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Applied Statistics and Multivariate Data Analysis for Business and Economics: A Modern Approach Using SPSS, Stata, and Excel Thomas Cleff
Engaging and accessible to students from a wide variety of mathematical backgrounds, Statistics Using Stata combines the teaching of statistical concepts with the acquisition of the popular Stata software package. It closely aligns Stata commands with numerous examples based on real data, enabling students to develop a deep understanding of statistics in a way that reflects statistical practice. Capitalizing on the fact that Stata has both a menu-driven “point and click” and program syntax interface, the text guides students effectively from the comfortable “point and click” environment to the beginnings of statistical programming. Its comprehensive coverage of essential topics gives instructors flexibility in curriculum planning and provides students with more advanced material to prepare them for future work. Online resources – including complete solutions to exercises, PowerPoint slides, and Stata syntax (Do-files) for each chapter – allow students to review independently and adapt codes to solve new problems, reinforcing their programming skills.
Sharon Lawner Weinberg is Professor of Applied Statistics and Psychology and former Vice Provost for Faculty Affairs at New York University. She has authored numerous articles, books, and reports on statistical methods, statistical education, and evaluation, as well as in applied disciplines, such as psychology, education, and health. She is the recipient of several major grants, including a recent grant from the Sloan Foundation to support her current work with NYU colleagues to evaluate the New York City Gifted and Talented programs.
Sarah Knapp Abramowitz is Professor of Mathematics and Computer Science at Drew University, where she is also Department Chair and Coordinator of Statistics Instruction. She is the coauthor, with Sharon Lawner Weinberg, of
Statistics Using IBM SPSS: An Integrative Approach, Third Edition (Cambridge University Press) and an Associate Editor of the Journal of Statistics Education.
STATISTICS USING STATA
An Integrative Approach
University
Sharon Lawner Weinberg New York
Sarah Knapp Abramowitz Drew University
32 Avenue of the Americas, New York NY 10013
Cambridge University Press is part of the University of Cambridge
It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence
www.cambridge.org
Information on this title: www.cambridge.org/9781107461185
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published 2016
Printed in the United States of America by Sheridan Books, Inc.
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloguing in Publication Data
Names: Weinberg, Sharon L | Abramowitz, Sarah Knapp, 1967–
Title: Statistics using stata: an integrative approach / Sharon Lawner Weinberg, New York University,
Sarah Knapp Abramowitz, Drew University
Description: New York NY: Cambridge University Press, 2016 | Includes index
Identifiers: LCCN 2016007888 | ISBN 9781107461185 (pbk )
Subjects: LCSH: Mathematical statistics – Data processing. | Stata.
LC record available at http://lccn.loc.gov/2016007888
ISBN 978-1-107-46118-5
Additional resources for this publication are at www cambridge org/Stats-Stata
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
To our families
Contents
Preface
Acknowledgments
1 Introduction
The Role of the Computer in Data Analysis
Statistics: Descriptive and Inferential
Variables and Constants
The Measurement of Variables
Discrete and Continuous Variables
Setting a Context with Real Data
Exercises
2 Examining Univariate Distributions
Counting the Occurrence of Data Values
When Variables Are Measured at the Nominal Level
Frequency and Percent Distribution Tables
Bar Charts
Pie Charts
When Variables Are Measured at the Ordinal, Interval, or Ratio Level
Frequency and Percent Distribution Tables
Stem-and-Leaf Displays
Histograms
Line Graphs
Describing the Shape of a Distribution
Accumulating Data
Cumulative Percent Distributions
Ogive Curves
Percentile Ranks
Percentiles
Five-Number Summaries and Boxplots
Modifying the Appearance of Graphs
Summary of Graphical Selection
Summary of Stata Commands in Chapter 2
Exercises
3 Measures of Location, Spread, and Skewness
Characterizing the Location of a Distribution
The Mode
The Median
The Arithmetic Mean
Interpreting the Mean of a Dichotomous Variable
The Weighted Mean
Comparing the Mode, Median, and Mean
Characterizing the Spread of a Distribution
The Range and Interquartile Range
The Variance
The Standard Deviation
Characterizing the Skewness of a Distribution
Selecting Measures of Location and Spread
Applying What We Have Learned
Summary of Stata Commands in Chapter 3
Helpful Hints When Using Stata
Online Resources
The Stata Command
Stata TIPS
Exercises
4 Reexpressing Variables
Linear and Nonlinear Transformations
Linear Transformations: Addition, Subtraction, Multiplication, and Division
The Effect on the Shape of a Distribution
The Effect on Summary Statistics of a Distribution
Common Linear Transformations
Standard Scores
z-Scores
Using z-Scores to Detect Outliers
Using z-Scores to Compare Scores in Different Distributions
Relating z-Scores to Percentile Ranks
Nonlinear Transformations: Square Roots and Logarithms
Nonlinear Transformations: Ranking Variables
Other Transformations: Recoding and Combining Variables
Recoding Variables
Combining Variables
Data Management Fundamentals – the Do-File
Summary of Stata Commands in Chapter 4
Exercises
5 Exploring Relationships between Two Variables
When Both Variables Are at Least Interval-Leveled Scatterplots
The Pearson Product Moment Correlation Coefficient
Interpreting the Pearson Correlation Coefficient
Judging the Strength of the Linear Relationship, The Correlation Scale Itself Is Ordinal, Correlation Does Not Imply Causation,
The Effect of Linear Transformations, Restriction of Range,
The Shape of the Underlying Distributions, The Reliability of the Data,
