Question

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.

Short answer

The researchers tested the hypothesis that the proportion of males and females who consume different quantity of water servings is the same against the alternative that the proportions vary. The degrees of freedom for this chi-square test are 4. At the significance level of 0.10, the researchers found significant evidence to reject the null hypothesis. However, at a significance level of 0.05, the researchers would not have found enough evidence to reject the null hypothesis.

Step-by-step solution

Understand the Hypotheses

For a chi-square test of homogeneity, the null hypothesis \((H_0)\) asserts that the populations are the same, or homogenous, across all groups. In this case, \((H_0)\) would be that the proportion of males and females who consume different quantity of water servings is the same. The alternative hypothesis \((H_A)\) is that at least one of the populations differs from the others. Hence, \((H_A)\) would be that the proportion of males and females who consume different quantity of water servings is different.

Calculate Degrees of Freedom

The degrees of freedom for a chi-square test are equal to \((r -1) * (c - 1)\) where r is the number of rows and c is the number of columns. Given that gender (male or female) provides 2 rows and the number of servings forms 5 columns (none, one, two to three, four to five, and six or more), plug these values into the equation to get the degrees of freedom = \((2-1) * (5-1) = 4\).

Analyze the Impact of Different Significance Levels

The researchers' p-value was 0.086 and was considered significant at a level of 0.10. This means that the null hypothesis would be rejected as the p-value is less than the significance level. However, if a significance level of 0.05 had been used, the p-value would not be considered significant as it is greater than the significance level. Therefore, the researchers would not have reached the same conclusion if a significance level of 0.05 had been used.