Question

The following data show the brand, price, and the overall score for six stereo headphones that were tested by consumer reports. The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 to 100.

Price Score

Bose 180 76

Skullcandy 150 71

Koss 85 61

Phillips/O'Neill 70 56

Denon 70 40

JVC 35 26

a). Develop the estimated regression equation that could be used to predict the score given the price.

b). Compute the coefficient of determination.

c). What is the value of the sample correlation coefficient?

d). Does the t test indicate a significant relationship between price and the overall score? What is your conclusion? Use α=0.05

e). Test for a significant relationship using the F test. What is your conclusion? Use α= 0.05

f). Develop a point estimate of the score for a headphone with a price of 100.

g). Develop a 95% confidence interval for the mean score for all headphones with a price of 100.

Answer 1

• Here price be the independent variable denoted by x and score be the dependent variable denoted by y.
• a)First we will calculate the sum of square of variable x and y
• =73150-(6*98.33²)=15133.33
• =19950.0-(6*55.0²)=1800.0
• =37145(6*98.33²*55.0²)=4695

• The slope is given by

• =4695/15133.33=0.3103

• The intercept is given by

• =24.492

• Hence the regression line is given by

• y=24.492-0.3103x

• b)The coefficient of determination is given by

• =0.8092

• =80.92%

• which means  the 80.92% of the variance in the score variable that is predictable from the variable.

• c)The coefficient of sample correlation is given by

• =0.8999

• To test for significant relationship between price and the overall score.

• d)HYPOTHESIS TO BE TESTED

• H0:r=0 vs H1:r0

• The test statistics is given by

• 

• Hence by substituting the respective values test statistics obtained is

• 4.127

• CONCLUSION

• The critical value at 5% significant level with 4 degrees of freedom is given by 2.132

• The test statistics obtained is greater than 5% critical value.

• Hence we have enough evidence to reject the null hypothesis.

• Hence we conclude that there is a linear relationship between and score and price variable