Question

The XQ3 scale of social functioning assesses the ability of an individual to interact or adapt in a normal or usual way to social environments.  Higher scores indicate better social functioning. The average score on the XQ3 for the US population of adults is 100 with a standard deviation of 20. These can be considered the national norms (i.e., population values). A researcher is interested in whether the social functioning of family members who are caregivers of patients who suffer from Alzheimer’s Disease differs from that of US population norms. He assesses a sample of 100 caregivers of Alzheimer’s patients. The sample mean is 96.2. Assume the sampling distribution of the mean is normal.

            a. Consider this study from the framework of statistical hypothesis testing. What are the null and alternative hypotheses?

            b. Conduct a statistical test of the researcher’s hypothesis. In addition to showing your work, be sure to include in your answer: the observed value of the test statistic, the critical value of the test statistic, and your decision about whether or not to reject the null hypothesis.

Answer 1

A) H₀:   = 100

H₁:   100

B) The test statistic z = ()/()

= (96.2 - 100)/(20/)

= -1.9

At alpha = 0.05, the critical values are +/- z_0.025 = +/- 1.96

Since the test statistic value is not less than the lower critical value (-1.9 > -1. 96), so we should not reject the null hypothesis.