Book cover for Statistics for Business and Economics

Statistics for Business and Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

ISBN #9780324365054

10th Edition

999 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers techniques for making inferences about the differences between two populations, whether comparing means or proportions. It illustrates methods for constructing confidence intervals and carrying out hypothesis tests under various scenarios—when population standard deviations are known, unknown, or when using matched samples. Emphasis is placed on computing and interpreting the standard error, selecting the correct statistical distribution (z or t), and recognizing the importance of sample design and assumptions in drawing valid inferences.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

The following results come from two independent random samples taken of two populations. $$\begin{array}{ll} \text { Sample } 1 & \text { Sample } 2 \\ n_{1}=50 & n_{2}=35 \\ \bar{x}_{1}=13.6 & \bar{x}_{2}=11.6 \\ \sigma_{1}=2.2 & \sigma_{2}=3.0 \end{array}$$ a. What is the point estimate of the difference between the two population means? b. Provide a $90 \%$ confidence interval for the difference between the two population means. c. Provide a $95 \%$ confidence interval for the difference between the two population means.

Example 2

Consider the following hypothesis test. $$\begin{array}{l} H_{0}: \mu_{1}-\mu_{2} \leq 0 \\ H_{\mathrm{a}}: \mu_{1}-\mu_{2}>0 \end{array}$$ The following results are for two independent samples taken from the two populations. $$\begin{array}{ll} \text { Sample 1 } & \text { Sample 2 } \\ n_{1}=40 & n_{2}=50 \\ \bar{x}_{1}=25.2 & \bar{x}_{2}=22.8 \\ \sigma_{1}=5.2 & \sigma_{2}=6.0 \end{array}$$ a. What is the value of the test statistic? b. What is the $p$ -value? c. With $\alpha=.05,$ what is your hypothesis testing conclusion?

Example 3

Consider the following hypothesis test. $$\begin{array}{l} H_{0}: \mu_{1}-\mu_{2}=0 \\ H_{\mathrm{a}}: \mu_{1}-\mu_{2} \neq 0 \end{array}$$ The following results are for two independent samples taken from the two populations. $$\begin{array}{ll} \text { Sample 1 } & \text { Sample 2 } \\ n_{1}=80 & n_{2}=70 \\ \bar{x}_{1}=104 & \bar{x}_{2}=106 \\ \sigma_{1}=8.4 & \sigma_{2}=7.6 \end{array}$$ a. What is the value of the test statistic? b. What is the $p$ -value? c. With $\alpha=.05,$ what is your hypothesis testing conclusion?

Example 4

Gasoline prices reached record high levels in 16 states during 2003 (The Wall Street Journal, March 7,2003 ). Two of the affected states were California and Florida. The American Automobile Association reported a sample mean price of $\$ 2.04$ per gallon in California and a sample mean price of $\$ 1.72$ per gallon in Florida. Use a sample size of 40 for the California data and a sample size of 35 for the Florida data. Assume that prior studies indicate a population standard deviation of .10 in California and .08 in Florida are reasonable. a. What is a point estimate of the difference between the population mean prices per gallon in California and Florida? b. At $95 \%$ confidence, what is the margin of error? c. What is the $95 \%$ confidence interval estimate of the difference between the population mean prices per gallon in the two states?

Example 5

The average expenditure on Valentine's Day was expected to be $\$ 100.89$ (USA Today February 13,2006 ). Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 40 male consumers was $\$ 135.67$, and the average expenditure in a sample survey of 30 female consumers was $\$ 68.64 .$ Based on past surveys, the standard deviation for male consumers is assumed to be $\$ 35,$ and the standard deviation for female consumers is assumed to be $\$ 20$ a. What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females? b. At $99 \%$ confidence, what is the margin of error? c. Develop a $99 \%$ confidence interval for the difference between the two population means.

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