Question

The article "Religion and Well-Being Among Canadian University Students: The Role of Faith Groups on Campus" (Journal for the Scientific Study of Religion \([1994]: 62-73\) ) compared the self-esteem of students who belonged to Christian clubs and students who did not belong to such groups. Each student in a random sample of \(n=169\) members of Christian groups (the affiliated group) completed a questionnaire designed to measure self-esteem. The same questionnaire was also completed by each student in a random sample of \(n=124\) students who did not belong to a religious club (the unaffiliated group). The mean self-esteem score for the affiliated group was \(25.08\), and the mean for the unaffiliated group was \(24.55 .\) The sample standard deviations weren't given in the article, but suppose that they were 10 for the affiliated group and 8 for the unaffiliated group. Is there evidence that the true mean self- esteem score differs for affiliated and unaffiliated students? Test the relevant hypotheses using a significance level of \(.01\).

Short answer

Based on the hypothesis testing with a significance level of 0.01, there was insufficient evidence to conclude that the true mean self-esteem score differs between the affiliated and unaffiliated groups. The test statistic (~0.553) did not exceed the critical values (~±2.58), hence the null hypothesis was not rejected.

Step-by-step solution

Set up the hypotheses

The null hypothesis (H0) is that the true mean self-esteem scores of the two groups are equal, and the alternative hypothesis (Ha) is that they are not equal. That is: \(H0: µ1 = µ2\) and \(Ha: µ1 ≠ µ2\) where µ1 and µ2 represent the true mean self-esteem scores of the affiliated and unaffiliated group, respectively.

Calculate the standard error

The standard error of the difference between the two sample means can be calculated using the formula \(\sqrt{{s1^2/n1 + s2^2/n2}}\), where \(s1 = 10\) (the standard deviation of the affiliated group), \(s2 = 8\) (the standard deviation of the unaffiliated group), \(n1 = 169\) (the sample size of the affiliated group), and \(n2 = 124\) (the sample size of the unaffiliated group). This equals to approximately 0.958.

Calculate the test statistic

The test statistic is calculated as the observed difference in sample means, divided by the standard error. Here, the observed difference in sample means (\(x1 - x2\)) is \(25.08 - 24.55 = 0.53\). So the test statistic is \(0.53/0.958 \approx 0.553\).

Determine the critical value

Using a significance level of 0.01, and given that this is a two-tailed test (because the alternative hypothesis is that the means are 'not equal'), the critical value from the t-distribution table is approximately ±2.58 (using degrees of freedom \(df = min(n1-1,n2-1) = min(168,123) = 123\)).

Make a decision

Since the test statistic does not fall in the critical region (i.e., it is not less than -2.58 or greater than 2.58), we will fail to reject the null hypothesis, meaning there is insufficient evidence at this significance level to conclude that the true mean self-esteem score differs for affiliated and unaffiliated students.