1) A study was done of personality characteristics of 100 students who were tested at the beginning and end of their first year of college. The researchers reported the results in the following table:

Personality Scale / Fall / Spring / Difference
M / SD / M / SD / M / SD
Anxiety / 16.82 / 4.21 / 15.32 / 3.84 / 1.50 / 1.85
Depression / 89.32 / 8.39 / 86.24 / 8.91 / 3.08 / 4.23
Introversion / 59.89 / 6.87 / 60.12 / 7.11 / 0.23 / 2.22
Neuroticism / 38.11 / 5.39 / 37.32 / 6.02 / 0.89 / 4.21

For each of the 4 personality scales, perform a t-test to determine whether there was a difference between Fall and Spring (use an alpha level of 0.05, and assume 2-tailed tests). State all of your hypotheses and conclusions.

Each of these situations use a repeated measures design. So the appropriate t-test to use is the related (dependent) samples t-test

Anxiety

H0: The anxiety scores do not differ from Fall to Spring: D = 0

HA: The anxiety scores do differ from Fall to Spring: D ≠ 0

The study predicted difference between Fall and Spring scores, this means that we should use a 2-tailed test. Also the problem states that  = 0.05. So that makes our tcrit (with df = n – 1 = 99) = ±2.0 (the on-line table doesn’t include 99, so I selected the number from df=60, the highest available without going over).

We’re given the difference scores and the standard deviation of the difference scores. Now we need to compute the estimated standard error based on the difference scores.

Now we need to compute the t-test

Now we compare this tobs with the tcrit

8.11 is in the critical region (which is scores beyond ±2.0), so we reject the null hypothesis and conclude that there is a difference in Anxiety between Fall and Spring semesters.

Depression

H0: The depression scores do not differ from Fall to Spring: D = 0

HA: The depression scores do not differ from Fall to Spring: D ≠ 0

The study predicted difference between Fall and Spring scores, this means that we should use a 2-tailed test. Also the problem states that  = 0.05. So that makes our tcrit (with df = n – 1 = 99) = ±2.0 (the on-line table doesn’t include 99, so I selected the number from df=60, the highest available without going over).

We’re given the difference scores and the standard deviation of the difference scores. Now we need to compute the estimated standard error based on the difference scores.

Now we need to compute the t-test

Now we compare this tobs with the tcrit

7.28 is in the critical region (which is scores beyond ±2.0), so we reject the null hypothesis and conclude that there is a difference in depression between Fall and Spring semesters.

Introversion

H0: The introversion scores do not differ from Fall to Spring: D = 0

HA: The introversion scores do not differ from Fall to Spring: D ≠ 0

The study predicted difference between Fall and Spring scores, this means that we should use a 2-tailed test. Also the problem states that  = 0.05. So that makes our tcrit (with df = n – 1 = 99) = ±2.0 (the on-line table doesn’t include 99, so I selected the number from df=60, the highest available without going over).

We’re given the difference scores and the standard deviation of the difference scores. Now we need to compute the estimated standard error based on the difference scores.

Now we need to compute the t-test

Now we compare this tobs with the tcrit

8.11 is not in either critical region (which is scores beyond ±2.0), so we fail to reject the null hypothesis and conclude that there is not a difference in introversion between Fall and Spring semesters.

Neuroticism

H0: The neuroticism scores do not differ from Fall to Spring: D = 0

HA: The neuroticism scores do not differ from Fall to Spring: D ≠ 0

The study predicted difference between Fall and Spring scores, this means that we should use a 2-tailed test. Also the problem states that  = 0.05. So that makes our tcrit (with df = n – 1 = 99) = ±2.0 (the on-line table doesn’t include 99, so I selected the number from df=60, the highest available without going over).

We’re given the difference scores and the standard deviation of the difference scores. Now we need to compute the estimated standard error based on the difference scores.

Now we need to compute the t-test

Now we compare this tobs with the tcrit

2.11 is in the critical region (which is scores beyond ±2.0), so we reject the null hypothesis and conclude that there is a difference in neuroticism between Fall and Spring semesters.