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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Features an integrated approach of statistical scenarios and simulations to aid readers in developing key intuitions needed to understand the wide ranging concepts and methods of statistics and inference
Illuminating Statistical Analysis Using Scenarios and Simulations presents the basic concepts of statistics and statistical inference using the dual mechanisms of scenarios and simulations. This approach helps readers develop key intuitions and deep understandings of statistical analysis. Scenario-specific sampling simulations depict the results that would be obtained by a very large number of individuals investigating the same scenario, each with their own evidence, while graphical depictions of the simulation results present clear and direct pathways to intuitive methods for statistical inference. These intuitive methods can then be easily linked to traditional formulaic methods, and the author does not simply explain the linkages, but rather provides demonstrations throughout for a broad range of statistical phenomena. In addition, induction and deduction are repeatedly interwoven, which fosters a natural "need to know basis" for ordering the topic coverage.
Examining computer simulation results is central to the discussion and provides an illustrative way to (re)discover the properties of sample statistics, the role of chance, and to (re)invent corresponding principles of statistical inference. In addition, the simulation results foreshadow the various mathematical formulas that underlie statistical analysis.
In addition, this book:
• Features both an intuitive and analytical perspective and includes a broad introduction to the use of Monte Carlo simulation and formulaic methods for statistical analysis
• Presents straight-forward coverage of the essentials of basic statistics and ensures proper understanding of key concepts such as sampling distributions, the effects of sample size and variance on uncertainty, analysis of proportion, mean and rank differences, covariance, correlation, and regression
• Introduces advanced topics such as Bayesian statistics, data mining, model cross-validation, robust regression, and resampling
• Contains numerous example problems in each chapter with detailed solutions as well as an appendix that serves as a manual for constructing simulations quickly and easily using Microsoft® Office Excel®
Illuminating Statistical Analysis Using Scenarios and Simulations is an ideal textbook for courses, seminars, and workshops in statistics and statistical inference and is appropriate for self-study as well. The book also serves as a thought-provoking treatise for researchers, scientists, managers, technicians, and others with a keen interest in statistical analysis.
Jeffrey E. Kottemann, Ph.D., is Professor in the Perdue School at Salisbury University. Dr. Kottemann has published articles in a wide variety of academic research journals in the fields of business administration, computer science, decision sciences, economics, engineering, information systems, psychology, and public administration. He received his Ph.D. in Systems and Quantitative Methods from the University of Arizona.
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Contents
Cover
Title Page
Copyright
Preface
Acknowledgements
Part I: Sample Proportions and the Normal Distribution
Chapter 1: Evidence and Verdicts
Chapter 2: Judging Coins I
Chapter 3: Brief on Bell Shapes
Chapter 4: Judging Coins II
Chapter 5: Amount of Evidence I
Chapter 6: Variance of Evidence I
Chapter 7: Judging Opinion Splits I
Chapter 8: Amount of Evidence II
Chapter 9: Variance of Evidence II
Chapter 10: Judging Opinion Splits II
Chapter 11: It Has Been the Normal Distribution All Along
A Note on Stricter Thresholds for Type I Error
Chapter 12: Judging Opinion Split Differences
Chapter 13: Rescaling to Standard Errors
Chapter 14: The Standardized Normal Distribution Histogram
Chapter 15: The z-Distribution
Chapter 16: Brief on Two-Tail Versus One-Tail
Chapter 17: Brief on Type I Versus Type II Errors
The Bigger Picture
Part II: Sample Means and the Normal Distribution
Chapter 18: Scaled Data and Sample Means
Chapter 19: Distribution of Random Sample Means
Chapter 20: Amount of Evidence
Chapter 21: Variance of Evidence
Variance and Standard Deviation
Chapter 22: Homing in on the Population Mean I
Chapter 23: Homing in on the Population Mean II
Chapter 24: Homing in on the Population Mean III
Chapter 25: Judging Mean Differences
Chapter 26: Sample Size, Variance, and Uncertainty
Chapter 27: The t-Distribution
Part III: Multiple Proportions and Means: The X2- and F-Distributions
Chapter 28: Multiple Proportions and the X2-Distribution
Chapter 29: Facing Degrees of Freedom
Chapter 30: Multiple Proportions: Goodness of Fit
A Note on Using Chi-squared to Test the Distribution of a Scaled Variable
Chapter 31: Two-Way Proportions: Homogeneity
Chapter 32: Two-Way Proportions: Independence
Chapter 33: Variance Ratios and the F-Distribution
Chapter 34: Multiple Means and Variance Ratios: ANOVA
Chapter 35: Two-Way Means and Variance Ratios: ANOVA
Part IV: Linear Associations: Covariance, Correlation, and Regression
Chapter 36: Covariance
Chapter 37: Correlation
Chapter 38: What Correlations Happen Just by Chance?
