Question

The article "Fish Oil Staves Off Schizophrenia" (USA Today, February 2,2010 ) describes a study in which 81 patients ages 13 to 25 who were considered at risk for mental illness were randomly assigned to one of two groups. Those in one group took four fish oil capsules daily. Those in the other group took a placebo. After 1 year, \(5 \%\) of those in the fish oil group and \(28 \%\) of those in the placebo group had become psychotic. Is it appropriate to use the twosample \(z\) test to test hypotheses about the difference in the proportions of patients receiving the fish oil and the placebo treatments who became psychotic? Explain why or why not.

Short answer

No, it isn't appropriate to use the two-sample z test in this case because the condition that \(n \cdot p > 5\) is not met for the fish oil group.

Step-by-step solution

Identifying the Proportions

Firstly, identify the proportions. The proportion of patients who became psychotic in the fish oil group is \(5%\) while in the placebo group it is \(28%\). This is represented as \(p1 = 0.05\) and \(p2 = 0.28\), respectively.

Recognize the Nature of the Sample

We know from the problem that the samples were randomly assigned, and hence, independent of each other. Therefore, the groups are independent, and we can consider \(p1\) and \(p2\) to be independent.

Using the Two-Sample Z Test Conditions

We need to verify whether all the conditions for a two-sample \(z\) test are met. Specifically, we need to ensure that: 1) The sample groups are independent. 2) For both groups, \(n \cdot p\) and \(n \cdot (1-p)\) need to be more significant than 5. The sample size \(n_1\) (fish oil) and \(n_2\) (placebo) is 81 each. Notice that \(n_1 \cdot p_1 = 81 \cdot 0.05 = 4.05\) and \(n_1 \cdot (1 - p_1) = 81 \cdot (1 - 0.05) = 76.95\). Similarly, \(n_2 \cdot p_2 = 81 \cdot 0.28 = 22.68\) and \(n_2 \cdot (1 - p_2) = 81 \cdot (1 - 0.28) = 58.32\). However, the condition \(n \cdot p > 5\) is not met for the fish oil group. Therefore, it is not suitable to use a two-sample \(z\) test in this case.