Question

The table below shows the scores of 25 Mathematics students in two mid-term tests and the Final test score.

Test 1	Test 2	Final Test
75	69	76
95	75	93
91	78	90
98	84	98
75	57	71
54	39	51
71	63	75
48	48	58
89	68	88
81	60	82
71	60	71
72	56	71
95	81	92
81	69	76
72	63	74
95	76	96
80	64	74
83	77	92
90	79	89
80	71	80
84	74	89
88	70	88
80	71	88
78	71	75
98	80	96

Tasks:

• Perform multiple linear regression analysis using Microsoft Excel, assuming final test score as the dependent variable and test 1 and test 2 scores as independent variables.
• Correctly state the regression equation.
• Evaluate the significance of the regression fit for the data.
• Determine if the regression coefficients are significant.
• Check if independent variables are collinear.
• Comment on the residual plots.
• Use the regression equation to predict final score for
  • test 1 score = 82, and
  • test 2 score = 69

Submission Details:

• Submit a 3–4 page Microsoft Word document, using APA style.
• The relevant Excel output should be copied to this document.

Answer 1

Regression Statistics
Multiple R	0.96081426
R Square	0.923164043
Adjusted R Square	0.916178956
Standard Error	3.460847936
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3165.936	1582.968	132.1621	5.511E-13
Residual	22	263.5043	11.97747
Total	24	3429.44
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	5.580586293	4.709803	1.184887	0.248704	-4.1869469	15.348119
X Variable 1	0.549573917	0.130968	4.196259	0.000374	0.2779637	0.8211841
X Variable 2	0.458689216	0.152598	3.00587	0.006505	0.1422207	0.7751578

Q. Correctly state the regression equation.

Ans: The required regression equation using the above excel output is

Final Test = 5.581 + 0.550 Test 1 + 0.459 Test 2

Q. Evaluate the significance of the regression fit for the data.

Ans: The p-value for the F-test from the above NAOVA table using excel output is 0.000. Hence, we can conclude that at least one independent variable has significant effect on the dependent variable Final Test.

Q. Determine if the regression coefficients are significant.

Ans:

	Coefficients	Standard Error	t Stat	P-value
Intercept	5.580586293	4.709803	1.184887	0.248704
X Variable 1	0.549573917	0.130968	4.196259	0.000374
X Variable 2	0.458689216	0.152598	3.00587	0.006505

The p-value for the significance test of slope coefficient for Test 1 indepemdnet avriable is 0.000374 and less than 0.05 significance level. Hence, we can conclude that Test 1 score has significant effect on Final Test score.

Again, the p-value for the significance test of slope coefficient for Test 2 indepemdnet avriable is 0.006505 and less than 0.05 significance level. Hence, we can conclude that Test 2 score has significant effect on Final Test score.

Q. Check if independent variables are collinear.

From the above scatter plot, increase the value of Test 1 increases the corresponding value of test 2 in a viceversa. Also, their correlation coefficient value is 0.9017. Hence, these two variables has high collinear

Q. Comment on the residual plots.

Ans:

The scatter plot of residuals VS Test 1 shows that these two variables are not correlated. Hence, the assumption of independence between the residulas VS independent variable is satisfied.

The scatter plot of residuals VS Test 2 shows that these two variables are not correlated. Hence, the assumption of independence between the residulas VS independent variable is satisfied.

The above normal probability plot is approximately form a straight line. Hence, we can conclude that the assumption of normal distribution on the residulas is satisfied.

• Use the regression equation to predict final score for
  • test 1 score = 82, and
  • test 2 score = 69

Ans: The The predicted Final score when test 1= 82 and test 2=69 is

Final Test = 5.581 + 0.550*82 + 0.459*69=82.352