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Chapter 11: Problem 18

Describe in your own words a test of independence and a test of homogeneity. Give one example of each.

Short Answer

Expert verified
A test of independence is used to determine whether two categorical variables are independent of each other, for instance testing if the grade level influences participation in school activities. A test of homogeneity checks if different population groups have same proportions for a categorical variable, like examining if the participation in school activities is the same across all grade levels.

Step by step solution

01

Describe Test of Independence

A test of independence is a statistical tool used to determine whether two categorical variables are independent of each other. This test primarily involves using a chi-square test on the contingency table of the observed frequencies of the variables.
02

Example of Test of Independence

For instance, a survey could be conducted in a school to see if there is a relationship between the grade level (9th, 10th, 11th, 12th) and participation in extracurricular activities (yes or no). The test of independence can help determine whether grade level and participation in activities are independent or linked.
03

Describe Test of Homogeneity

A test of homogeneity, on the other hand, is used to determine if different populations (or groups) have the same proportions for a categorical variable. This test also involves using a chi-square test, but applied on different categories of population samples to check if they show similar behavior.
04

Example of Test of Homogeneity

For example, a test of homogeneity might be used to determine if the proportion of students who participate in extracurricular activities is the same across all grade levels (9th, 10th, 11th, 12th). This test can verify whether grade level influences the activity participation rate or not.

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Most popular questions from this chapter

A random sample of 1000 Americans was taken, and these adults were asked if experience in politics was necessary for a candidate to be president of America. The following table presents the results of the survey. $$ \begin{array}{lccc} \hline & \text { Necessary } & \text { Not Necessary } & \text { No Opinion } \\\ \hline \text { Men } & 260 & 220 & 70 \\ \text { Women } & 230 & 180 & 40 \\ \hline \end{array} $$ Test at a \(1 \%\) significance level whether gender and opinions are related.

Consider the following contingency table, which records the results obtained for four samples of fixed sizes selected from four populations. $$ \begin{array}{lcccc} \hline & \multicolumn{4}{c} {\text { Sample Selected From }} \\ \cline { 2 - 5 } & \text { Population 1 } & \text { Population 2 } & \text { Population 3 } & \text { Population 4 } \\ \hline \text { Row 1 } & 24 & 81 & 60 & 121 \\ \text { Row 2 } & 46 & 64 & 91 & 72 \\ \text { Row 3 } & 20 & 37 & 105 & 93 \\ \hline \end{array} $$ a. Write the null and alternative hypotheses for a test of homogeneity for this table. b. Calculate the expected frequencies for all cells assuming that the null hypothesis is true. c. For \(\alpha=.025\), find the critical value of \(\chi^{2}\). Show the rejection and nonrejection regions on the chi-square distribution curve. d. Find the value of the test statistic \(\chi^{2}\). e. Using \(\alpha=.025\), would you reject the null hypothesis?

Construct the \(95 \%\) confidence intervals for the population variance and standard deviation for the following data, assuming that the respective populations are (approximately) normally distributed. a. \(n=10, s^{2}=7.2\) b. \(n=18, s^{2}=14.8\)

Two random samples, one of 95 blue-collar workers and a second of 50 white- collar workers, were taken from a large company. These workers were asked about their views on a certain company issue. The following table gives the results of the survey. $$ \begin{array}{lccc} \hline & \multicolumn{3}{c} {\text { Opinion }} \\ \cline { 2 - 4 } & \text { Favor } & \text { Oppose } & \text { Uncertain } \\\ \hline \text { Blue-collar workers } & 44 & 39 & 12 \\ \text { White-collar workers } & 21 & 26 & 3 \\ \hline \end{array} $$ Using a \(2.5 \%\) significance level, test the null hypothesis that the distributions of opinions are homogeneous for the two groups of workers.

How is the expected frequency of a category calculated for a goodness-of-fit test? What are the degrees of freedom for such a test?
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