When at Least One Variable Is Ordinal and the Other Is at Least Ordinal: The Spearman Rank Correlation Coefficient
When at Least One Variable Is Dichotomous: Other Special Cases of the Pearson Correlation Coefficient
The Point Biserial Correlation Coefficient: The Case of One at Least Interval and One Dichotomous Variable
The Phi Coefficient: The Case of Two Dichotomous Variables
Other Visual Displays of Bivariate Relationships
Selection of Appropriate Statistic/Graph to Summarize a Relationship
Summary of Stata Commands in Chapter 5
Exercises
6 Simple Linear Regression
The “Best-Fitting” Linear Equation
The Accuracy of Prediction Using the Linear Regression Model
The Standardized Regression Equation
R as a Measure of the Overall Fit of the Linear Regression Model
Simple Linear Regression When the Independent Variable Is Dichotomous
Using r and R as Measures of Effect Size
Emphasizing the Importance of the Scatterplot
Summary of Stata Commands in Chapter 6
Exercises
7 Probability Fundamentals
The Discrete Case
The Complement Rule of Probability
The Additive Rules of Probability
First Additive Rule of Probability
Second Additive Rule of Probability
The Multiplicative Rule of Probability
The Relationship between Independence and Mutual Exclusivity
Conditional Probability
The Law of Large Numbers
Exercises
8 Theoretical Probability Models
The Binomial Probability Model and Distribution
The Applicability of the Binomial Probability Model
The Normal Probability Model and Distribution
Using the Normal Distribution to Approximate the Binomial Distribution
Summary of Chapter 8 Stata Commands
Exercises
9 The Role of Sampling in Inferential Statistics
Samples and Populations
Random Samples
Obtaining a Simple Random Sample
Sampling with and without Replacement
Sampling Distributions
Describing the Sampling Distribution of Means Empirically
Describing the Sampling Distribution of Means Theoretically: The Central Limit Theorem
Central Limit Theorem (CLT)
Estimators and BIAS
Summary of Chapter 9 Stata Commands
Exercises
10 Inferences Involving the Mean of a Single Population When σ Is Known
Estimating the Population Mean, µ, When the Population Standard Deviation, σ, Is Known
Interval Estimation
Relating the Length of a Confidence Interval, the Level of Confidence, and the Sample Size
Hypothesis Testing
The Relationship between Hypothesis Testing and Interval Estimation
Effect Size
Type II Error and the Concept of Power
Increasing the Level of Significance, α
Increasing the Effect Size, δ
Decreasing the Standard Error of the Mean,
Closing Remarks
Summary of Chapter 10 Stata Commands
Exercises
11 Inferences Involving the Mean When σ Is Not Known: One- and Two-Sample Designs
Single Sample Designs When the Parameter of Interest Is the Mean and σ Is Not Known
The t Distribution
Degrees of Freedom for the One Sample t-Test
Violating the Assumption of a Normally Distributed Parent
Population in the One Sample t-Test
Confidence Intervals for the One Sample t-Test
Hypothesis Tests: The One Sample t-Test
Effect Size for the One Sample t-Test
Two Sample Designs When the Parameter of Interest Is µ, and σ Is Not Known
Independent (or Unrelated) and Dependent (or Related) Samples
Independent Samples t-Test and Confidence Interval
The Assumptions of the Independent Samples t-Test
Effect Size for the Independent Samples t-Test
Paired Samples t-Test and Confidence Interval
The Assumptions of the Paired Samples t-Test
Effect Size for the Paired Samples t-Test
The Bootstrap
Summary
Summary of Chapter 11 Stata Commands
Exercises
12 Research Design: Introduction and Overview
Questions and Their Link to Descriptive, Relational, and Causal
Research Studies
The Need for a Good Measure of Our Construct, Weight
The Descriptive Study
From Descriptive to Relational Studies
From Relational to Causal Studies
The Gold Standard of Causal Studies: The True Experiment and Random Assignment
Comparing Two Kidney Stone Treatments Using a Non-randomized
Controlled Study
Including Blocking in a Research Design
Underscoring the Importance of Having a True Control Group Using Randomization
Analytic Methods for Bolstering Claims of Causality from Observational Data (Optional Reading)
Quasi-Experimental Designs
Threats to the Internal Validity of a Quasi-Experimental Design
Threats to the External Validity of a Quasi-Experimental Design
Threats to the Validity of a Study: Some Clarifications and Caveats
Threats to the Validity of a Study: Some Examples
Exercises
13 One-Way Analysis of Variance
The Disadvantage of Multiple t-Tests
The One-Way Analysis of Variance
A Graphical Illustration of the Role of Variance in Tests on Means
ANOVA as an Extension of the Independent Samples t-Test
Developing an Index of Separation for the Analysis of Variance
Carrying Out the ANOVA Computation
The Between Group Variance (MSB)
The Within Group Variance (MSW)
The Assumptions of the One-Way ANOVA
Testing the Equality of Population Means: The F-Ratio
How to Read the Tables and Use Stata Functions for the F-
Distribution
ANOVA Summary Table
Measuring the Effect Size
Post-Hoc Multiple Comparison Tests
The Bonferroni Adjustment: Testing Planned Comparisons
The Bonferroni Tests on Multiple Measures
Summary of Stata Commands in Chapter 13
Exercises
14 Two-Way Analysis of Variance
The Two-Factor Design
The Concept of Interaction
The Hypotheses That Are Tested by a Two-Way Analysis of Variance
Assumptions of the Two-Way Analysis of Variance
Balanced versus Unbalanced Factorial Designs
Partitioning the Total Sum of Squares
Using the F-Ratio to Test the Effects in Two-Way ANOVA
Carrying Out the Two-Way ANOVA Computation by Hand
Decomposing Score Deviations about the Grand Mean
Modeling Each Score as a Sum of Component Parts
Explaining the Interaction as a Joint (or Multiplicative) Effect
Measuring Effect Size
Fixed versus Random Factors
Post-hoc Multiple Comparison Tests
Summary of Steps to be Taken in a Two-Way ANOVA Procedure
Summary of Stata Commands in Chapter 14
Exercises
15 Correlation and Simple Regression as Inferential Techniques
The Bivariate Normal Distribution
