Question

The following data are from a random sample of 10 students who participated in a study undertaken to investigate the effect of sleep time (measured in average number of hours of sleep per night) on GPA (grade point average, measured on a 4-point scale).

Student	Sleep time	GPA
1	7	3.28
2	9	3.16
3	8	3.75
4	6	2.50
5	4	2.45
6	8	2.91
7	7	3.53
8	6	3.02
9	3	2.30
10	8	3.48

a. Find the equation between GPA (y) as function of sleep time (x).

b. What is the estimated GPA of a student who averages 5 hours of sleep per night?

c. What is the coefficient of determination?

*Please don't use the EXCEL from.

Answer 1

a.

Sum of X = 66
Sum of Y = 30.38
Mean X = 6.6
Mean Y = 3.038
Sum of squares (SS_X) = 32.4
Sum of products (SP) = 6.542

Regression Equation = ŷ = bX + a

b = SP/SS_X = 6.54/32.4 = 0.2019

a = M_Y - bM_X = 3.04 - (0.2*6.6) = 1.7054

ŷ = 0.2019X + 1.7054

b. For x=5, ŷ = (0.2019*5) + 1.7054=2.7149

c.

X Values
∑ = 66
Mean = 6.6
∑(X - M_x)² = SS_x = 32.4

Y Values
∑ = 30.38
Mean = 3.038
∑(Y - M_y)² = SS_y = 2.214

X and Y Combined
N = 10
∑(X - M_x)(Y - M_y) = 6.542

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 6.542 / √((32.4)(2.214)) = 0.7723

So r^2=0.7723^2=0.5964

Hence coefficient of determination is 0.5964

So 59.64% of variation in y is explained by x