Special Considerations: Confidence Intervals for Sample Correlations
Chapter 39: Judging Correlation Differences
Special Considerations: Sample Correlation Differences
Chapter 40: Correlation with Mixed Data Types
Chapter 41: A Simple Regression Prediction Model
Chapter 42: Using Binomials Too
Getting More Sophisticated #1
Getting More Sophisticated #2
Chapter 43: A Multiple Regression Prediction Model
Getting More Sophisticated
Chapter 44: Loose End I (Collinearity)
Chapter 45: Loose End II (Squaring R)
Chapter 46: Loose End III (Adjusting R-Squared)
Chapter 47: Reality Strikes
Part V: Dealing with Unruly Scaled Data
Chapter 48: Obstacles and Maneuvers
Chapter 49: Ordered Ranking Maneuver
Chapter 50: What Rank Sums Happen Just by Chance?
Chapter 51: Judging Rank Sum Differences
Chapter 52: Other Methods Using Ranks
Chapter 53: Transforming the Scale of Scaled Data
Chapter 54: Brief on Robust Regression
Chapter 55: Brief on Simulation and Resampling
Part VI: Review and Additional Concepts
Chapter 56: For Part I
Chapter 57: For Part II
Chapter 58: For Part III
Chapter 59: For Part IV
Chapter 60: For Part V
Appendices
A: Data Types and Some Basic Statistics
Some Basic Statistics (Primarily for Scaled and Binomial Variables)
B: Simulating Statistical Scenarios
Random Variation
General Guidelines
Scenario-Specific Instructions
C: Standard Error as Standard Deviation
D: Data Excerpt
E: Repeated Measures
F: Bayesian Statistics
A Note on Priors
Getting More Sophisticated
G: Data Mining
Index
End User License Agreement
List of Tables
Table E.1
Table E.2
Table F.1
Table F.2
Table F.3
Table F.4
Table F.5
Table G.1
Table 1.1
Table 2.1
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 7.1
Table 8.1
Table 8.2
Table 9.1
Table 10.1
Table 12.1
Table 12.2
Table 13.1
Table 13.2
Table 15.1
Table 15.2
Table 15.3
Table 17.1
Table 17.2
Table 17.3
Table 17.4
Table 26.1
Table 27.1
Table 28.1
Table 30.1
Table 31.1
Table 31.2
Table 31.3
Table 32.1
Table 32.2
Table 32.3
Table 32.4
Table 32.5
Table 33.1
Table 33.2
Table 35.1
Table 35.2
Table 35.3
Table 35.4
Table 35.5
Table 35.6
Table 35.7
Table 42.1
Table 42.2
Table 43.1
Table 44.1
Table 44.2
Table 44.3
Table 44.4
Table 44.5
Table 47.1
Table 47.2
Table 49.1
Table 51.1
Table 51.2
Table 56.1
Table 56.2
Table 57.1
Table 57.2
Table 58.1
Table 58.2
Table 58.3
Table 58.4
Table 58.5
Table 58.6
Table 58.7
Table 59.1
Table 59.2
Table 59.3
Table 59.4
Table 59.5
Table 59.6
Table 59.7
Table 60.1
Table 60.2
Table 60.3
List of Illustrations
Figure A.1
Figure B.1
Figure B.2
Figure C.1
Figure E.1
Figure F.1
Figure G.1
Figure G.2
Figure G.3
Figure G.4
Figure G.5
Figure G.6
Figure G.7
Figure G.8
Figure 2.1
Figure 2.2
Figure 3.1
Figure 3.2
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 6.1
Figure 6.2
Figure 6.3
Figure 7.1
Figure 8.1
Figure 9.1
Figure 9.2
Figure 9.3
Figure 10.1
Figure 12.1
Figure 14.1
Figure 14.2
Figure 14.3
Figure 15.1
Figure 15.2
Figure 15.3
Figure 15.4
Figure 16.1
Figure 16.2
Figure 16.3
