Question

The American Psychiatric Association (APA) claims that there is a difference in the proportions of doctors and lawyers who see a therapist. In a random sample of 160 doctors and 120 lawyers, they find that 25% of doctors and 35% of lawyers see a therapist. Test the APA’s claim at a 5% significance level.

a) Define Population 1 and Population 2.

b) Define the parameter and random variable of interest.

c) State the null and alternative hypotheses, and identify the claim.

d) Determine the distribution of the test statistic. (Check the relevant criteria.)

e) Calculate the test statistic.

f) Find the p-value.

g) State your decision.

h) State your conclusion.

Answer 1

In this hypothesis test, we are comparing the proportions of doctors and lawyers who see a therapist. The null hypothesis is that the proportions are equal, while the alternative hypothesis is that they are not equal. We use a test statistic and p-value to make a decision and draw a conclusion at a 5% significance level.

Step-by-step explanation:

a) Population 1 and Population 2:

Population 1 refers to the population of doctors, while Population 2 refers to the population of lawyers.

b) Parameter and Random Variable:

The parameter of interest is the proportion of doctors and lawyers who see a therapist. The random variable is the number of doctors and lawyers in the sample who see a therapist.

c) Null and Alternative Hypotheses:

Null hypothesis (H0): The proportions of doctors and lawyers who see a therapist are equal. Alternative hypothesis (Ha): The proportions of doctors and lawyers who see a therapist are not equal. The APA's claim is that there is a difference in the proportions.

d) Distribution of Test Statistic:

Since the sample sizes are large enough (160 doctors and 120 lawyers), we can assume that the sampling distribution of the proportions is approximately normal.

e) Test Statistic:

Calculate the test statistic using the formula: z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2)), where p1 is the proportion of doctors who see a therapist, p2 is the proportion of lawyers who see a therapist, p is the pooled proportion, n1 is the sample size of doctors, and n2 is the sample size of lawyers.

f) P-value:

Find the p-value associated with the test statistic using the standard normal distribution table or a statistical calculator.

g) Decision:

Compare the p-value to the significance level (5%). If the p-value is less than 0.05, reject the null hypothesis. If the p-value is greater than or equal to 0.05, fail to reject the null hypothesis.

h) Conclusion:

State the conclusion in the context of the problem. For example, if the null hypothesis is rejected, it can be concluded that there is sufficient evidence to support the APA's claim that there is a difference in the proportions of doctors and lawyers who see a therapist.