Testing Whether the Population Pearson Product Moment Correlation
Equals Zero
Using a Confidence Interval to Estimate the Size of the Population
Correlation Coefficient, ρ
Revisiting Simple Linear Regression for Prediction
Estimating the Population Standard Error of Prediction, σY|X
Testing the b-Weight for Statistical Significance
Explaining Simple Regression Using an Analysis of Variance Framework
Measuring the Fit of the Overall Regression Equation: Using R and R2
Relating R2 to σ2 Y|X
Testing R2 for Statistical Significance
Estimating the True Population R2: The Adjusted R2
Exploring the Goodness of Fit of the Regression Equation: Using Regression Diagnostics
Residual Plots: Evaluating the Assumptions Underlying Regression
Detecting Influential Observations: Discrepancy and Leverage
Using Stata to Obtain Leverage
Using Stata to Obtain Discrepancy
Using Stata to Obtain Influence
Using Diagnostics to Evaluate the Ice Cream Sales Example
Using the Prediction Model to Predict Ice Cream Sales
Simple Regression When the Predictor is Dichotomous
Summary of Stata Commands in Chapter 15 Exercises
16 An Introduction to Multiple Regression
The Basic Equation with Two Predictors
Equations for b, β, and RY12 When the Predictors Are Not Correlated
Equations for b, β, and RY.12 When the Predictors Are Correlated
Summarizing and Expanding on Some Important Principles of Multiple Regression
Testing the b-Weights for Statistical Significance
Assessing the Relative Importance of the Independent Variables in the Equation
Measuring the Drop in R2 Directly: An Alternative to the Squared Semipartial Correlation
Evaluating the Statistical Significance of the Change in R2
The b-Weight as a Partial Slope in Multiple Regression
Multiple Regression When One of the Two Independent Variables is Dichotomous
The Concept of Interaction between Two Variables That Are at Least Interval-Leveled
Testing the Statistical Significance of an Interaction Using Stata
Centering First-Order Effects to Achieve Meaningful Interpretations of b-Weights
Understanding the Nature of a Statistically Significant Two-Way Interaction
Interaction When One of the Independent Variables Is Dichotomous and the Other Is Continuous
Summary of Stata Commands in Chapter 16
Exercises
17 Nonparametric Methods
Parametric versus Nonparametric Methods
Nonparametric Methods When the Dependent Variable Is at the Nominal Level
The Chi-Square Distribution (χ2)
The Chi-Square Goodness-of-Fit Test
The Chi-Square Test of Independence
Assumptions of the Chi-Square Test of Independence
Fisher’s Exact Test
Calculating the Fisher Exact Test by Hand Using the Hypergeometric Distribution
Nonparametric Methods When the Dependent Variable Is OrdinalLeveled
Wilcoxon Sign Test
The Mann-Whitney U Test
The Kruskal-Wallis Analysis of Variance
Summary of Stata Commands in Chapter 17
Exercises
Appendix A Data Set Descriptions
Appendix B Stata .do Files and Data Sets in Stata Format
Appendix C Statistical Tables
Appendix D References
Appendix E Solutions
Index
Preface
This text capitalizes on the widespread availability of software packages to create a course of study that links good statistical practice to the analysis of real data, and the many years of the authors’ experience teaching statistics to undergraduate students at a liberal arts university and to graduate students at a large research university from a variety of disciplines including education, psychology, health, and policy analysis. Because of its versatility and power, our software package of choice for this text is the popularly used Stata, which provides both a menu-driven and command line approach to the analysis of data, and, in so doing, facilitates the transition to a more advanced course of study in statistics. Although the choice of software is different, the content and general organization of the text derive from its sister text, now in its third edition, Statistics Using IBM SPSS: An Integrative Approach, and as such, this text also embraces and is motivated by several important guiding principles found in the sister text.
First, and perhaps most important, we believe that a good data analytic plan must serve to uncover the story behind the numbers, what the data tell us about the phenomenon under study. To begin, a good data analyst must know his/her data well and have confidence that it satisfies the underlying assumptions of the statistical methods used. Accordingly, we emphasize the usefulness of diagnostics in both graphical and statistical form to expose anomalous cases, which might unduly influence results, and to help in the selection of appropriate assumption-satisfying transformations so that ultimately we may have
confidence in our findings. We also emphasize the importance of using more than one method of analysis to answer fully the question posed and understanding potential bias in the estimation of population parameters.
Second, because we believe that data are central to the study of good statistical practice, the textbook’s website contains several data sets used throughout the text. Two are large sets of real data that we make repeated use of in both worked-out examples and end-of-chapter exercises. One data set contains forty-eight variables and five hundred cases from the education discipline; the other contains forty-nine variables and nearly forty-five hundred cases from the health discipline. By posing interesting questions about variables in these large, real data sets (e.g., Is there a gender difference in eighth graders’ expected income at age thirty?), we are able to employ a more meaningful and contextual approach to the introduction of statistical methods and to engage students more actively in the learning process. The repeated use of these data sets also contributes to creating a more cohesive presentation of statistics; one that links different methods of analysis to each other and avoids the perception that statistics is an often-confusing array of so many separate and distinct methods of analysis, with no bearing or relationship to one another.