Figure 17.1
Figure 17.2
Figure 17.3
Figure 19.1
Figure 20.1
Figure 20.2
Figure 21.1
Figure 21.2
Figure 22.1
Figure 23.1
Figure 24.1
Figure 24.2
Figure 25.1
Figure 25.2
Figure 26.1
Figure 26.2
Figure 26.3
Figure 26.4
Figure 26.5
Figure 26.6
Figure 27.1
Figure 27.2
Figure 27.3
Figure 27.4
Figure 27.5
Figure 28.1
Figure 28.2
Figure 28.3
Figure 28.4
Figure 28.5
Figure 28.6
Figure 30.1
Figure 33.1
Figure 33.2
Figure 33.3
Figure 33.4
Figure 34.1
Figure 34.2
Figure 34.3
Figure 34.4
Figure 34.5
Figure 36.1
Figure 36.2
Figure 36.3
Figure 36.4
Figure 38.1
Figure 38.2
Figure 38.3
Figure 38.4
Figure 38.5
Figure 38.6
Figure 39.1
Figure 39.2
Figure 39.3
Figure 40.1
Figure 41.1
Figure 41.2
Figure 42.1
Figure 43.1
Figure 43.2
Figure 45.1
Figure 45.2
Figure 47.1
Figure 47.2
Figure 47.3
Figure 47.4
Figure 48.1
Figure 48.2
Figure 48.3
Figure 48.4
Figure 50.1
Figure 53.1
Figure 56.1
Figure 57.1
Figure 57.2
Figure 59.1
Figure 59.2
Figure 59.3
Figure 59.4
Figure 59.5
Figure 59.6
Figure 60.1
Guide
Cover
Table of Contents
Begin Reading
Part 1
Chapter 1
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Illuminating Statistical Analysis Using Scenarios and Simulations
Jeffrey E Kottemann Ph.D.
Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Names: Kottemann, Jeffrey E.
Title: Illuminating statistical analysis using scenarios and simulations/Jeffrey E Kottemann, Ph.D.
Description: Hoboken, New Jersey: John Wiley & Sons, Inc. [2017], | Includes index.
Identifiers: LCCN 2016042825| ISBN 9781119296331 (cloth) | ISBN 9781119296362 (epub)
Subjects: LCSH: Mathematical statistics. | Distribution (Probability theory)
Classification: LCC QA276 .K676 2017 | DDC 519.5—dc23 LC record available at https://lccn.loc.gov/2016042825
Preface
The goal of this book is to help people develop an assortment of key intuitions about statistics and inference and use those intuitions to make sense of statistical analysis methods in a conceptual as well as a practical way. Moreover, I hope to engender good ways of thinking about uncertainty. The book is comprised of a series of short, concise chapters that build upon each other and are best read in order. The chapters cover a wide range of concepts and methods of classical (frequentist) statistics and inference. (There are also appendices on Bayesian statistics and on data mining techniques.)
Examining computer simulation results is central to our investigation. Simulating pollsters, for example, who survey random people for responses to an agree or disagree opinion question not only mimics reality but also has the added advantage of being able to employ 1000 independent pollsters simultaneously. The results produced by such simulations provide an eye-opening way to (re)discover the properties of sample statistics, the role of chance, and to (re)invent corresponding principles of statistical inference. The simulation results also foreshadow the various mathematical formulas that underlie statistical analysis.
