Question

4. Data were collected on student teachers relative to their use of certain teaching strategies that had been presented to them in preservice education. There were 28 student teachers who had learned to use the strategies (9 in 1979, 9 in 1980, and 10 in 1981). In 1978 there were 6 teachers who did not learn to use the strategies, and they were used as a control group. The investigator recorded the average number of strategies used per week by each of the student teachers during their student teaching assignments. The investigator wanted to know whether the number of strategies used by the student teachers was different among the years.

Average Number of Different Strategies Used
Control 1978	1979	1980	1981
6.88	7.25	10.85	7.29
5.4	10.5	7.43	14.38
16	8.43	6.71	6
9.8	8.63	7.6	5
7.63	8.63	7.6	5.38
5	7	5.57	14.14
	11.13	8.71	9.25
	7.25	5.86	5.71
	10.38	7.2	7.35
			10.75

a. Write the linear statistical model for this study, and explain the model components.

b. State the assumptions necessary for an analysis of variance of the data.

c. Compute the analysis of variance for the data.

d. Compute the least squares means and their standard errors for each treatment.

e. Compute the 95 % confidence interval estimates of the treatment means .

f. Test the hypothesis of no differences among means of the four treatments with the F test at the .05 level of significance.

g. Write the normal equations for the data.

Answer 1

1. For the linear statistical model, you need one dependent variable and one or more independent variables. According to the data given, we can see that, the dependent variable is the average number of strategies used let us consider it as S, and the dependent variable would be the year in which strategies are framed Let us consider it as Y

Here, S is a numerical value, whereas Y is a categorical variable which is a year. For categorical variable, we need to have dummy variables for all the categories in it. Therefore the four dummy variables can be Y1, Y2, Y3 and Y4 for the years 1978, 1979, 1980 and 1981.

Therefore the linear statistical model can be as below

S =

Where as is the error

2. The underlying assumptions for ANOVA for the data are as below

a) Normality of the data, the data in each group i.e., under each year should be normal

b) The variance among groups should be similia

c) The data taken under different groups/years should be independent

3. ANOVA output conducted between different years is as shown

Since F value is < F-criticle, we accept the null hypothesis, that all means of different years are equal

So from this we can conclude that, means of different years are significantly similar