Question

Consider the following data on price ($) and the overall score for six stereo headphones tested by a certain magazine. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest).

Brand	Price ($)	Score
A	180	78
B	150	71
C	95	63
D	70	58
E	70	42
F	35	24

(a)

The estimated regression equation for this data is

ŷ = 23.124 + 0.329x,

where x = price ($) and y = overall score. Does the t test indicate a significant relationship between price and the overall score? Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₀ ≠ 0
H_a: β₀ = 0H₀: β₁ = 0
H_a: β₁ ≠ 0    H₀: β₁ ≥ 0
H_a: β₁ < 0H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₀ = 0
H_a: β₀ ≠ 0

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

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

p-value =

What is your conclusion?

Reject H₀. We conclude that the relationship between price ($) and overall score is significant.Reject H₀. We cannot conclude that the relationship between price ($) and overall score is significant.     Do not reject H₀. We cannot conclude that the relationship between price ($) and overall score is significant.Do not reject H₀. We conclude that the relationship between price ($) and overall score is significant.

(b)

Test for a significant relationship using the F test. Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₁ ≥ 0
H_a: β₁ < 0H₀: β₁ = 0
H_a: β₁ ≠ 0    H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₀ = 0
H_a: β₀ ≠ 0H₀: 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 =

What is your conclusion?

Do not reject H₀. We conclude that the relationship between price ($) and overall score is significant.Do not reject H₀. We cannot conclude that the relationship between price ($) and overall score is significant.     Reject H₀. We cannot conclude that the relationship between price ($) and overall score is significant.Reject H₀. We conclude that the relationship between price ($) and overall score is significant.

(c)

Show the ANOVA table for these data. (Round your p-value to three decimal places and all other values to two decimal places.)

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p-value
Regression
Error
Total

Answer 1

a)

H₀: β₁ = 0
H_a: β₁ ≠ 0

std error of slope =se(β1) =s/√Sxx=		0.0782
test stat t =(b1-β1)/se(β1)=		4.202
p value =	0.0137	(from excel:tdist(4.202,4,2)

Reject H₀. We conclude that the relationship between price ($) and overall score is significant

b)

H₀: β₁ = 0
H_a: β₁ ≠ 0  

value of the test statistic F=17.65

p value =0.014

Reject H₀. We conclude that the relationship between price ($) and overall score is significant.

c)

Source	SS	df	MS	F	p value
regression	1615.87	1	1615.87	17.65	0.014
Residual error	366.13	4	91.53
Total	1982.00	5