Mathematics used in the book involves basic algebra. Of particular relevance is interpreting the relationships found in formulas. Take, for example, . As increases, increases because is the numerator of the fraction. And as increases, decreases because is the denominator. Going one step further, we could have . Here, as increases, decreases, increases, so increases. These functional forms mirror most of the statistical formulas we will encounter.
Acknowledgements
My thanks go to Dan Dolk, Gene Hahn, Fati Salimian, and Kathie Wright for their feedback and encouragement. At John Wiley & Sons, thanks go to Susanne Steitz-Filler, Kathleen Pagliaro, Vishnu Narayanan, and Shikha Pahuja.
Part ISample Proportions and the Normal Distribution
1Evidence and Verdicts
Before we focus in on using statistics as evidence to be used in making judgments, let's take a look at a widely used “verdict outcomes framework.” This general framework is useful for framing judgments in a wide range of situations, including those encountered in statistical analysis.
Anytime we use evidence to arrive at a judgment, there are four generic outcomes possible, as shown in Table 1.1. Two outcomes correspond to correct judgments and two correspond to incorrect judgments, although we rarely know whether our judgments are correct or incorrect. Consider a jury trial in U.S. criminal court. Ideally, the jury is always correct, judging innocent defendants not guilty and judging guilty defendants guilty. Evidence is never perfect, though, and so juries will make erroneous judgments, judging innocent defendants guilty or guilty defendants not guilty.
Table 1.1 Verdict outcomes framework.
In U.S. criminal court, the presumption is that a defendant is innocent until “proven” guilty. Further, convention in U.S. criminal court has it that we are more afraid of punishing an innocent person (type I error) than we are of letting a guilty person go unpunished (type II error). Because of this fear, the threshold for a guilty verdict is set high: “Beyond a reasonable doubt.” So, convicting an innocent person should be a relatively unlikely outcome. In U.S. criminal court, we are willing to have a greater chance of letting a guilty person go unpunished than we are of punishing an innocent person. In short, we need to be very sure before we reject the presumption of innocence and render a verdict of guilty in U.S. criminal court.
We can change the relative chances of the two types of error by changing the threshold. Say we change from “beyond a reasonable doubt” to “a preponderance of evidence.” The former is the threshold used in U.S. criminal court, and the latter is the threshold used in U.S. civil court. Let's say that the former corresponds to being 95% sure before judging a defendant guilty and that the latter corresponds to being 51% sure before judging a defendant guilty. You can imagine cases where the same evidence results in different verdicts in criminal and civil court, which indeed does happen. For example, say that the evidence leads to the jury being 60% sure of the defendant's guilt. The jury verdict in criminal court would be not guilty (60% < 95%) but the jury verdict in civil court would be guilty (60% > 51%). Compared to criminal court, civil court is more likely to declare an innocent person guilty (type I error), but is also less likely to declare a guilty person not guilty (type II error).
Changing the verdict threshold either decreases type I error while increasing type II error (criminal court) or increases type I error while decreasing type II error (civil court)
Statistical analysis is conducted as if in criminal court. Below are a number of jury guidelines that have parallels in statistical analysis, as we'll see repeatedly.
Erroneously rejecting the presumption of innocence (type I error) is feared most.
The possible verdicts are “guilty” and “not guilty.” There is no verdict of “innocent.”
Reasons other than perceived innocence can lead to a not guilty verdict, such as insufficient evidence.
To reject the presumption of innocence and render a guilty verdict, there must be a sufficient amount of (unbiased) evidence.
To reject the presumption of innocence and render a guilty verdict, the pieces of evidence must be sufficiently consistent (not at variance with each other).
Statistical analysis formally evaluates evidence in order to determine whether to reject or not reject a stated presumption, and it is primarily concerned with limiting the chances of type I error. Further, the amount of evidence and the variance of evidence are key characteristics of evidence that are formally incorporated into the evaluation process. In what follows, we'll see how this is accomplished.
2Judging Coins I
Let's start with the simplest statistical situation: that of judging whether a coin is fair or not fair. Later we'll see that this situation is statistically equivalent to agree or disagree opinion polling. A coin is fair if it has a 50% chance of coming up heads, and a 50% chance of coming up tails when you flip it. Adjusting the verdict table to the coin-flipping context gives us Table 2.1.
