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Chapter 15: Problem 27

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.

Short Answer

Expert verified
The Chi-square test of homogeneity can be used in a situation where a company wants to understand the preference for different soda flavors in different cities. The populations sampled would be the residents of the three cities, and the variable recorded would be the preferred soda flavor.

Step by step solution

01

Understanding Chi-Square Test of Homogeneity

A chi-square test of homogeneity is applied when there's a need to determine whether different populations have the same distribution for one variable. It's a non-parametric statistical test. Non-parametric means it makes no assumptions about population parameters and the distribution's shape. Hence, it could be used in various situations, including market research, clinical trials, voter preferences, among others.
02

Example Scenario

Let's consider an example scenario in market research. Suppose a company sells soda drink in three different flavors: cola, lemon, and orange, and is planning to launch a new flavor into three different cities (city A, city B, city C). Before investing further in the production of the new flavor, the company must first understand which existing flavor is most preferred in each city. The company will conduct taste tests in different cities and record the flavor each participant prefers.
03

Describing the Populations and Variable

In this situation, the populations sampled would be the residents of the three different cities (city A, city B, city C). The variable recorded would be the preferred soda flavor (cola, lemon, or orange). Once the data is collected, a chi-square test of homogeneity can be performed to analyze if there is a significant difference among the populations in their preference for soda flavors.

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Most popular questions from this chapter

Think about how you would answer the following question. Next Wednesday's meeting has been moved forward two days. What day is the meeting now that it has been rescheduled? This question is ambiguous, as "moved forward" can be interpreted in two different ways. Would you have answered Monday or Friday? The authors of the paper "Even Abstract Motion Influences the Understanding of Time" (Metaphor and Symbol [2011]\(: 260-271)\) wondered if the answers Monday and Friday would be provided an equal proportion of the time. Each student in a random sample of students at Stanford University was asked this question, and the responses are summarized in the following table. The authors of the paper used a chi-square goodness-of-fit test to test the null hypothesis \(H_{0}: p_{1}=0.50, p_{2}=0.50\), where \(p_{1}\) is the proportion who would respond Monday, and \(p_{2}\) is the proportion who would respond Friday. They reported \(X^{2}=11.00\) and \(P\) -value \(<0.001\). What conclusion can be drawn from this test?

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.

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types \(1,2,3,\) and 4 are advertised to be \(40 \%, 30 \%, 20 \%,\) and \(10 \%,\) respectively. A random sample of nuts is selected, and each one is categorized by type. a. If the sample size is 200 and the resulting test statistic value is \(X^{2}=19.0,\) what conclusion would be appropriate for a significance level of \(0.001 ?\) b. If the random sample had consisted of only 40 nuts, would you use the chi- square goodness-of-fit test? Explain your reasoning.

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types \(1,2,3,\) and 4 are advertised to be \(40 \%, 30 \%, 20 \%,\) and \(10 \%,\) respectively. A random sample of nuts is selected, and each one is categorized by type. a. If the sample size is 200 and the resulting test statistic value is \(X^{2}=19.0,\) what conclusion would be appropriate for a significance level of \(0.001 ?\) b. If the random sample had consisted of only 40 nuts, would you use the chi- square goodness-of-fit test? Explain your reasoning.
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