Third, to facilitate the analysis of these data, and to provide students with a platform for actively engaging in the learning process associated with what it means to be a good researcher and data analyst, we have incorporated the latest version of Stata (version 14), a popular statistical software package, into the presentation of statistical material using a highly integrative approach that reflects practice. Students learn Stata along with each new statistical method covered, thereby allowing them to apply their newly-learned knowledge to the real world of applications. In addition to demonstrating the use of Stata within each chapter, all chapters have an associated .do-file, designed to allow students not only to replicate all worked out examples within a chapter but also to
reproduce the figures embedded in a chapter, and to create their own.do-files by extracting and modifying commands from them. Emphasizing data workflow management throughout the text using the Stata.do-file allows students to begin to appreciate one of the key ingredients to being a good researcher. Of course, another key ingredient to being a good researcher is content knowledge, and toward that end, we have included in the text a more comprehensive coverage of essential topics in statistics not covered by other textbooks at the introductory level, including robust methods of estimation based on resampling using the bootstrap, regression to the mean, the weighted mean, and potential sources of bias in the estimation of population parameters based on the analysis of data from quasi-experimental designs, Simpson’s Paradox, counterfactuals, other issues related to research design.
Fourth, in accordance with our belief that the result of a null hypothesis test (to determine whether an effect is real or merely apparent) is only a means to an end (to determine whether the effect is important or useful) rather than an end in itself, we stress the need to evaluate the magnitude of an effect if it is deemed to be real, and of drawing clear distinctions between statistically significant and substantively significant results. Toward this end, we introduce the computation of standardized measures of effect size as common practice following a statistically significant result. Although we provide guidelines for evaluating, in general, the magnitude of an effect, we encourage readers to think more subjectively about the magnitude of an effect, bringing into the evaluation their own knowledge and expertise in a particular area.
Finally, we believe that a key ingredient of an introductory statistics text is a lively, clear, conceptual, yet rigorous approach. We emphasize conceptual understanding through an exploration of both the mathematical principles underlying statistical methods and real world applications. We use an easygoing, informal style of writing that we have found gives readers the impression
that they are involved in a personal conversation with the authors. And we sequence concepts with concern for student readiness, reintroducing topics in a spiraling manner to provide reinforcement and promote the transfer of learning.
Another distinctive feature of this text is the inclusion of a large bibliography of references to relevant books and journal articles, and many endof-chapter exercises with detailed answers on the textbook’s website. Along with the earlier topics mentioned, the inclusion of linear and nonlinear transformations, diagnostic tools for the analysis of model fit, tests of inference, an in-depth discussion of interaction and its interpretation in both two-way analysis of variance and multiple regression, and nonparametric statistics, the text provides a highly comprehensive coverage of essential topics in introductory statistics, and, in so doing, gives instructors flexibility in curriculum planning and provides students with more advanced material for future work in statistics.
The book, consisting of seventeen chapters, is intended for use in a one- or two-semester introductory applied statistics course for the behavioral, social, or health sciences at either the graduate or undergraduate level, or as a reference text as well. It is not intended for readers who wish to acquire a more theoretical understanding of mathematical statistics. To offer another perspective, the book may be described as one that begins with modern approaches to Exploratory Data Analysis (EDA) and descriptive statistics, and then covers material similar to what is found in an introductory mathematical statistics text, designed, for example, for undergraduates in math and the physical sciences, but stripped of calculus and linear algebra and grounded instead in data examples. Thus, theoretical probability distributions, The Law of Large Numbers, sampling distributions and The Central Limit Theorem are all covered, but in the context of solving practical and interesting problems.
Acknowledgments
This book has benefited from the many helpful comments of our New York University and Drew University students, too numerous to mention by name, and from the insights and suggestions of several colleagues. For their help, we would like to thank (in alphabetical order) Chris Apelian, Ellie Buteau, Jennifer Hill, Michael Karchmer, Steve Kass, Jon Kettenring, Linda Lesniak, Kathleen Madden, Isaac Maddow-Zimet, Meghan McCormick, Joel Middleton, and Marc Scott. Of course, any errors or shortcomings in the text remain the responsibility of the authors.
Chapter One Introduction ◈
Welcome to the study of statistics! It has been our experience that many students face the prospect of taking a course in statistics with a great deal of anxiety, apprehension, and even dread. They fear not having the extensive mathematical background that they assume is required, and they fear that the contents of such a course will be irrelevant to their work in their fields of concentration.
Although it is true that an extensive mathematical background is required at more advanced levels of statistical study, it is not required for the introductory level of statistics presented in this book. Greater reliance is placed on the use of the computer for calculation and graphical display so that we may focus on issues of conceptual understanding and interpretation. Although hand computation is deemphasized, we believe, nonetheless, that a basic mathematical background – including the understanding of fractions, decimals, percentages, signed (positive and negative) numbers, exponents, linear equations, and graphs – is essential for an enhanced conceptual understanding of the material.
As for the issue of relevance, we have found that students better comprehend the power and purpose of statistics when it is presented in the context of a substantive problem with real data. In this information age, data are available on a myriad of topics. Whether our interests are in health, education,
psychology, business, the environment, and so on, numerical data may be accessed readily to address our questions of interest. The purpose of statistics is to allow us to analyze these data to extract the information that they contain in a meaningful way and to write a story about the numbers that is both compelling and accurate.
