Question

The article "Fish Oil Staves Off Schizophrenia" (USA Today, February 2, $2\mathrm{O}1\mathrm{O}$ ) describes a study in which 81 patients age 13 to 25 who were considered atrisk for mental illness were randomly assigned to one of two groups. Those in one group took four fish oil capsules daily. The other group took a placebo. After 1 year, $5\mathrm{\%}$ of those in the fish oil group and $28\mathrm{\%}$ of those in the placebo group had become psychotic. Is it appropriate to use the two-sample $z$ test of this section 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 is not appropriate to use the two-sample z-test because the condition for normality isn't satisfied. The sample doesn't meet the criteria requiring at least 10 successes and 10 failures.

Step-by-step solution

Checking the Randomization Condition

Randomization is a basic condition for most statistical tests. In the problem, it's stated that the patients were randomly assigned to the fish oil or placebo groups, hence satisfying this condition.

Checking the Normal Condition

The normal condition requires that we should expect at least 10 successes and 10 failures in our sample. For the fish oil group, there were $81\times 5\mathrm{\%}=4.05$ psychotic incidents (successes) and $81\times 95\mathrm{\%}=76.95$ non-psychotic incidents (failures). In the placebo group, there were $81\times 28\mathrm{\%}=22.68$ psychotic incidents (successes) and $81\times 72\mathrm{\%}=58.32$ non-psychotic incidents (failures). Hence, both conditions are not met as the number of successes and failures are not all above 10.

Conclusion

Since the normal condition isn't met, it isn't appropriate to utilize the two-sample z-test for this problem.

Key concepts

Two-sample z-test

The two-sample z-test is a statistical method used to determine if there is a significant difference between the proportions of two independent groups. In simple terms, this test compares the difference in success rates between two groups to see if this difference is due to chance or if it reflects a real difference in behavior or outcome.
When conducting a two-sample z-test, we assume that both samples (or groups) follow a normal distribution. This means that the data should reflect the typical bell-shaped curve when plotted.

• The test requires the data to meet certain conditions, such as random sampling and adequate sample size.
• It uses a z-distribution (which is why it's named a z-test) to estimate the probability of observing the given difference if the null hypothesis is true.
• Common null hypothesis assumes no difference between the groups' proportions.

In the case of comparing the effects of fish oil and a placebo in preventing psychosis, the two-sample z-test would analyze if there is a statistically significant difference in the proportion of psychotic cases between the two groups.

Randomization

Randomization is the process of randomly assigning participants to different groups in an experiment to ensure that each group is similar across all characteristics except for the factor being tested. This helps to eliminate bias and confounding variables, which can skew results and conclusions.
In the fish oil study, participants were randomly assigned to either the fish oil group or the placebo group. This step is crucial as it ensures that any differences in outcomes between groups can be attributed to the treatment effect rather than other variables.

• Randomization improves the reliability of the study results.
• It helps in balancing both known and unknown factors across the groups.
• It allows for the generalization of the findings to a broader population.

With random assignment, the study guarantees that any psychological or physiological differences between participants do not systematically affect the outcome, thereby fulfilling the randomization condition needed for statistical tests like the two-sample z-test.

Normal Condition

The normal condition refers to the assumption that the sampling distribution of the difference between two proportions is approximately normal, which is crucial for the accuracy of the z-test. For this approximation to be valid, we need each group's expected number of successes and failures to be at least 10.
In mathematical terms, if

• $n_1\times p_1\ge 10$ and $n_1\times (1-p_1)\ge 10$
• $n_2\times p_2\ge 10$ and $n_2\times (1-p_2)\ge 10$

are true, where $n_1$ and $n_2$ are the sample sizes, and $p_1$ and $p_2$ are the sample proportions.
In the study mentioned, the fish oil group does not meet the normal condition since it has less than 10 expected successes. Due to this, the sampling distribution might not be normal, which suggests that the two-sample z-test may not be appropriate. This highlights the importance of checking assumptions before conducting statistical tests.

Proportion Comparison

Proportion comparison involves determining whether there is a significant difference between the proportions of success (or failure) in two different groups. It is a common analysis in research, especially in fields like medicine and social sciences, where understanding group differences is crucial.
For example, in the fish oil study, researchers want to compare the proportion of psychotic outcomes between participants taking fish oil and those taking a placebo. The proportion for each group is

• Fish oil group: $5\mathrm{\%}$
• Placebo group: $28\mathrm{\%}$

The goal is to see if the seemingly lower psychotic rate in the fish oil group is statistically significant. The difficulty arises in deciding whether the observed difference is due to the treatment effect or merely a product of random variation. Statistical tests like the two-sample z-test are used to clarify this distinction under appropriate conditions. However, it is crucial to evaluate whether these conditions — such as normality and randomization — are met before proceeding with such tests.