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Chapter 6: Problem 191

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether people with a specific genetic marker are more likely to have suffered from clinical depression than people without the genetic marker, using the information that \(38 \%\) of the 42 people in a sample with the genetic marker have had clinical depression while \(12 \%\) of the 758 people in the sample without the genetic marker have had clinical depression.

Short Answer

Expert verified
First calculate the sample proportions and the pooled proportion. Formulate the null and alternate hypothesis. Finally, conduct the hypothesis test: calculate the standard error, the Z-score, and the associated p-value. If the p-value is less than the significance level, then we reject the null hypothesis 鈥� there is enough evidence to suggest that those with the genetic marker are more likely to have had depression. Otherwise, there is not enough evidence to suggest this.

Step by step solution

01

Calculate the relevant sample proportions and the pooled proportion

First, find the proportion of the sample with the genetic marker who have had clinical depression. This is given by the number with depression (38% of 42) divided by the total number with the genetic marker (42). Similarly, the proportion of the sample without the genetic marker who have had clinical depression is given by the number without depression (12% of 758) divided by the total number without the genetic marker (758). The pooled proportion is the total number with depression (from both samples) divided by the total number in both samples.
02

Formulate the null and alternate hypothesis

The null hypothesis is that there is no difference between the proportions 鈥� that is, the proportion of those with the genetic marker who have had depression is equal to the proportion of those without the genetic marker who have had depression. The alternative hypothesis is what we are trying to prove 鈥� that the proportion of those with the genetic marker who have had depression is greater than the proportion of those without the genetic marker who have had depression.
03

Complete the hypothesis test

Assume the null hypothesis is true. Using the normal distribution, calculate the standard error of the pooled proportion. Then calculate the Z-score, which is the difference between the sample proportion (the sample with the genetic marker) and the pooled proportion, divided by the standard error. If the Z-score corresponds to a probability (p-value) less than the significance level (usually 0.05), then reject the null hypothesis. If not, fail to reject the null hypothesis.

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