Question

In: Statistics and Probability

Consider the following data on price ($) and the overall score for six stereo headphones tested...

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

Solutions

Expert Solution

a)

H0: β1 = 0
Ha: 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
Ha: 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

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