Book cover for Essentials of Modern Business Statistics

Essentials of Modern Business Statistics

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

ISBN #9780357131626

8th Edition

810 Questions

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Example Problems
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Summary

Chapter 12 covers three main chi-square tests used in categorical data analysis: the Goodness of Fit Test, the Test of Independence, and the Test for Equality of Multiple Proportions. Each test relies on comparing observed frequencies to those expected under a specified hypothesis. Accurate computation of expected frequencies is critical to these analyses, and both p-value and critical value approaches can be used to determine significance. When significant differences are found, additional procedures such as the Marascuilo procedure allow for detailed pairwise comparisons.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

Test the following hypotheses for a multinomial probability distribution by using the $\chi^{2}$ goodness of fit test. $$ H_{0}: p_{\mathrm{A}}=.40, p_{\mathrm{B}}=.40, \text { and } p_{\mathrm{C}}=.20 $$ $H_{0}:$ The probabilities are not $$ p_{\mathrm{A}}=.40, p_{\mathrm{B}}=.40, \text { and } p_{\mathrm{C}}=.20 $$ A sample of size 200 yielded 60 in category $\mathrm{A}, 120$ in category $\mathrm{B},$ and 20 in category $\mathrm{C}$. Use $\alpha=.01$ and test to see whether the probabilities are as stated in $H_{0}$ a. Use the $p$ -value approach. b. Repeat the test using the critical value approach.

Example 2

Suppose we have a multinomial population with four categories: $\mathrm{A}, \mathrm{B}, \mathrm{C},$ and $\mathrm{D} .$ The null hypothesis is that the proportion of items is the same in every category. The null hypothesis is $$ H_{0}: p_{\mathrm{A}}=p_{\mathrm{B}}=p_{\mathrm{C}}=p_{\mathrm{D}}=.25 $$ A sample of size 300 yielded the following results. $$ \begin{array}{llll} \text { A: } 85 & \text { B: } 95 & \text { C: } 50 & \text { D: } 70 \end{array} $$ Use $\alpha=.05$ to determine whether $H_{0}$ should be rejected. What is the $p$ -value?

Example 3

During the first 13 weeks of the television season, the Saturday evening 8:00 P.M. to 9: 00 P.M. audience proportions were recorded as $\mathrm{ABC} 29 \%, \mathrm{CBS} 28 \%, \mathrm{NBC} 25 \%,$ and Independents $18 \% .$ A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and Independents 46 homes. Test with $\alpha=.05$ to determine whether the viewing audience proportions changed.

Example 4

Mars, Inc. manufactures M\&M's, one of the most popular candy treats in the world. The milk chocolate candies come in a variety of colors including blue, brown, green, orange, red, and yellow (M\&M website). The overall proportions for the colors are .24 blue, .13 brown, .20 green, .16 orange, .13 red, and 14 yellow. In a sampling study, several bags of M\&M milk chocolates were opened and the following color counts were obtained. $$ \begin{array}{cccccc} \text { Blue } & \text { Brown } & \text { Green } & \text { Orange } & \text { Red } & \text { Yellow } \\ 105 & 72 & 89 & 84 & 70 & 80 \end{array} $$ Use a .05 level of significance and the sample data to test the hypothesis that the overall proportions for the colors are as stated above. What is your conclusion?

Example 5

The Harris Poll tracks the favorite sport of Americans who follow at least one sport. Results of the poll show that professional football is the favorite sport of $33 \%$ of Americans who follow at least one sport, followed by baseball at $15 \%,$ men's college football at $10 \%$, auto racing at $6 \%,$ men's professional basketball at $5 \%,$ and ice hockey at $5 \%,$ with other sports at $26 \% .$ Consider a survey in which 344 college undergraduates who follow at least one sport were asked to identify their favorite sport produced the following results: $$ \begin{array}{cccccc} \text { Professional Football } & \text { Baseball } & \text { Men's College Football } & \text { Auto Racing } & \text { Men's Professional Basketball } & \text { Ice Hockey } & \text { Other Sports } \\ 111 & 39 & 46 & 14 & 6 & 20 & 108 \end{array} $$ Do college undergraduate students differ from the general public with regard to their favorite sports? Use $\alpha=.05$
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