Table 2.1 Coin flipping outcomes.
Statistical Scenario–Coins #1
You want to see if a coin is fair or not. To gather evidence, you plan on flipping it 10 times to see how many heads come up. Beforehand, you want to set the verdict rule.
Where do you draw the two lines for declaring the coin to be not fair?
0
1
2
3
4
5
6
7
8
9
10
Number of heads
Draw one line toward the left for your threshold of “too few heads,” and another line toward the right for your threshold of “too many heads.”
Where you draw the lines represents your choice of thresholds.
Just use your intuition; don't do any arithmetic.
Intuitively, it seems extremely unlikely for a fair coin to come up heads only 0 or 1 times out of 10, and most people would arrive at the verdict that the coin is not fair. Likewise, it seems extremely unlikely for a fair coin to come up heads 9 or 10 times out of 10, and most people would arrive at the verdict that the coin is not fair. On the other hand, it seems fairly likely for a fair coin to come up heads 4, 5, or 6 times out of 10, and so most people would say that the coin seems fair. But what about 2, 3, 7, or 8 heads? Let's experiment.
Shown in Figure 2.1 is a histogram of what actually happened (in simulation) when 1000 people each flipped a fair coin 10 times. This shows us how fair coins tend to behave. The horizontal axis is the number of heads that came up out of 10. The vertical axis shows the number of people out of the 1000 who came up with the various numbers of heads.
Figure 2.1
Appendix B gives step-by-step instructions for constructing this simulation using common spreadsheet software; guidelines are also given for constructing additional simulations found in the book.
Sure enough, fair coins very rarely came up heads 0, 1, 9, or 10 times. And, sure enough, they very often came up heads 4, 5, or 6 times. What about 2, 3, 7, or 8 heads?
Notice that 2 heads came up a little less than 50 times out of 1000, or near 5% of the time. Same with 8 heads. And, 3 heads came up well over 100 times out of 1000, or over 10% of the time. Same with 7 heads.
Where do you want to draw the threshold lines now?
0
1
2
3
4
5
6
7
8
9
10
Number of heads
How about the lines shown on the histogram in Figure 2.2?
Figure 2.2
Eyeballing the histogram, we can add up the frequencies for 0, 1, 2, 8, 9, and 10 heads out of the 1000 total. Fair coins came up with 0, 1, 2, 8, 9, or 10 heads a total of about and of the time.
So, a “verdict rule” that has about an 11% chance of type I error can be stated as:
If the number of heads flipped is outside the interval
Number of heads ≥3 and number of heads ≤7
Then reject the presumption that the coin is fair.
Fair coins will be inside the interval about 89% of the time, and we will correctly judge them to be fair coins.
Fair coins will be outside the interval about 11% of the time, and we will incorrectly judge them to be unfair coins.
The “verdict rule” defines an interval outside of which we will reject the presumption.
Type I error
occurs when a fair coin's head count is outside our interval.
Type II error
occurs when an unfair coin's head count is inside our interval.
3Brief on Bell Shapes
Statistical Scenario—Distribution of Outcomes.
Why is the histogram bell-shaped?
Before expanding the previous Statistical Scenario let's briefly explore why the histogram, reproduced in Figure 3.1, is shaped the way it is: bell-shaped. It tapers off symmetrically on each side from a single peak in the middle.
Figure 3.1
Since each coin flip has two possible outcomes and we are considering ten separate outcomes together, there are a total of unique possible patterns (permutations) of heads and tails with 10 flips of a coin. Of these, there is only one with 0 heads and only one with 10 heads. These are the least likely outcomes.
