Question

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4.14. The survival rate (in percentage) of bull semen after storage is measured at various combinations of concentrations of three materials (additives) that are thought to increase the chance of survival. The data listed below are given in the file bsemen % Survival % Weight 1 % Weight 2 % Weight 3 (x3) (x2) (xi) 1.74 6.32 25.5 5.30 10.80 31.2 5.42 9.40 8.41 4.63 11.60 6.22 25.9 7.20 38.4 10.52 8.50 1.19 1.22 18.4 9.40 26.7 9.90 5.85 % Survival % Weight 1 % Weight 2 % Weight 3 (x2) (xi) (x3) 4.10 6.62 26.4 8.00 6.32 25.9 32.0 9.10 8.72 4.42 4.08 8.70 7.60 4.83 4.15 25.2 9.20 10.15 1.72 39.7 9.40 3.12 35.9 7.60 26.5 1.70 5.30 8.20 Assume the model y = At Axi + Ax2 + d. Construct a 90% confidence interval for i. the predicted mean value of y when Xi-3, X2-8, and X3-9; ii. the predicted individual value of y when Xi-3, X2-8, and X3-9 e. Construct the ANOVA table and test for a significant linear relationship between y and the three predictor variables.

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.95357488
R Square	0.909305052
Adjusted R Square	0.879073403
Standard Error	2.107691293
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	3	400.8510444	133.617	30.07792	5.07165E-05
Residual	9	39.98126328	4.442363
Total	12	440.8323077
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	39.48151839	5.985544416	6.596145	9.97E-05	25.94127621	53.02176056
X Variable 1	1.0092246	0.194088687	5.199812	0.000564	0.570165487	1.448283714
X Variable 2	-1.87265715	0.271797595	-6.8899	7.15E-05	-2.487506028	-1.257808277
X Variable 3	-0.36669973	0.62737472	-0.5845	0.573243	-1.785919946	1.052520486

d. У-39.48 15 + 1.0092x1-1.8727T2-0.3667x3. where, βι = 1.0092, standard error of β1 = se(A) = 0.1941; B2 =-1.8727, standard error of β2 = se(3) = 0.2718; Въ--03667, standard error of A-se(33)-0.6274; i, 90% C.I. for the predicted mean value oy y when x,-3, x2-8, x3- 9 is MSE3 0.05,9 MSE3 13 (22.7427, 25.7117), where, yo 39.4815 +1.0092 3 1.8727 8-0.3667 9 24.2272, MSE = mean surn of square error = 4.4424 (see ANOVA table) T4.5715, 2 6.2938, T3-8.8769, t9 1.8331

(ii) 90% P.I. for the y when x1-3, x2-8, x3-9 is 90% С.Г. for the predicted mean value oy y when x1-3, x2-8, x3-9 is MSE3 MSE ri Ti se (21.8684, 26.5860)

e.

ANOVA
	df	SS	MS	F	Significance F
Regression	3	400.8510444	133.617	30.07792	5.07165E-05
Residual	9	39.98126328	4.442363
Total	12	440.8323077

Since p-value<0.05 so we conclude that linear relationship between y and the three predictors is significant.