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 section explains how to statistically compare two groups by examining differences in proportions or means. The key steps include defining hypotheses, checking relevant assumptions, calculating the standard error of the difference, and constructing confidence intervals or conducting hypothesis tests using two-sample z-tests or t-tests. It is important to understand the underlying conditions, to distinguish between practical and statistical significance, and to consider the effect sizes and power of the test. Mastery of these techniques is essential for drawing informed conclusions in research and applied settings.

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

Parameters and hypotheses For each of the following situations, define the parameter (proportion or mean) and write the null and alternative hypotheses in terms of parameter values. Example: We want to know if the proportion of up days in the stock market is $50 \%$. Answer: Let $p=$ the proportion of up days. $\mathrm{H}_{0}: p=0.5 \mathrm{vs} . \mathrm{H}_{\mathrm{A}}: p \neq 0.5$ a) A casino wants to know if their slot machine really delivers the 5 in 100 win rate that it claims. b) Last year, customers spent an average of $\$ 36.95$ per visit to the company's website. Based on a random sample of purchases this year, the company wants to know if the mean this year has changed. c) A pharmaceutical company wonders if their new drug has a cure rate different from the $20 \%$ reported by the placebo. d) A bank wants to know if the percentage of customers using their website has changed from the $30 \%$ that used it before their system crashed last week.

Example 2

Hypotheses and parameters As in Exercise $1,$ for each of the following situations, define the parameter and write the null and altemative hypotheses in terms of parameter values. a) Seat-belt compliance in Massachusetts was $65 \%$ in $2008 .$ The state wants to know if it has changed. b) Last year, a survey found that $45 \%$ of the employees were willing to pay for on-site day care. The company wants to know if that has changed. c) Regular card customers have a default rate of $6.7 \%$. A credit card bank wants to know if that rate is different for their Gold card customers. d) Regular card customers have been with the company for an average of 17.3 months. The credit card bank wants to know if their Gold card customers have been with the company on average the same amount of time.

Example 3

P-values Which of the following are true? If false, explain briefly. a) A very high P-value is strong evidence that the null hypothesis is false. b) A very low P-value proves that the null hypothesis is false. c) A high P-value shows that the null hypothesis is true. d) A P-value below 0.05 is always considered sufficient evidence to reject a null hypothesis.

Example 4

More P-values Which of the following are true? If false, explain briefly. a) A very low P-value provides evidence against the null hypothesis. b) A high P-value is strong evidence in favor of the null hypothesis. c) A P-value above 0.10 shows that the null hypothesis is true. d) If the null hypothesis is true, you can't get a P-value below 0.01

Example 5

Hypotheses For each of the following, write out the null and alternative hypotheses, being sure to state whether the alternative is one-sided or two-sided. a) A company reports that last year, $40 \%$ of their reports in accounting were on time. From a random sample this year, they want to know if that proportion has changed. b) A company wants to know if average click-through rates (in minutes) are shorter than the 5.4 minutes that customers spent on their old website before making a purchase. c) Last year, $42 \%$ of the employees enrolled in at least one wellness class at the company's site. Using a survey, they want to know if a greater percentage is planning to take a wellness class this year. d) A political candidate wants to know from recent polls if she's going to garner a majority of votes in next week's election.
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Step-by-Step Explanations

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Common Mistakes

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