TTTTTTTTTT
HHHHHHHHHH
There are ten with 1 head, and ten with 9 heads:
HTTTTTTTTT
THHHHHHHHH
THTTTTTTTT
HTHHHHHHHH
TTHTTTTTTT
HHTHHHHHHH
TTTHTTTTTT
HHHTHHHHHH
TTTTHTTTTT
HHHHTHHHHH
TTTTTHTTTT
HHHHHTHHHH
TTTTTTHTTT
HHHHHHTHHH
TTTTTTTHTT
HHHHHHHTHH
TTTTTTTTHT
HHHHHHHHTH
TTTTTTTTTH
HHHHHHHHHT
Since there are 10 times more ways to get 1 or 9 heads than 0 or 10 heads, we expect to flip 1 or 9 heads 10 times more often than 0 or 10 heads.
Further, there are 45 ways to get 2 or 8 heads, 120 ways to get 3 or 7 heads, and 210 ways to get 4 or 6 heads. Finally, there are 252 ways to get 5 heads, which is the most likely outcome and therefore the most frequently expected outcome. Notice how the shape of the histogram of simulation outcomes we saw in Figure 3.1 closely mirrors the number of ways (#Ways) chart that is shown in Figure 3.2.
Figure 3.2
You don't need to worry about calculating #ways. Soon we won't need such calculations. Just for the record, the formula for the #ways is where is the number of flips, h is the number of heads you are interested in, and ! is the factorial operation (example: ). In official terms, #ways is the number of combinations of things taken at a time.
4Judging Coins II
Let's revisit Statistical Scenario–Coins #1, now with additional information on each of the possible outcomes. Table 4.1 summarizes this additional information. As noted, there are a total of different unique patterns of heads & tails possible when we flip a coin 10 times. For any given number of heads, as we have just seen, there are one or more ways to get that number of heads.
Statistical Scenario—Coins #2
Using Table 4.1, where do you draw the two lines for declaring the coin to be unfair?
Table 4.1 Coin flipping details.
#Heads
#Ways
Expected relative frequency
Probability
as Percent
Rounded
0
1
1/1024
0.00098
0.098%
0.1%
1
10
10/1024
0.00977
0.977%
1.0%
2
45
45/1024
0.04395
4.395%
4.4%
3
120
120/1024
0.11719
11.719%
11.7%
4
210
210/1024
0.20508
20.508%
20.5%
5
252
252/1024
0.24609
24.609%
24.6%
6
210
210/1024
0.20508
20.508%
20.5%
7
120
120/1024
0.11719
11.719%
11.7%
8
45
45/1024
0.04395
4.395%
4.4%
9
10
10/1024
0.00977
0.977%
1.0%
10
1
1/1024
0.00098
0.098%
0.1%
Totals:
1024
1024/1024
1.0
100%
100%
The #ways divided by 1024 gives us the expectedrelative frequency for that number of heads expressed as a fraction. For example, we expect to get 5 heads 252/1024ths of the time. The fraction can also be expressed as a decimal value. This decimal value can be viewed as the probability that a certain number of heads will come up in 10 flips. For example, the probability of getting 5 heads is approximately 0.246. We can also express this as a percentage, 24.6%.
A probability of 1 (100%) means something will always happen and a probability of 0 (0%) means something will never happen. A probability of 0.5 (50%) means something will happen half the time. The sum of the probabilities of the entire set of possible outcomes is the sum of all the probabilities and always equals 1 (100%). The probability of a subset of possible outcomes can be calculated by summing the probabilities of each of the outcomes. For example, using the rounded percentages from the table, the probability of 2 or fewer heads is .
Notice how the bars of our simulation histogram, reproduced in Figure 4.1, reflect the corresponding probabilities in Table 4.1.
Figure 4.1
Say someone gives you a coin to test. When you flip the coin 10 times, you are sampling the coin's behavior 10 times. The number of heads you toss is your evidence. Based on this evidence you must decide whether to reject your presumption of fairness and judge the coin as not fair.
What happens if you make your “verdict rule” to be:
Verdict “coin is not fair” if outside the interval #heads ≥ 1 and ≤ 9 as shown in Table 4.2 and the accompanying Figure 4.2?