Throughout this course of study we make use of a series of real data sets that are located in the website for this book and may be accessed by clicking on the Resources tab using the URL, www.cambridge.org/Stats-Stata. We will pose relevant questions and learn the appropriate methods of analysis for answering such questions. Students will learn that more than one statistic or method of analysis typically is needed to address a question fully. Students also will learn that a detailed description of the data, including possible anomalies, and an ample characterization of results, are critical components of any data analytic plan. Through this process we hope that this book will help students come to view statistics as an integrated set of data analytic tools that when used together in a meaningful way will serve to uncover the story contained in the numbers.
The Role of the Computer in Data Analysis
From our own experience, we have found that the use of a statistics software package to carry out computations and create graphs not only enables a greater emphasis on conceptual understanding and interpretation but also allows students to study statistics in a way that reflects statistical practice. We have selected the latest version of Stata available to us at the time of writing, version 14, for use with this text. We have selected Stata because it is a well-established comprehensive package with a robust technical support infrastructure that is widely used by behavioral and social scientists. In addition, not only does Stata include a menu-driven “point and click” interface, making it accessible to the new user, but it also includes a command line or program syntax interface, allowing students to be guided from the comfortable “point and click” environment to the beginnings of statistical programming. Like MINITAB, JMP, Data Desk, Systat, SPSS, and SPlus, Stata is powerful enough to handle the analysis of large data sets quickly. By the end of the course, students will have obtained a conceptual understanding of statistics as well as an applied, practical skill in how to carry out statistical operations.
Statistics: Descriptive and Inferential
The subject of statistics may be divided into two general branches: descriptive and inferential. Descriptive statistics are used when the purpose of an investigation is to describe the data that have been (or will be) collected. Suppose, for example, that a third-grade elementary school teacher is interested in determining the proportion of children who are firstborn in her class of thirty children. In this case, the focus of the teacher’s question is her own class of thirty children and she will be able to collect data on all of the students about whom she would like to draw a conclusion. The data collection operation will involve noting whether each child in the classroom is firstborn or not; the statistical operations will involve counting the number who are, and dividing that number by thirty, the total number of students in the class, to obtain the proportion sought. Because the teacher is using statistical methods merely to describe the data she collected, this is an example of descriptive statistics.
Suppose, by contrast, that the teacher is interested in determining the proportion of children who are firstborn in all third-grade classes in the city where she teaches. It is highly unlikely that she will be able to (or even want to) collect the relevant data on all individuals about whom she would like to draw a conclusion. She will probably have to limit the data collection to some randomly selected smaller group and use inferential statistics to generalize to the larger group the conclusions obtained from the smaller group. Inferential statistics are used when the purpose of the research is not to describe the data that have been collected but to generalize or make inferences based on it. The smaller group on which she collects data is called the sample, whereas the larger group to whom conclusions are generalized (or inferred), is called the population. In general,
two major factors influence the teacher’s confidence that what holds true for the sample also holds true for the population at large. These two factors are the method of sample selection and the size of the sample. Only when data are collected on all individuals about whom a conclusion is to be drawn (when the sample is the population and we are therefore in the realm of descriptive statistics), can the conclusion be drawn with 100 percent certainty. Thus, one of the major goals of inferential statistics is to assess the degree of certainty of inferences when such inferences are drawn from sample data. Although this text is divided roughly into two parts, the first on descriptive statistics and the second on inferential statistics, the second part draws heavily on the first.
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The Project Gutenberg eBook of Atomic!
This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Atomic!
Author: Henry Kuttner
Illustrator: Virgil Finlay
Release date: May 25, 2022 [eBook #68167]
Language: English
Original publication: United States: Standard Magazines, Inc, 1947
Credits: Greg Weeks, Mary Meehan, Alex White & the online Distributed Proofreaders Canada team at https://www.pgdpcanada.net
ATOMIC!
By HENRY KUTTNER
Illustrated by Virgil Finlay.
[Transcriber's Note: This etext was produced from Thrilling Wonder Stories, August 1947.
Extensive research did not uncover any evidence that the U.S. copyright on this publication was renewed.]
What nuclear war may do to the world we know is a closed book to mankind—but here’s what coming eras may bring!
CHAPTER I
The Eye
The alarm went off just after midnight. The red signal showed emergency. But it was always emergency at first. We all knew that. Ever since the arachnid tribe in the Chicago Ring had mutated we’d known better than to take chances. That time the human race had very nearly gone under. Not many people knew how close we’d been to extinction. But I knew.
Everybody in Biological Control Labs knew. To anyone who lived before the Three-Hour War such things would have sounded incredible. Even to us now they sound hard to believe. But we know.
There are four hundred and three Rings scattered all over the world and every one of them is potentially deadly.
Our Lab was north of what had been Yonkers and was a deserted, ruinous wilderness now. The atomic bomb of six years ago hadn’t hit Yonkers of course. What it struck was New York. The radiation spread far enough to wipe out Yonkers and the towns beyond it, and inland as far as White Plains—but everyone who lived through the Three-Hour War knows what the bomb did in the New York area.
The war ended incredibly fast. But what lingered afterward made the real danger, the time-bomb that may quite easily lead to the wiping out of our whole civilization. We don’t know yet. All we can do is keep the Labs going and the planes out watching.
That’s the menace—the mutations.
It was familiar stuff to me. I recorded the televised report on the office ticker, punched a few buttons and turned around to look at Bob Davidson, the new hand. He’d been here for two weeks, mostly learning the ropes.
