Question

4. [35 points] The following table contains the ACT scores and the GPA for ten college students. Student GPA ACT 2.821 3.4 24 3.026 3.5 27 3.629 3.025 3.225 3.5 28 2.520 3.7 30 10 (a) [10 points] Estimate the relationship between GPA and ACT; that is, obtain the intercept and slope estimates in the equation (You may use a calculator or a spreadsheet program, but show your work.) (b) [2 points] Does the intercept have a useful interpretation here? Explain. (c) [3 points] How much higher is GPA predicted to be if the ACT score is increased (d) [10 points] Compute the fitted values and residuals for each observation and list (e) [5 points] What is the predicted value of GPA when ACT - 20? Does this seem (f) [5 points] How much of the variation in GPA for these ten students is explained by five points? them in a table. Verify that the residuals sum to zero (approximately) like a reasonable predicted value? by ACT? Explain.

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9011
R Square	0.8119
Adjusted R Square	0.7884
Standard Error	0.1786
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1.10	1.10	34.53	0.00
Residual	8	0.26	0.03
Total	9	1.36
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.4676	0.4718	0.9912	0.3506	-0.6203	1.5555
ACT	0.1079	0.0184	5.8765	0.0004	0.0656	0.1503

a) GPA = 0.4676+0.1079*ACT

b) As the value of the intercept is low and the p-value is less than 0.05, it is not significant and a chance factor

c) GPA predicted increases by 0.5395=(0.1079*5)

d) Yes, the residuals sum to zero

Student	GPA	ACT	Predicted	Residuals
1	2.8	21	2.7	-0.1
2	3.4	24	3.1	-0.3
3	3	26	3.3	0.3
4	3.5	27	3.4	-0.1
5	3.6	29	3.6	0.0
6	3	25	3.2	0.2
7	3.2	25	3.2	0.0
8	3.5	28	3.5	0.0
9	2.5	20	2.6	0.1
10	3.7	30	3.7	0.0
			Sum	0.1

e) GPA =0.4676+0.1079*20 = 2.63. yes as predicted value is close to actual value

f) We need to use R square value to understand the variation in GPA. As the value is 0.8119, 81.19 percent of variation in GPA is explained by ACT.