Question

The sales of a company (in million dollars) for each year are shown in the table below. Is there evidence of a linear relationship between the two variables, and are they positively correlated?

x (years since 2010)	5	6	7	8	9
y (sales in millions)	12	19	29	37	45

                                             

a. Assign variables and sketch the scatter gram

b. State the Null and Alternative Hypotheses as if a complete hypothesis test had been conducted.

c. Find the linear correlation coefficient.

d. At what level is there evidence of linear correlation?

e. State the least squares equation and sketch it on the scatter gram.

f. Estimate the income in the year 2021

Answer 1

X	Y	XY	X²	Y²
5	12	60	25	144
6	19	114	36	361
7	29	203	49	841
8	37	296	64	1369
9	45	405	81	2025

Ʃx =	35
Ʃy =	142
Ʃxy =	1078
Ʃx² =	255
Ʃy² =	4740
Sample size, n =	5
x̅ = Ʃx/n = 35/5 =	7
y̅ = Ʃy/n = 142/5 =	28.4
SSxx = Ʃx² - (Ʃx)²/n = 255 - (35)²/5 =	10
SSyy = Ʃy² - (Ʃy)²/n = 4740 - (142)²/5 =	707.2
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 1078 - (35)(142)/5 =	84

a)

b) Null and alternative hypothesis:  

Ho: ρ = 0 ; Ha: ρ ≠ 0  

c) Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 84/√(10*707.2) = 0.9989

Test statistic :    

t = r*√(n-2)/√(1-r²) = 0.9989 *√(5 - 2)/√(1 - 0.9989²) = 36.3731

df = n-2 = 3

p-value = T.DIST.2T(ABS(36.3731), 3) = 0.0000

Conclusion:  

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

e) Slope, b = SSxy/SSxx = 84/10 =    8.4

y-intercept, a = y̅ -b* x̅ = 28.4 - (8.4)*7 =    -30.4

Regression equation :   

ŷ = -30.4 + (8.4) x  

f) Predicted value of y at x = 16

ŷ = -30.4 + (8.4) * 16 = 104