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Chapter 15: Problem 30

Explain the difference between situations that would lead to a chi-square test for homogeneity and those that would lead to a chi-square test for independence.

Short Answer

Expert verified
The difference primarily lies in the purpose and setup of the experiment. Chi-square test for homogeneity is used when you're comparing the same variable across different groups within a population, while Chi-square test for independence is used when you're observing two categorical variables within the same population to see if they are related.

Step by step solution

01

Define Chi-square test for Homogeneity

Firstly, the chi-square test for homogeneity is used when the goal is to compare the distribution of outcomes in different groups. These groups are usually a subset of the same population and each group is surveyed only once. Consider a scenario where three flavours of a soda (cherry, lemon, and vanilla) are tested in three different cities to see if the preference is the same in all cities. This is a case for the chi-square test for homogeneity as it is testing whether different groups (cities) follow the same distribution (soda preference)
02

Define Chi-square test for Independence

Now, the chi-square test for independence is used when you have two categorical variables from the same population. You want to see if there is a significant relationship between those two variables. Consider a scenario where a researcher wants to test whether gender is independent of career choice. This involves a single population (all people) but looks at two distinct factors (gender and career choice). Thus, this is a case for using a chi-square test for independence as it checks if two factors (career choice and gender) are independent.
03

Summary of Differences

In summary, your use of chi-square test for homogeneity versus chi-square test for independence is based on whether you're comparing different categorical variables from the same group (independence) or comparing the same variable from different groups (homogeneity). Make sure to appropriately define your populations and groups when choosing a test.

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Most popular questions from this chapter

Give an example of a situation where it would be appropriate to use a chi- square test of homogeneity. Describe the populations that would be sampled and the variable that would be recorded.

The press release titled "Nap Time" (pewresearch.org, July 2009) described results from a nationally representative survey of 1,488 adult Americans. The survey asked several demographic questions (such as gender, age, and income) and also included a question asking respondents if they had taken a nap in the past 24 hours. The press release stated that \(38 \%\) of the men surveyed and \(31 \%\) of the women surveyed reported that they had napped in the past 24 hours. For purposes of this exercise, suppose that men and women were equally represented in the sample. a. Use the given information to fill in observed cell counts for the following table: b. Use the data in the table from Part (a) to carry out a hypothesis test to determine if there is an association between gender and napping. c. The press release states that more men than women nap. Although this is true for the people in the sample, based on the result of your test in Part ( \(b\) ), is it reasonable to conclude that this holds for adult Americans in general? Explain.

The following passage is from the paper "Gender Differences in Food Selections of Students at a Historically Black College and University" (College Student Journal \([2009]: 800-806):\) Also significant was the proportion of males and their water consumption ( 8 oz. servings) compared to females \(\left(X^{2}=8.166, P=.086\right) .\) Males came closest to meeting recommended daily water intake ( 64 oz. or more) than females \((29.8 \%\) vs. \(20.9 \%)\) This statement was based on carrying out a chi-square test of homogeneity using data in a two-way table where rows corresponded to gender (male, female) and columns corresponded to number of servings of water consumed per day, with categories none, one, two to three, four to five, and six or more. a. What hypotheses did the researchers test? What is the number of degrees of freedom associated with the reported value of the \(X^{2}\) statistic? b. The researchers based their statement on a test with a significance level of 0.10 . Would they have reached the same conclusion if a significance level of 0.05 had been used? Explain.

A certain genetic characteristic of a particular plant can appear in one of three forms (phenotypes). A researcher has developed a theory, according to which the hypothesized proportions are \(p_{1}=0.25, p_{2}=0.50,\) and \(p_{3}=0.25 .\) A random sample of 200 plants yields \(X^{2}=4.63\). a. Carry out a test of the null hypothesis that the theory is correct, using level of significance \(\alpha=0.05\). b. Suppose that a random sample of 300 plants had resulted in the same value of \(X^{2}\). How would your analysis and conclusion differ from those in Part (a)?

What is the approximate \(P\) -value for the following values of \(X^{2}\) and df? a. \(X^{2}=14.44, \mathrm{df}=6\) b. \(X^{2}=16.91, \mathrm{df}=9\) c. \(X^{2}=32.32, \mathrm{df}=20\)
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