My assistant, Williams, was due for a vacation and I had about decided to take young Davidson on as a substitute.
“Want to go out and look it over, Dave?” I asked.
“Sure. That’s a red alarm, isn’t it? Emergency?”
I pulled a mike forward.
“Send up relief men,” I ordered, “and wake Williams to take over. Get the recon copter ready. Red flight.” Then I turned to Davidson.
“It’ll be routine,” I told him, “unless something unexpected happens. Not much data yet. The sky-scanners showed a cave-in and some activity around it. May be nothing but we can’t take chances. It’s Ring Seventy-Twelve.”
“That’s where the air liner crashed last week, isn’t it?” Dave asked, looking up with renewed interest. “Any dope yet on what became of the passengers?”
“Nothing. The radiations would have got them if nothing else did. That’s in the closed file now, poor devils. Still, we might spot the ship.” I stood up. “The whole thing may be a wild-goose chase but we never take any chances with the Rings.”
“It ought to be interesting, anyhow,” Dave said and followed me out.
We could see it from a long way off. Four hundred and three of them dot the world now, but in the days before the War no one could have imagined such a thing as a Ring and it would be hard to make anyone visualize one through bare description. You have to feel the desolation as you fly over that center of bare, splashed rock in which nothing may ever grow again until the planet itself disintegrates, and see around that dead core the violently boiling life of the Ring.
It was a perimeter of life brushed by the powers of death. The sunforces unleashed by the bombs gave life, a new, strange, mutable life that changed and changed and changed and would go on changing until a balance was finally struck again on this world which for three hours reeled in space under the blows of an almost cosmic disaster. We were still shuddering beneath the aftermath of those blows. The balance was not yet.
From time to time we work them over with flame throwers
When the hour of balance comes, mankind may no longer be the dominant race. That’s why we keep such a close watch on all the Rings. From time to time we work them over with flame-throwers. Only atomic power, of course, would quiet that seething life permanently—which is no solution. We’ve got Rings enough right now without resorting to more atom bombs.
It’s a hydra-headed problem without an answer. All we can do is watch, wait, be ready....
The world was still dark. But the Ring itself was light, with a strange, pale luminous radiance that might mean anything. It was new. That was all we knew about it yet.
“Let’s have the scanner,” I said to Davidson. He handed me the mask and I pushed the head-clips past my ears and settled the monocular view-plate before my eyes, expecting to see the darkness melt into the reversed vision of the night-scanner.
It melted, all right—the part that didn’t matter. I could see the negative images of trees and ruined houses standing ghostly pale against the dark. But within the Ring—nothing.
It wasn’t good. It could be very bad indeed. In silence I pulled off the mask and handed it to Davidson, watched him look down. When he turned I could see his troubled frown through the monocular lens even before he lowered the scanner. He looked a little pale in the light of the instrument board.
“Well?” he asked.
“Looks as if they’d hit on something good this time,” I said.
“They?”
“Who knows? Could be anything this time. You know how the lifeforms shoot up into mutations without the least warning. Something’s done it again down there. Maybe something that’s been quietly working away underground for a long time, just waiting for the right moment. Whatever it is they can stop the scanners and that isn’t an easy thing to do.”
“The first boys over reported a cave-in,” Davidson said, peering futilely down. “Could you see anything?”
“Just the luminous fog. Nothing inside. Total blackout. Well, maybe daylight will show us what’s up. I hope so.”
It didn’t. A low sea of yellow-gray fog billowed slowly in a vast circle over the entire Ring as far as we could see. Dead central core and outer circle of unnatural life had vanished together into that mist which no instrument we had could penetrate—and we’ve developed
a lot of stuff for seeing through fog and darkness. This was solid. We couldn’t crack it.
“We’ll land,” I told Davidson finally “Something’s going on behind that shield, something that doesn’t want to be spied on. And somebody’s got to investigate—fast! It might as well be us.”
We wore the latest development in the way of lead-suits, flexible and easy on the body. We snapped our face-plates shut as the ground came up to meet us and the little Geiger-counter each of us carried began to tick erratically, like a sort of Morse code mechanically spelling out the death in the air we sank through.
I was measuring the ground below for a landing when Davidson grabbed my shoulder suddenly, pointing down.
“Look!” His voice came tinnily through the ear-diaphragms in my helmet. I looked.
Now this is where the story gets difficult to tell.
I know what I saw. That much was clear to me from start to finish. I saw an eye looking up through the pale mist at us. But whether it was an enormous lens far below or a normal-sized eye close to us I couldn’t have said just then. My distance-sense had stopped functioning.
I stared into the Eye....
The next thing I remember is sitting in the familiar lab office across the desk from Williams, hearing myself speaking.
“... no signs of activity anywhere in the Ring. Perfectly normal—”
“There’s that lake, of course,” Davidson interrupted in a conscientious voice. I looked at him. He was turning his cap over and over in his hands as he sat there by the wall. His pink-cheeked face was haggard and there was something strained and dazed in the glance he turned to meet mine. I knew I looked dazed too.
It was like waking out of a dream, knowing you’ve dreamed, knowing you’re awake now—but having the dream go on—being powerless to stop it. I wanted to jump up and slam my fist on the desk and shout that all this was phony.
I couldn’t.
Something like a tremendously powerful psychic inhibition held me down. The room swam before me for a moment with my effort to break free and I met Davidson’s eyes and saw the same swimming strain in them.
It wasn’t hypnosis.
We don’t win our posts in Bio Control until we’ve been through exhaustive tests and a lot of heavy training. None of us are hypnosis-prone. We can’t afford to be. It’s been tried.
