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

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

Ans:

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

Please see the discriptive summary with 95% confidence interval below and treatment means are highlited in yellow color.

	1978	1979	1980	1981
Mean	8.452	8.800	7.503	8.525
Standard Error	1.665	0.514	0.525	1.110
Median	7.255	8.630	7.430	7.320
Mode	#N/A	7.250	7.600	#N/A
Standard Deviation	4.078	1.543	1.575	3.509
Sample Variance	16.633	2.382	2.480	12.312
Kurtosis	2.567	-1.450	1.928	-0.662
Skewness	1.586	0.357	1.113	0.876
Range	11.000	4.130	5.280	9.380
Minimum	5.000	7.000	5.570	5.000
Maximum	16.000	11.130	10.850	14.380
Sum	50.710	79.200	67.530	85.250
Count	6.000	9.000	9.000	10.000
Largest(1)	16.000	11.130	10.850	14.380
Smallest(1)	5.000	7.000	5.570	5.000
Confidence Level(95.0%)	4.280	1.186	1.210	2.510

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

Please see the Ftest summary at .05 level of significance below

F-Test	1978	1979	1980	1981
Mean	8.452	8.800	7.503	8.525
Variance	16.633	2.382	2.480	12.312
Observations	6.000	9.000	9.000	10.000
df	5.000	8.000	8.000	9.000
F	6.983		0.201
P(F<=f) one-tail	0.009		0.017
F Critical one-tail	3.687		0.295

g. Write the normal equations for the data?

Normal Equation is an analytical approach to Linear Regression with a Least Square Cost Function. We can directly find out the value of θ without using Gradient Descent. Following this approach is an effective and a time-saving option when are working with a dataset with small features.

Normal Equation is a follows :

In the above equation,
θ : hypothesis parameters that define it the best.
X : Input feature value of each instance.
Y : Output value of each instance.