Question

Suppose you are interested in buying a new Toyota Corolla. You are standing on the sales lot looking at a model with different options. The list price is on the vehicle. As a salesperson approaches, you wonder what the dealer invoice price is for this model with its options. The following data are based on a random selection of Toyota Corollas of different models and options. Let y be the dealer invoice (in thousands of dollars) for the given vehicle.

x	12.9	13.0	12.8	13.6	13.4	14.2
y	11.6	11.9	11.5	12.2	12.0	12.8

(a) Verify that Σx = 79.9, Σy = 72, Σx² = 1065.41, Σy² = 865.1, Σxy = 960.02, and r ≈ 0.980.

Σx	=
Σy	=
Σx2	=
Σy2	=
Σxy	=
r	=

(b) Use a 1% level of significance to test the claim that ρ > 0. (Use 2 decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis, there is sufficient evidence that ρ > 0.

Reject the null hypothesis, there is insufficient evidence that ρ > 0.    

Fail to reject the null hypothesis, there is insufficient evidence that ρ > 0.

Fail to reject the null hypothesis, there is sufficient evidence that ρ > 0.

(c) Verify that S_e ≈ 0.1039, a ≈ 0.464, and b ≈ 0.866.

Se	=
a	=
b	=

(d) Find the predicted dealer invoice when the list price is x = 13.6 (thousand dollars). (Use 2 decimal places.)


(e) Find a 90% confidence interval for y when x = 13.6 (thousand dollars). (Use 2 decimal places.)

lower limit	=
upper limit	=

(f) Use a 1% level of significance to test the claim that β > 0. (Use 2 decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis, there is sufficient evidence that β > 0.

Reject the null hypothesis, there is insufficient evidence that β > 0.    

Fail to reject the null hypothesis, there is insufficient evidence that β > 0.

Fail to reject the null hypothesis, there is sufficient evidence that β > 0.

(g) Find a 90% confidence interval for β and interpret its meaning. (Use 2 decimal places.)

lower limit	=
upper limit	=

Interpretation

For every $1,000 increase in list price, the dealer price increases by an amount that falls within the confidence interval.

For every $1,000 increase in list price, the dealer price increases by an amount that falls outside the confidence interval.    

For every $1,000 increase in list price, the dealer price decreases by an amount that falls within the confidence interval.

For every $1,000 increase in list price, the dealer price decreases by an amount that falls outside the confidence interval.

Answer 1

a)

X	Y	XY	X²	Y²
12.9	11.6	149.64	166.41	134.56
13.0	11.9	154.7	169	141.61
12.8	11.5	147.2	163.84	132.25
13.6	12.2	165.92	184.96	148.84
13.4	12.0	160.8	179.56	144
14.2	12.8	181.76	201.64	163.84
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
79.9	72	960.02	1065.41	865.1

Sample size, n =	6
x̅ = Ʃx/n =	13.3167
y̅ = Ʃy/n =	12
SSxx = Ʃx² - (Ʃx)²/n =	1.40833
SSyy = Ʃy² - (Ʃy)²/n =	1.1
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	1.22

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 0.9802 =0.980

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b) Null and alternative hypothesis:  

Ho: ρ = 0, Ha: ρ > 0  

α = 0.01

Test statistic :    

t = r*√(n-2)/√(1-r²) = 0.9802 *√(6 - 2)/√(1 - 0.9802²) = 9.90

df = n-2 = 4

Critical value, t_c = T.INV(0.01, 4) = 3.75

Conclusion:  

Reject the null hypothesis, there is sufficient evidence that ρ > 0.

-------------------------------------

(c) Sum of Square error, SSE = SSyy -SSxy²/SSxx = 0.043147929

Standard error, se = √(SSE/(n-2)) = 0.1039

Slope, b = SSxy/SSxx = 0.866

y-intercept, a = y̅ -b* x̅ = 0.464

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(d) ŷ = 0.464 + (0.866) * 13.6 = 12.25

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(e) Significance level, α = 1 - 0.90 = 0.10

df = n-2 = 4

Critical value, t_c = T.INV.2T(0.10, 4) = 2.1318  

90% confidence interval for y when x = 13.6

Lower limit = 12.00

Upper limit = 12.49

-------------------------------------

(f) Null and alternative hypothesis:  

Ho: β₁ = 0, Ha: β₁ > 0  

Test statistic:  

t = b /(se/√SSxx) = 9.90

df = n-2 =    4

Critical value, t_c = T.INV(0.01, 4) = 3.75

Conclusion

Reject the null hypothesis, there is sufficient evidence that β > 0.

(g) Significance level, α = 0.10

Critical value, t_c = T.INV.2T(0.1, 4) = 2.1318  

90% confidence interval for β :

Interpretation

For every $1,000 increase in list price, the dealer price increases by an amount that falls within the confidence interval.