We can’t be hypnotized except under very special circumstances safeguarded by Bio Control itself.
No, the answer wasn’t that easy. It seemed to lie in—myself. Some door had slammed in the center of my brain, to shut in vital information that must not escape—yet—under any circumstances at all.
The minute I hit on that analogy I knew I was on the right trail. I felt safer and surer of myself. Whatever had happened in that blank space just passed my instinct was in control now. I could trust that instinct.
“... break-through, just as the boys reported,” Davidson was saying. “That must be what started the lake pouring up. Nothing stirring there now, though. I suppose the regular sky-scanners are watching it?”
His glance crossed mine and I knew he was right. I knew he was talking to me, not Williams. Of course the lake couldn’t be hidden now that it was out in plain sight. We couldn’t make a worse mistake
than to rouse interest in ourselves and the lake by telling obvious lies about it....
What lake?
Like a mirage, swimming slowly back through my mind, the single memory came. Ourselves, standing on the raw, bare rock of the deathly Ring-center, looking through a rift of mist like a broad, low window a mile long and not very high.
The lake was incredibly blue in the dawn, incredibly calm. Beyond it a wall of cliff stretched left and right beyond our vision, a wall like a great curtain of rock hanging in majestic folds, pink in the pink dawn, looming about its perfect image reflected in the mirror of the lake.
The mirage dissolved. That much I could remember—no more. There was a lake. We had stood on its rocky shore. And then— what? Reason told me we must have seen something, or heard or learned something, that made the lake a deadly danger to mankind.
I knew that feel of naked terror deep in my mind must have a cause. But all I could do now was follow my instinct. The basic human instincts, I told myself, are self preservation and preservation of the species. If I rely on that foundation I can’t go wrong....
But—I didn’t know how long I’d been back here. I didn’t know how much I’d said, or how little—what orders I’d given to my subordinates, or whether anything in my outward aspect had roused any suspicion yet.
I looked around—and this time gave a perfectly genuine start of surprise. Except for Williams and myself the office was quite empty In this last bout with my daydreaming memory I must really have lost touch with things.
Williams was looking at me with—curiosity? Suspicion?
I rubbed my eyes, put weariness in my voice.
“I’m tired,” I said. “Almost dozed off, didn’t I? Well—”
The sound of the ticker behind Williams interrupted my alibi. I knew in a moment what was happening. A televised report had come into my own office which my secretary was switching to the ticker for me. That meant it was important. It also meant—as I had reason to hope an instant later—that the visor was shut off in my office and the news clicking directly here for our eyes alone.
Leaning over Williams’ shoulder, I read the tape feeding through. It read—
UNIDENTIFIED ACTIVITIES IN PROGRESS AROUND NEW RING LAKE. SUGGEST DESTROYERS WORK OVER AREA.
FITZGERALD.
The bottom dropped out of my stomach. Only one thing stood clear in my mind’s confusion—this must not happen. There was some terrible, some deadly danger to the whole fabric of civilization if Fitzgerald’s message reached any other eyes than ours. I had to do something, fast.
Williams was rereading the tape. He glanced up at me across his shoulder.
“Fitz is right,” he said. “Of course. Can’t let anything get started down there. Better wipe it out right now, hadn’t we?”
I said, “No!” so explosively that he froze in the act of reaching for the interoffice switch.
“Why not?” He stared at me in surprise.
I opened my mouth and closed it again hopelessly, knowing the right words wouldn’t come. To me it seemed so self-evident I couldn’t even explain why we must disregard the message. It would be like trying to tell a man why he mustn’t touch off an atom bomb out of sheer exuberance—the reasons were so many and so obvious I couldn’t choose among them.
“You weren’t there. You don’t know.” My voice sounded thick and unsteady even to me. “Fitz is wrong. Let that lake alone, Williams!”
“You ought to know.” He gave me a strange look. “Still, I’ve got to record the report. Headquarters will make the final decision.” And he reached again for the switch.
I’m not sure how far I would have gone toward stopping him. Instinct deeper than all reason seemed to explode in me in the urgent forward surge that brought me to my feet. I had to stop him—now— without delay—taking no time to delve into my mind and dredge up a reason he would accept as valid.
But the decision was taken out of our hands.
A burst of soundless white fire flashed blindingly across my eyes. It blotted out Williams, it blotted out the ticker with its innocent, deadly message. I was aware of a killing pain in the very center of my skull....
CHAPTER II
The Other Peril
Someone was shaking me.
I sat up dizzily, meeting a stare that I recognized only after what seemed infinities of slow waking. Davidson, his pink face frightened, shook me again.
“What happened? What was it? Jim, are you all right? Wake up, Jim! What was it?”
I let him help me to my feet. The room began to steady around me but it reeled sharply again when I saw what lay before the ticker, the tape looping down about him—face down on the floor, blood still crawling from the bullet hole in his back....
Williams never saw who got him. It must have been the same flash that blinded me. I felt my cheek for the powder burn that must have
scorched it as the unseen killer fired past my face. I felt only numbness. I was numb all over, even my brain. But one thing had to be settled in a hurry.
How much time had elapsed? Had that deadly message gone out while I lay here helpless? I made it to the ticker in two unsteady strides. The tape that looped the fallen Williams still bore its dangerous message.
Whoever fired past my cheek had fired for another reason, then, than this message. Of course, for how could anyone else have known its importance? There was a bewildering mystery here but I had no time to think about it.
