Question

Jensen Tire & Auto is in the process of deciding whether to purchase a maintenance contract for its new computer wheel alignment and balancing machine. Managers feel that maintenance expense should be related to usage, and they collected the following information on weekly usage (hours) and annual maintenance expense (in hundreds of dollars).

Weekly Usage (hours)	Annual Maintenance Expense
13	17.0
10	22.0
20	30.0
28	37.0
32	47.0
17	30.5
24	32.5
31	39.0
40	51.5
38	40.0

(a)

Develop the estimated regression equation that could be used to predict the annual maintenance expense (in hundreds of dollars) given the weekly usage (in hours). (Round your numerical values to two decimal places.)

ŷ =

(b)

Test the significance of the relationship in part (a) at a 0.05 level of significance. (Use the F test.)

State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0

H₀: β₁ ≥ 0
H_a: β₁ < 0    

H₀: β₁ = 0
H_a: β₁ ≠ 0

H₀: β₀ ≠ 0
H_a: β₀ = 0

H₀: β₀ = 0
H_a: β₀ ≠ 0

Find the value of the test statistic. (Round your answer to two decimal places.)

_____

Find the p-value. (Round your answer to three decimal places.)

p-value = _____

State your conclusion.

Do not reject H₀. We conclude that the relationship between weekly usage (hours) and annual maintenance expense (in hundreds of dollars) is significant.

Do not reject H₀. We cannot conclude that the relationship between weekly usage (hours) and annual maintenance expense (in hundreds of dollars) is significant.     

Reject H₀. We conclude that the relationship between weekly usage (hours) and annual maintenance expense (in hundreds of dollars) is significant.

Reject H₀. We cannot conclude that the relationship between weekly usage (hours) and annual maintenance expense (in hundreds of dollars) is significant.

(c)

Jensen expects to use the new machine 29 hours per week. Develop a 95% prediction interval for the company's annual maintenance expense (in hundreds of dollars). (Round your answers to two decimal places.)

$ ______ hundred to $_____ hundred

(d)

If the maintenance contract costs $3,200 per year, would you recommend purchasing it? Why or why not? (Round your answer to two decimal places.)

The expected expense of a machine being used 29 hours per week is $_______ hundred. Since this expected expense is ( ---Select--- greater less than) the cost of the maintenance contract, the maintenance contract  ---Select--- (is , (is not advisable.)

Answer 1

X	Y	XY	X²	Y²
13	17.0	221	169	289
10	22.0	220	100	484
20	30.0	600	400	900
28	37.0	1036	784	1369
32	47.0	1504	1024	2209
17	30.5	518.5	289	930.25
24	32.5	780	576	1056.25
31	39.0	1209	961	1521
40	51.5	2060	1600	2652.25
38	40.0	1520	1444	1600
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
253	346.5	9668.5	7347	13010.75

Sample size, n =	10
x̅ = Ʃx/n = 253/10 =	25.3
y̅ = Ʃy/n = 346.5/10 =	34.65
SSxx = Ʃx² - (Ʃx)²/n = 7347 - (253)²/10 =	946.1
SSyy = Ʃy² - (Ʃy)²/n = 13010.75 - (346.5)²/10 =	1004.525
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 9668.5 - (253)(346.5)/10 =	902.05

a)

Slope, b = SSxy/SSxx = 902.05/946.1 = 0.9534404

y-intercept, a = y̅ -b* x̅ = 34.65 - (0.95344)*25.3 = 10.527957

Regression equation :

ŷ = 10.53 + (0.95) x

b)

Null and alternative hypothesis:

Ho: β₁ = 0

Ha: β₁ ≠ 0

n=10

α = 0.05

SSE = SSyy -SSxy²/SSxx = 1004.525 - (902.05)²/946.1 = 144.47

SSR = SSxy²/SSxx = (902.05)²/946.1 = 860.0509

Test statistic:

F = SSR/(SSE/(n-2)) = 860.0509/(144.4741/8) = 47.62

p-value = F.DIST.RT(47.6238, 1, 8) = 0.000

Conclusion:

Reject H₀. We conclude that the relationship between weekly usage (hours) and annual maintenance expense (in hundreds of dollars) is significant.

c)

Predicted value of y at x = 29

ŷ = 10.528 + (0.9534) * 29 = 38.1777

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1004.525 - (902.05)²/946.1 = 144.4740514

Standard error, se = √(SSE/(n-2)) = √(144.47405/(10-2)) = 4.24962

Critical value, t_c = T.INV.2T(0.05, 8) = 2.306

95% Confidence interval :

Lower limit = ŷ - tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 38.1777 - 2.306*4.2496*√((1/10) + ((29 - 25.3)²/(946.1))) = 34.86

Upper limit = ŷ + tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 38.1777 + 2.306*4.2496*√((1/10) + ((29 - 25.3)²/(946.1))) = 41.49

d)

The expected expense of a machine being used 29 hours per week is $38.18 hundred. Since this expected expense is less than the cost of the maintenance contract, the maintenance contract is advisable.