Book cover for Stats Data and Models

Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

ISBN #9780321986498

4th Edition

1,406 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter focuses on constructing and interpreting confidence intervals for population proportions using sample data. It describes the formula for the standard error, explains the significance of the margin of error and critical value, and demonstrates how to determine appropriate sample sizes. Emphasis is placed on checking assumptions (independence, 10% condition, success/failure condition) to ensure the validity of the Normal approximation. Correct interpretation avoids misstatements; the interval reflects uncertainty about the fixed parameter rather than variability in the parameter itself.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Analysis of Variance (ANOVA)

A collection of statistical techniques used to compare the means of three or more groups by partitioning total variability into components due to differences among groups (between-group) and within groups.

Example Problems

Example 1

Pew Research, in November $2011,$ polled a random sample of 799 U.S. teens about Internet use. $49 \%$ of those teens reported that they had misrepresented their age online to gain access to websites and online services. a) Explain the meaning of $\hat{p}=0.49$ in the context of this situation. b) Calculate the standard error of $\hat{p}$ c) Explain what this standard error means in the context of this situation.

Example 2

Gallup regularly conducts a poll asking people to imagine a ladder with 10 rungs. Rung 0 represents the worst possible life and rung 10 represents the best possible life. Respondents are asked what rung they would say they are on. Responses are classified as "Thriving" (standing on rung 7 or higher, and expecting to be on rung 8 or higher five years from now), "Suffering" (standing on rung 4 or lower and expecting to be on rung 4 or lower 5 years from now), or "Struggling" (not thriving or suffering). The poll for November $22-25,2011,$ found $4 \%$ of respondents to be suffering. The polls all use a sample size of approximately 1500 people. (www.gallup .com/poll/152033/Fewer-Americans-Thriving-2011- 2010.aspx) a) Explain the meaning of $" \hat{p}=0.04$ " in the context of this situation. b) Calculate the standard error of $\hat{p}$. c) Explain what this standard error means in the context of this situation.

Example 3

The $95 \%$ confidence interval for the number of teens in Exercise 1 who reported that they had misrepresented their age online is from $45.6 \%$ to $52.5 \%$ a) Interpret the interval in this context. b) Explain the meaning of "95\% confident" in this context.

Example 4

In a 2010 Pew Research study on trends in marriage and family, $4 \%$ of randomly selected respondents said that the trend of more single women having children is "a good thing." The $95 \%$ confidence interval is from $2.9 \%$ to $5.1 \%(n=1229)$. a) Interpret this interval in this context. b) Explain the meaning of "95\% confident" in this context.

Example 5

St. Norbert's College in Green Bay, Wisconsin, and Wisconsin Public Radio conduct an annual poll of Wisconsinites about political opinions. The Fall 2011 survey asked a random sample of 402 adult Wisconsin residents whether they think things in the country are going in the right direction or in the wrong direction. $66 \%$ said that things were going in the wrong direction. a) Calculate the margin of error for the proportion of all adult Wisconsin residents who think things are going in the wrong direction for $90 \%$ confidence. b) Would the margin of error be larger or smaller for $95 \%$ confidence? Explain.
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