I tore off the tape, crumpled it into my pocket. I flipped the ticker switch and sent a reverse message out as fast as my shaking hand could operate the machine.
FITZGERALD URGENT URGENT MEET ME AT RING POST 27 AM LEAVING HEADQUARTERS NOW DO NOTHING UNTIL I ARRIVE URGENT SIGNED J. OWEN.
Davidson watched me, round-eyed, as I vised for a helicopter. He put out his hand as I turned toward the door. I forced myself to stop and think.
“Well?” I said.
He didn’t speak. He only glanced at Williams’ body on the floor.
“No,” I said. “I didn’t kill him. But I might have if that had turned out to be the only way. There’s trouble at the lake.” I hesitated. “You were there too, Dave. Do you know what I mean?” I wasn’t quite sure what I was trying to find out. I waited for his answer.
“You’re the boss,” was all he said. “Still, it wasn’t any mutation that did—this. It was a bullet. You’ve got to know who shot him, Jim.”
“I don’t though. I blanked out. Something ...” My mind whirled and then steadied again with a sudden idea. I put a hand to my forehead, dizzy with trying to remember things still closed to me.
“Maybe something like a mutation had a part in it at that,” I conceded. “Maybe we’re not alone in wanting to—to keep the lake quiet. I wonder—could something from the Ring have blanked me out deliberately, so I wouldn’t see Williams killed?”
But there wasn’t time to follow even that speculation through. I said impatiently, “The point is, Dave, one man’s death doesn’t mean a thing right now. The Ring....” I stopped unable to go on. I didn’t need to.
“What do you want me to do?” Davidson asked. That was better. I knew I could depend on him, and I might need someone dependable very soon.
“Take over here,” I said. “I’m going to see Fitzgerald. And listen, Dave, this is urgent. Hold any messages Fitzgerald sends. Any! Understand?”
“Check,” he said. His eyes were still asking questions as I went out. Neither of us could answer them—yet.
The desolation spun past below me, aftermath of the Three-Hour War, ruined buildings, ruined fields, ruined woods. Far off I could catch a pale gleam of water beyond the seething edge of the Ring.
I’d been en route long enough to make some sort of order in my mind—but I hadn’t done it. Evidently more than time would be required to open the closed doors in my brain. I had been in the Ring today—I had seen something or learned something there—and whatever I learned had been of such vital and terrible import that memory of it was wiped from Davidson’s mind and mine until the hour came for action.
I didn’t know what hour or what action. But I knew with a deep certainty that when the time for decision came I would not falter. Along with the terror and the blackness in my mind went that one abiding knowledge upon which all my actions now were based. I could trust that instinct.
Fitzgerald’s copter was waiting. I could see his lead-suited figure, tiny and far below, pacing up and down impatiently as I dropped
toward him. My copter settled lightly earthward. And for a moment another thought crossed my mind.
Williams! A man murdered, a man I knew and had worked with. A man I liked. That should have affected me much more deeply than it did. I knew why it hadn’t. Williams’ death was unimportant— completely trivial in the face of the—the other peril that loomed namelessly, in all its invisible menace, like a shrouded ghost rising from the lake beyond us.
Fitzgerald was a big blond man with blue eyes and a scar puckering his forehead, souvenir of our last battle with mutated marmosa in the Atlanta Ring. His transmitter-disc vibrated tinnily as I got out of the copter.
“Hello, chief. You got my second message?”
“No. What was it?”
“More funny stuff.” He gestured toward the Ring. “In the lake this time—signs of life. I can’t make anything out of it.”
I drew a deep breath of relief. Davidson would have stopped that message. It was up to me now to find a way to keep Fitzgerald quiet.
“We’ll take a look at the lake, then,” I said. “What’s your report?”
“Well....” He shifted uneasily from one foot to the other, glancing at me through his face-plate as if he didn’t quite expect me to believe him. “It’s a funny place, that lake. I got the impression it was—well, watching me.
“I know it sounds silly but I have to tell you. It could be important, I suppose. And then when I was making a second turn over the water I saw something, in the lake.” He paused. “People,” he added after a moment.
“What kind of people?”
“I—they weren’t human.”
“How do you know?”
“They weren’t wearing lead suits,” he said simply, glad of a chance to pin his story down with facts. “I figured they were either not human or else insane. They heard my ship. And they went into the lake.”
“Swimming?”
“They walked in. Right under the water. And they stayed there.”
“What did they look like?”
“I didn’t get a close look,” he said evasively, his eyes troubled as they avoided mine.
I was aware of a strange, mounting excitement that swelled in my throat until I could hardly speak. I jerked my head toward the lake.
“Come on,” I said.
There lay the blue water, moving gently in the breeze. The cliffs like folded curtains rose beyond it. There was no sign of life in sight as we crossed the bare, pitted rocks. Fitzgerald eyed me askance as we clumped toward the water in our heavy lead-lined boots. I knew he expected doubt from me.
But I knew also that he had told the truth. The lost memory of danger sent its premonitory shadows through my mind and I believed, dimly, that I too had seen those aquatic people, sometime in that immediate past which had been expunged from my brain.
We were halfway across the rocks, our Geiger-counters clicking noisy warning of the death in the air all around us, when the first of the lake people rose up before us from behind a ledge of rock.
He was a perfectly normal looking man—except that he stood there in khaki trousers and shirt, sleeves rolled up, in the bath of potent destruction which was the very air of the Ring. He looked at us with a blankness impossible to describe and yet with a strangely avid interest in his eyes.
When we were half a dozen paces away he raised his arm and, without changing expression, in a voice totally without inflection, he spoke.
