Question

Chapter Three Exercises
Exercise 3.5
At a used dealership, let X be an independent variable representing the age in years of a motorcycle and Y be the dependent variable representing the selling price of used motorcycle. The data is now given to you.

X = {5, 10, 12, 14, 15}

Y = {500, 400, 300, 200, 100}

A.) Construct a 95% confidence interval for B₁.
B.) Do the data provide sufficient evidence to indicate that X contributes information to the prediction of Y? (hint: t-test)

Answer 1

X	Y	XY	X²	Y²
5	500	2500	25	250000
10	400	4000	100	160000
12	300	3600	144	90000
14	200	2800	196	40000
15	100	1500	225	10000
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
56	1500	14400	690	550000

Sample size, n =	5
x̅ = Ʃx/n =	11.2
y̅ = Ʃy/n =	300
SSxx = Ʃx² - (Ʃx)²/n =	62.8
SSyy = Ʃy² - (Ʃy)²/n =	100000
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	-2400

Slope, b₁ = SSxy/SSxx = -38.2166

Sum of Square error, SSE = SSyy -SSxy²/SSxx =    8280.254777

Standard error, se = √(SSE/(n-2)) =   52.53651

Standard error for slope, se(_b1) = se/√SSxx = 6.6295

a) Significance level, α = 0.05

Critical value, t_c = T.INV.2T(0.05, 3) = 3.1824  

95% Confidence interval for slope, B1:  

Lower limit = β₁ - tc*se_b1 = -59.3146

Upper limit = β₁ + tc*se_b1 = -17.1185

b) Null and alternative hypothesis:  

Ho: β₁ = 0 ; Ha: β₁ ≠ 0  

Test statistic:  

t = b₁ /se(_b1) = -5.7646

df = n-2 = 3

Critical value, t_c = T.INV.2T(0.05, 3) =    3.182446305

p-value, t_c = T.DIST.2T(ABS(-5.7646), 3) = 0.0104

Conclusion:  

p-value < α Reject the null hypothesis.

There is sufficient evidence to indicate that X contributes information to the prediction of Y.