From the Statistical ScenarioTable 4.1, we can see that a fair coin will come up 0 heads about 0.1% of the time, and 10 heads about 0.1% of the time. The sum is about 0.2% of the time, or about 2 out of 1000. So, it will be extremely rare for us to make a type I error and erroneously call a fair coin unfair because fair coins will almost never come up with 0 or 10 heads. However, what about 1 head or 9 heads? Our rule says not to call those coins unfair. But a fair coin will only come up 1 head or 9 heads about of the time. Therefore, we may end up misjudging many unfair coins that come up heads one or nine times because we'll declare them to be fair coins. That is type II error.
Table 4.2 First verdict rule scenario.
Figure 4.2
Determining the chance of type II error is too involved for discussion now (that is Chapter 17), but recall from Chapter 1 that increasing the chance of type I error decreases the chance of type II error, and vice versa.
To lower the chances of type II error, we can narrow our “verdict rule” interval to #heads ≥ 2 and ≤ 8 as shown in Table 4.3 and Figure 4.3. Now the probability of making a type I error is about . This rule will decrease the chances of type II error, while increasing the chances of type I error from 0.2 to 2.2%.
If we narrow our “verdict rule” interval even more to #heads ≥ 3 and ≤ 7, we get Table 4.4 and Figure 4.4.
Table 4.3 Second verdict rule scenario.
Figure 4.3
Now the probability of making a type I error is about because a fair coin will come up 0, 1, 2, 8, 9, or 10 heads about 11% of the time. We can express this uncertainty by saying either that there will be an 11% chance of a type I error, or that we are 89% confident that there will not be a type I error. Notice that this is what we came up with earlier by simply eyeballing the histogram of actual simulation outcomes in Chapter 2.
Table 4.4 Third verdict rule scenario.
Figure 4.4
As noted earlier, type I error is feared most. And an 11% chance of type I error is usually seen as excessive. So, we can adopt this rule:
Verdict:
If outside the interval #Heads ≥ 2 and ≤ 8, Judge Coin to be Not Fair.
This gives us about a 2% chance of type I error.
From now on, we'll typically use the following threshold levels for type I error: 10% (0.10), 5% (0.05), and 1% (0.01). We'll see the effects of using various thresholds as we go along. Also as we go along we'll need to replace some common words with statistical terminology. Below are statistical terms to replace the common words we have been using.
Alpha-Level is the threshold probability we stipulate for the occurrence of type I error. Commonly used alpha-levels are 0.05 and 0.01. Alpha-levels used in sciences such as physics are typically much lower.
Confidence level is the complement of alpha-level and is the threshold percentage we stipulate for the nonoccurrence of type I error. 95% confidence corresponds to an alpha-level of 0.05. 99% confidence corresponds to an alpha-level of 0.01.
Null hypothesis is the presumption we have been referring to above. Example: the coin is not unusual, it is a fair coin. Most of the simulations in this book simulate outcomes to expect when the null hypothesis is true: like a fair coin.
Binomial is a variable with only two possible values, such as heads or tails. The term comes from the Latin for “two names.” Other examples of binomials are agree or disagree, male or female. When we represent the binomial values as 0 or 1, then the average will give us the proportion of 1s. For example, the average of 0, 1, 0, and 1 is 0.5 and the average of 1, 1, 0, and 1 is 0.75. (Appendix A overviews all the types of data we'll be working with as the book progresses.)
Sample is the set of observations we have, which is the set of heads and tails flipped for the above coin flipping cases. A sample constitutes evidence.
Sample size is the number of separate observations, which equals 10 for the above cases. The italicized letter is often used: the sample .
Sample statistic is a calculated value that serves to summarize the sample. They are summarizations of evidence. Examples: number of heads in the sample or Proportion of heads in the sample (number of heads divided by sample size). Counts and proportions are the basic sample statistics for binomial variables.
Sampling distributions are illustrated by the histograms of simulation results. They reflect the distributions of the values for sample statistics we get with repeated sampling.
It is important to emphasize that simulation histograms represent sampling distributions that tell us what to expect when the null hypothesis is true. We'll look at many, many sampling distribution histograms in this book. For the remainder of Part I, we'll switch from using counts as our sample statistic to using proportions as our sample statistic. The sampling distribution histogram in Figure 4.5 shows the use of proportions rather than counts on the horizontal axis.
Figure 4.5
