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 the use of the chi-square distribution to construct confidence intervals and conduct hypothesis testing for a population variance. It details the derivation of the formula used to create a confidence interval for ?², shows how to test hypotheses about the variance using the computed chi-square statistic, and extends the discussion to compare two variances using the F distribution. Mastery of these methods is essential for quality control, process improvement, and interpreting variability in a variety of practical contexts.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

Find the following chi-square distribution values from Table 11.1 or Table 3 of Appendix $\mathrm{B}$. a. $\quad \chi_{.05}^{2}$ with $d f=5$ b. $\quad \chi_{.025}^{2}$ with $d f=15$ c. $\quad \chi_{.975}^{2}$ with $d f=20$ d. $\quad \chi_{.01}^{2}$ with $d f=10$ e. $\quad \chi_{.95}^{2}$ with $d f=18$

Example 2

A sample of 20 items provides a sample standard deviation of 5 a. Compute the $90 \%$ confidence interval estimate of the population variance. b. Compute the $95 \%$ confidence interval estimate of the population variance. c. Compute the $95 \%$ confidence interval estimate of the population standard deviation.

Example 3

A sample of 16 items provides a sample standard deviation of $9.5 .$ Test the following hypotheses using $\alpha=.05 .$ What is your conclusion? Use both the $p$ -value approach and the critical value approach. \[ \begin{array}{l} H_{0}: \sigma^{2} \leq 50 \\ H_{\mathrm{a}}: \sigma^{2}>50 \end{array} \]

Example 4

The variance in drug weights is critical in the pharmaceutical industry. For a specific drug, with weights measured in grams, a sample of 18 units provided a sample variance of $s^{2}=.36$ a. Construct a $90 \%$ confidence interval estimate of the population variance for the weight of this drug. b. Construct a $90 \%$ confidence interval estimate of the population standard deviation.

Example 5

The daily car rental rates for a sample of eight cities follow. a. Compute the variance and the standard deviation for these data. b. What is the $95 \%$ confidence interval estimate of the variance of car rental rates for the population? c. What is the $95 \%$ confidence interval estimate of the standard deviation for the population?
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Common Mistakes

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