In: Statistics and Probability
Additionally, researchers hypothesized that depression would increase after divorce. To test this hypothesis, they collected data on depression from 71 individuals at two time points: 1 year before divorce and 1 year after divorce. They observed that the average depression at time 1 was 2.1 and the average depression at time 2 was 3.4. The sum of squares for the difference scores (SSD) was 230. Use the critical-value approach for a dependent samples t-test (where α = .001) to determine if there is a significant increase in depression after divorce. 8. Identify the alternative hypothesis 9. List your degrees of freedom 10. List your estimated standard error 11. List your t test statistic (where: Time 2 – Time 1) 12. List your critical t value(s) 13. Is the increase in depression following divorce statistically significant? (Yes/No) 14. Calculate Cohen’s d for the difference in depression A group of researchers also wants to determine if divorce can affect satisfaction in other domains, such as job satisfaction. To test this hypothesis, they collected data on job satisfaction from 195 individuals at two time points: 1 year before divorce and 1 year after divorce. They observed that the average job satisfaction at time 1 was 6.7 and the average job satisfaction at time 2 was 6.2. The sum of squares for the difference scores (SSD) was 350. Additionally, life satisfaction and job satisfaction were strongly related (r = .46). Use a confidence interval (where α = .01) to determine if there are any significant differences in job satisfaction following divorce. 15. Identify the alternative hypothesis 16. List your estimated standard error 17. List the t-value used to construct your confidence interval 18. List the lower bound of your confidence interval (where: Time 2 – Time 1) 19. List the upper bound of your confidence interval (where: Time 2 – Time 1) 20. Is the difference in job satisfaction following divorce statistically significant? (Yes/No) 21. Calculate Cohen’s d for the difference in job satisfaction 22. Determine the t value for the correlation between job and life satisfaction
8.
Alternative Hypothesis H0: Average depression after divorce is greater than that before divorce.
9.
Degree of freedom = n - 1 = 71 - 1 = 70
10.
Standard deviation of difference = SSD / (n - 1) = 230 / 70 = 3.2857
Standard error of mean difference = 3.2857 / = 0.39
11.
t test statistic = (t2 - t1) / standard error = (3.4 - 2.1) / 0.39 = 3.33
12.
Critical t value for df = 70 and = 0.001 is, 3.21
13.
Since the test statistic t is greater than the critical value, we reject null hypothesis and conclude that there is significant evidence that average depression after divorce is greater than that before divorce.
Answer is Yes, increase in depression following divorce is statistically significant.
14.
Cohen’s d = (t2 - t1) / standard deviation = (3.4 - 2.1) / 3.2857 = 0.3957
15.
Alternative Hypothesis H0: Average job satisfaction after divorce is different than that before divorce.
16.
Standard deviation of difference = SSD / (n - 1) = 350 / (195 - 1) = 1.804124
Standard error of mean difference = 1.804124 / = 0.1292
17.
Degree of freedom = n - 1 = 195 - 1 = 194
For two tail test, Critical t value for df = 194 and = 0.01/2 = 0.005 is, 2.60
18.
Estimated t2 - t1 = 6.2 - 6.7 = -0.5
Lower bound = -0.5 - 2.60 * 0.1292 = -0.83592
19.
Upper bound = -0.5 + 2.60 * 0.1292 = -0.16408
20,
Since the confidence interval does not contain the value 0, the difference in job satisfaction following divorce is statistically significant. Answer is Yes.
21.
Cohen’s d = |t2 - t1| / std deviation = |6.2 - 6.7| / 1.804124 = 0.2771428
22.
t value = r * sqrt(n-2) / sqrt(1 - r^2)
= 0.46 * sqrt(195 - 2) / sqrt(1 - 0.46^2)
= 7.20