Question

To examine changes in teaching self-efficacy, 10 teachers were measured on their self-efficacy toward teaching at the beginning of their teaching career and at the end of their first year and third years of teaching. The teaching self-efficacy scale ranged from 0 to 100 with higher scores reflecting greater teaching self-efficacy. The data are shown here.

A. Conduct a one-factor repeated measures ANOVA to determine mean differences across time, using alpha = .05.

B. Then, use the Bonferroni method to detect if and/or where the differences are among the time points.

Give necessary tables/figures for each.

Subject	Beginning Year 1	End Year 1	End Year 3
1	35	50	45
2	50	75	82
3	42	51	56
4	70	72	71
5	65	50	81
6	92	42	69
7	80	82	88
8	78	76	79
9	85	60	83
10	64	71	89

Answer 1

1. This is a simple problem related ot ascertaining the significance of the differences of the means of three samples given one factor repeated measures ANOVA.

THe details of the test conducted along with its steps are:-

Sample is

Subject	Beginning Year 1	End Year 1	End Year 3
1	35	50	45
2	50	75	82
3	42	51	56
4	70	72	71
5	65	50	81
6	92	42	69
7	80	82	88
8	78	76	79
9	85	60	83
10	64	71	89

Thus we observe that the the difference of means is NOT siginificant at 95 % confidence.

B.Benferroni test for testing the significance of differences

We follow the following steps in at order to perform the berferroni test.

1. First perform the tukey's post hoc test to determine if the samples are different pair wise. to do so , We first establish the critical value of the Tukey-Kramer HSD Q statistic based on the k=3 treatments and ν=27 degrees of freedom for the error term, for significance level α= 0.01 and 0.05 (p-values)

The Q (critical) α= 0.05 =3.5058

This will be our reference point.

Next, we establish a Tukey test statistic from our sample columns to compare with the appropriate critical value

The Tukey-Kramer HSD Q-statistic, or simply the Tukey HSD Q-statistic is given as:

where the denominator in the above expression is:

Thus we determine the Q statistic for each pair (# nos. )

They are shown below along with Q statisitc ,corresponding p values and p critical.

Clearly non of the pairs are siginificantly different as per tukey's test.

2. Now define a new statistic T given for i ,j pair as

3. The test statistic T obtained above, along with the number of contrasts (pairs) q (3 in this case) being simultaneously compared, leads to the Bonferroni formula corresponding to an observed value of T in the context of simultaneous comparison of q contrasts

4. Keeping values get the berferroni p value for each pair and compare.

The detailed berferroni table along with indication of significance is given below.

Clearly non of the two pairs are siginificant in difference.