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	69
C	95	63
D	70	58
E	70	38
F	35	24

1. The estimated regression equation for this data is  ŷ = 21.990 + 0.330x, 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.

1a. State the null and alternative hypotheses.

(a) H₀: β₀ ≠ 0
H_a: β₀ = 0

(b) H₀: β₁ = 0
H_a: β₁ ≠ 0    

(c) H₀: β₁ ≠ 0
H_a: β₁ = 0

(d) H₀: β₀ = 0
H_a: β₀ ≠ 0

(e) H₀: β₁ ≥ 0
H_a: β₁ < 0

1b. Find the value of the test statistic. (Round your answer to three decimal places.)

1c. Find the p-value. (Round your answer to four decimal places.)

p-value =

1d. What is your conclusion?

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

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

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

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

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

2a. State the null and alternative hypotheses.

(a) H₀: β₁ = 0
H_a: β₁ ≠ 0

(b) H₀: β₀ = 0
H_a: β₀ ≠ 0    

(c) H₀: β₁ ≠ 0
H_a: β₁ = 0

(d) H₀: β₀ ≠ 0
H_a: β₀ = 0

(e) H₀: β₁ ≥ 0
H_a: β₁ < 0

2b. Find the value of the test statistic. (Round your answer to two decimal places.)

2c. Find the p-value. (Round your answer to three decimal places.)

p-value =

2d. What is your conclusion?

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

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

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

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

3,. 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

SSE =Syy-(Sxy)2/Sxx=

418.955

	s2 =SSE/(n-2)=	104.7387
std error σ              =	=se =√s2=	10.2342

estimated std error of slope =se(β1) =s/√Sxx=

0.084

1a)

(b) H₀: β₁ = 0
H_a: β₁ ≠ 0    

1b)

test stat t =

(bo-β1)/se(β1)=

=

3.944

1c)

p value =0.0169

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

2a)

(b) H₀: β₀ = 0
H_a: β₀ ≠ 0

2b)

value of the test statistic =15.55

2c)

p value =0.017
(b) Reject H₀. We conclude that the relationship between price ($) and overall score is significant.  

Source	SS	df	MS	F	p value
regression	1629.05	1	1629.05	15.55	0.017
Residual error	418.95	4	104.74
Total	2048.00	5