A graduate student is performing a study on a new antidepressant. The drug is supposed to reduce depression, but the graduate student realizes that it may do nothing or even increase depression, so she decides to formulate nondirectional hypotheses and conduct a two-tailed test. First, she uses mu_general population to denote the average score on the depression inventory for the population of all depressed people, and mu_antidepressant to denote the average score on the depression inventory for the population of depressed people who take the new antidepressant. She knows that the average score for all depressed people is mu_general population = 25, with a standard deviation of sigma = 6. The graduate student identifies the null and alternative hypotheses as: H_0: mu_antidepressant = mu_general population H_1: mu_antidepressant notequalto mu_general population The sample of 36 depressed people who tried out the new antidepressant scored an average of 23.3. Since the graduate student knows the standard deviation of the scores on the depression inventory for the population of people who are depressed, she intends to use a hypothesis test that uses the z-score of the sample mean as the test statistic (also known as the z test). First, she wants to make sure all the required assumptions are satisfied. Which of the following conditions is not a required assumption for the z test?
The scores on the depression inventory follow a normal distribution or the sample size is large (at least 30).
The standard deviation of the scores on the depression inventory is the same for those who take the antidepressant and those who don't.
Each observation is independent of every other observation.
The sample size must be very large.
Members of the sample are selected randomly.
Use the Distributions tool to find the critical region for alpha = .05. The critical z-scores (the values for z-scores that separate the tails from the main body of the distribution, forming the critical regions) are___
Calculate the z statistic, and use the Distributions tool to evaluate the null hypothesis. The z statistic is The z statistic_ lie in the critical region for a two-tailed hypothesis test. Therefore, the null hypothesis is__ .

Since the sample size is 30, we do not need the condition and it's enough for the Z test provided that the population is not normally distributed. If the population is distributed normally, then any sample size is valid:

Then, the test statistics can be computed as follows:

[tex]Z = \dfrac{X - \mu}{\dfrac{\sigma}{\sqrt{n}} }[/tex]

[tex]Z = \dfrac{23.3 - 25}{\dfrac{6}{\sqrt{36}} }[/tex]

[tex]Z = \dfrac{23.3 - 25}{\dfrac{6}{6} }[/tex]

[tex]Z = -1.7[/tex]

At the level of significance 0.05

Since the test is two-tailed, the critical value [tex]Z_{\alpha/2} = Z_{0.025} = 1.96[/tex]

Provided that the Z value is not less than the critical value;

Then, we fail to reject the null hypothesis.

We conclude that there is no enough evidence that the drug reduces depression.