Question

For each problem, compute the appropriate statistical test and report your conclusion. Your answer should include the following:

a. State the null hypothesis.

b.   Identify the appropriate statistical test

c.   Calculate the appropriate test statistic. Make sure to report the degrees of freedom for the statistical test (if appropriate). You must show your work to receive full credit.

d.   State your conclusions (use a two-tailed test with a = .05 for all tests).

e.   Compute an effect size.

A therapist would like to evaluate an intervention for treating depression. A sample of n = 20 depressed clients is obtained, and each person’s level of depression is measured using a standardized questionnaire before they begin the therapy program. Two weeks after therapy, each client’s level of depression is measured again. The average level of depression dropped by 4.0 points following therapy. The difference scores had a variance of 64.

Answer 1

Solution :

a) The null and alternative hypotheses are as follows :

i.e. The mean depression levels before and after the therapy program are equal.

i.e. The mean depression levels before and after the therapy program are not equal.

b) To test the hypothesis we shall use paired t-test. The test statistic is given as follows :

Where, is sample mean of the differences, s² is sample variance of differences, n is sample size.

c) We have,  

The value of the test statistic is 2.2361.

Degrees of freedom = (n - 1) = (20 - 1) = 19

d) Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic. The two-tailed p-value is given as follows :

p-value = 2.P(T > t)

p-value = 2.P(T > 2.2361)

p-value = 0.0375

The p-value is 0.0375.

Decision :

p-value = 0.0375

Significance level = 0.05

(0.0375 < 0.05)

Since, p-value is less than the significance level of 0.05, therefore we shall reject the null hypothesis (H​​​​​​​​​​​0) at 0.05 significance level.

Conclusion : At 0.05 significance level, there is sufficient evidence to conclude that the therapy program is effective.

e) The effect size is given as follows :

Where, Where, is sample mean of the difference, s is sample standard deviation of differences.

We have,  

The effect size is 0.50.

Please rate the answer. Thank you.