Question

Listed below are alumnus’s contribution (in dollars) and the years the alumnus has been out
of school for a small college.

Years, x	1	5	3	10	7	6
Contribution y, $	500	100	300	50	75	80

a) Find the value of the linear correlation coefficient r and use a significance level of α = 0.05 to
determine whether there is a significant linear correlation between the two variables.

b) Find the best predicted value for the contribution of an alumnus after 4 years.

Answer 1

X	Y	XY	X²	Y²
1	500	500	1	250000
5	100	500	25	10000
3	300	900	9	90000
10	50	500	100	2500
7	75	525	49	5625
6	80	480	36	6400

Ʃx =	32
Ʃy =	1105
Ʃxy =	3405
Ʃx² =	220
Ʃy² =	364525
Sample size, n =	6
x̅ = Ʃx/n = 32/6 =	5.333333333
y̅ = Ʃy/n = 1105/6 =	184.1666667
SSxx = Ʃx² - (Ʃx)²/n = 220 - (32)²/6 =	49.33333333
SSyy = Ʃy² - (Ʃy)²/n = 364525 - (1105)²/6 =	161020.8333
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 3405 - (32)(1105)/6 =	-2488.33333

a)

Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= -2488.33333/√(49.33333*161020.83333) = -0.8829

Null and alternative hypothesis:

Ho: ρ = 0 ; Ha: ρ ≠ 0

Test statistic :  

t = r*√(n-2)/√(1-r²) = -0.8829 *√(6 - 2)/√(1 - -0.8829²) = -3.76

df = n-2 = 4

p-value = T.DIST.2T(ABS(-3.76), 4) = 0.0198

Conclusion:

p-value < α Reject the null hypothesis. There is a correlation between x and y.

b)

Slope, b = SSxy/SSxx = -2488.33333/49.33333 = -50.43919

y-intercept, a = y̅ -b* x̅ = 184.16667 - (-50.43919)*5.33333 = 453.17568

Regression equation :

ŷ = 453.1757 + (-50.4392) x

Predicted value of y at x = 4

ŷ = 453.1757 + (-50.4392) * 4 = 251.4189