectives.

C.2.

Is the level of common-cause variation something that a company is necessarily stuck with? Why or why not?

Answer: Common cause variability is the variability observed in a process that is in control. But as in the previous answer, “in control” doesn’t mean “good.” If the level of common cause variability is too high, a manager should (and usually can) change the process in some way to reduce the level of variability. Then, this “better” process should be monitored to ensure that it stays in control.

C.3.

The range of a small set of measurements has traditionally been used as a measure of variability for X and R charts. Why isn’t standard deviation used instead? And why are two charts necessary, an X chart and an R chart?

Answer: The range is used, rather than the standard deviation, for the simple reason that it is easier for people on the shop floor to calculate. This is undoubtedly less of an issue in many of today’s situations where calculations are performed with dedicated software. The reason for two charts, X and R, is that the latter checks for changes in variability over time, while the former checks for changes in mean level over time.

Either of these could signal an out-of-control condition.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 19: Statistical Process Control

C.4.

In an X chart, is the center line where the company wants the average of its measurements to be? Are the lower and upper control limits the bounds beyond which the produced parts fail to meet specifications?

Answer: The center line and the control limits are simply what the data are telling us.

The center line indicates the average of the observations, and the control limits show us how low or high the averages might be. These are not necessarily related to what we want the process to be doing. For example, the center line might be at 5 inches in diameter, even though a “good” process would average 3 inches in diameter. Again, this gets back to the answer in #1 above: a predictable process is not necessarily a good process.

C.5.

If a company estimates the average run length (ARL) to be 430, explain what this really means.

Answer: The average run length is the expected time (i.e., number of subsamples) until an out-of-control condition will appear in the control chart, even if the process remains in control. The idea is that an extreme observation will occur sooner or later, just by luck.

C.6.

If you apply the ideas of hypothesis testing to control charts, what does a type II error represent?

Answer: The null hypothesis is that the process is in control, and, as always, a type II error means not rejecting the null when you should. In this case, it means not spotting that the process has gone out of control when, in fact, it has.

C.7.

When creating X and R charts, it is customary to measure a few observations at nearly the same time, such as 5 parts every 15 minutes. Each set of 5 parts is called a subsample. Why is it done this way? Why not just observe 1 part every 15 minutes?

Answer: The reason for taking the observations in a subsample all at the same time is that this ensures nothing has changed in the process during these observations. The reason for taking several observations, not just one, is that several observations provide an estimate of process variability at that point in time. With only one observation, you wouldn’t have any idea about process variability at that time.

C.8.

What was the point of Deming’s funnel experiment? What was the point of his red bead experiment?

Answer: Deming’s funnel experiment illustrates, in a very concrete way, that when you tamper with a process that’s already in control, you only tend to make it worse by increasing its variability. The purpose of the red bead experiment is to show, again in a very concrete way, that “employee of the week” awards are typically based totally on short-term chance, not on any employee really being better in the long run.

C.9.

How do control charts for variables ( X and R charts) differ from control charts for attributes (p charts)? More specifically, why are they relevant for different situations?

Answer: Control charts for variables are relevant when you have measurements, such as the width of a rod. Control charts for attributes are relevant when each outcome is

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Solution and Answer Guide: Business Analytics: Data Analysis and Decision Making, 8e 2025, 9780357984581; Chapter 19: Statistical Process Control

categorical: good or bad (or non-defective or defective). In the first, you are interested in averages and measures of variability. In the second, you are interested in proportions.

C.10. How are control limits in typical control charts different from specification limits in process capability analysis?

Answer: See the answer to question #4 above for more on this. In short, the control limits indicate how the process is behaving. The specification limits indicate how you want it to behave.

C.11.

Why is it virtually impossible to measure whether a process is capable (in the sense of having a high probability of being within specification limits) if it isn’t in control?

Answer: Again, being out of control implies a lack of predictability. So if a process isn’t predictable, you can’t really measure whether it is behaving the way you want it to behave (in terms of specification limits).

C.12. In the formula for the Cp index, why does the term 6σ s in the denominator imply that this index equals 1 for a “3-sigma” process and equals 2 for a “6-sigma” process? You can assume that the mean of the process is midway between LSL and USL. Answer: The numerator for Cp is the difference between the specification limits, USL – LSL. By definition, a k-sigma process is one in which the distance from the mean to each specification limit is kσ . (This assumes that each specification limit is the same distance from the process mean, as we’ve assumed here.) Therefore, if k equals 3, the numerator and denominator are both 6σ , and Cp equals 1. If k equals 6, then the numerator is twice as large as the denominator, and Cp equals 2.

© 2025 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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