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
Fall
Spring
Difference
Scale
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 ttest 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.
sX 
sD
1.85

 0.185
nD
100
Now we need to compute the t-test
t obs 
D   D 1.50  0.0

 8.11
sD
0.185
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.
sX 
sD
4.23

 0.423
nD
100
Now we need to compute the t-test
t obs 
D   D 3.08  0.0

 7.28
sD
0.423
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.
sX 
sD
2.22

 0.222
nD
100
Now we need to compute the t-test
t obs 
D   D 0.23  0.0

 1.04
sD
0.222
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.
sX 
sD
4.21

 0.421
nD
100
Now we need to compute the t-test
t obs 
D  D 0.89  0.0

 2.11
sD
0.421
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.