Question

Study Time and Exam Score

An elementary statistics instructor is interested in determining how well the amount of time students spend studying for her class predicts their results on exam. The instructor asks her students to keep track of the number of hours they spent working on their statistics course between the first and second exam (including in class time, tutoring time, computer time, etc.) She then recorded their score on the second exam and the results are shown below.

Study Time	Exam Score
30	72
40	85
30	75
35	78
45	89
15	58
15	71
50	94
30	78
0	10
20	75
10	43
15	62
20	65
25	68
25	60
25	70
30	68
40	82
35	75

(A) Name the explanatory (predictor) and response variables for this analysis.

(B) What is the slope of the regression line? Interpret this value in context.

(C) What is the y-intercept of the regression line? Interpret this value in context.

(D) Determine the regression line.

(E) Use the equation of the regression line to predict a student's score when they study:

10 hours _____

20 hours _____

30 hours ____

(F) What is the residual for a person that studies 10 hours?

(G) What is the value of the correlation coefficient? Interpret this value.

Answer 1

Study Time, X	Exam Score, Y	XY	X²	Y²
30	72	2160	900	5184
40	85	3400	1600	7225
30	75	2250	900	5625
35	78	2730	1225	6084
45	89	4005	2025	7921
15	58	870	225	3364
15	71	1065	225	5041
50	94	4700	2500	8836
30	78	2340	900	6084
0	10	0	0	100
20	75	1500	400	5625
10	43	430	100	1849
15	62	930	225	3844
20	65	1300	400	4225
25	68	1700	625	4624
25	60	1500	625	3600
25	70	1750	625	4900
30	68	2040	900	4624
40	82	3280	1600	6724
35	75	2625	1225	5625
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
535	1378	40575	17225	101104

Sample size, n =	20
x̅ = Ʃx/n = 535/20 =	26.75
y̅ = Ʃy/n = 1378/20 =	68.9
SSxx = Ʃx² - (Ʃx)²/n = 17225 - (535)²/20 =	2913.75
SSyy = Ʃy² - (Ʃy)²/n = 101104 - (1378)²/20 =	6159.8
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 40575 - (535)(1378)/20 =	3713.5

a)

Explanatory (predictor) variable = Study time

Response variables = Exam score

b)

Slope, b = SSxy/SSxx = 3713.5/2913.75 = 1.274474474

With a unit increase in study time the exam score increases by 1.2745 unit.

c)

y-intercept, a = y̅ -b* x̅ = 68.9 - (1.27447)*26.75 = 34.80780781

The y-intercept is the exam score when x =0.

d)

Regression equation :

ŷ = 34.8078 + (1.2745) x

e)

Predicted value of y at x = 10

ŷ = 34.8078 + (1.2745) * 10 = 47.5526

Predicted value of y at x = 20

ŷ = 34.8078 + (1.2745) * 20 = 60.2973

Predicted value of y at x = 30

ŷ = 34.8078 + (1.2745) * 30 = 73.042

f)

Residual = y - ŷ = 43 - 47.5526 = -4.5526

g)

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 3713.5/√(2913.75*6159.8) = 0.8765

There is a strong positive relationship between Study time and Exam score.