Question

Consumer Reports provided extensive testing and ratings of 24 treadmills. They provided an overall score based on ease of use, ergonomics, exercise range, and quality. Higher scores indicated higher performance. The following data gives Price (in dollars), an overall categorical quality rating (Good, Very Good, Excellent), and the performance score (Consumer Reports, February 2006). Basic summary statistics for the numerically-scaled variables are also provided.

Brand & Model	Price	Quality	Score
Landice L7	2900	Excellent	86
NordicTrack S3000	3500	Very good	85
SportsArt 3110	2900	Excellent	82
Precor	3500	Excellent	81
True Z4 HRC	2300	Excellent	81
Vision Fitness T9500	2000	Excellent	81
Precor M 9.31	3000	Excellent	79
Vision Fitness T9200	1300	Very good	78
Star Trac TR901	3200	Very good	72
Trimline T350HR	1600	Very good	72
Schwinn 820p	1300	Very good	69
Bowflex 7-Series	1500	Excellent	83
NordicTrack S1900	2600	Very good	83
Horizon Fitness PST8	1600	Very good	82
Horizon Fitness 5.2T	1800	Very good	80
Evo by Smooth Fitness FX30	1700	Very good	75
ProForm 1000S	1600	Very good	75
Horizon Fitness CST4.5	1000	Very good	74
Keys Fitness 320t	1200	Very good	73
Smooth Fitness 7.1HR Pro	1600	Very good	73
NordicTrack C2300	1000	Good	70
Spirit Inspire	1400	Very good	70
ProForm 750	1000	Good	67
Image 19.0 R	600	Good	66

a.) Use these data to develop an estimated regression equation that could be used to estimate the overall score given the price.

b.) Use a=.05 to test for overall significance

c.) To incorporate the effect of quality, a categorical variable with these levels, we used two dummy variables: Quality-E and Quality-VG. Each variables was coded 0 or 1 as follows.

Quality-E; 1 if quality rating is excellent, 0 otherwise

Quality-VG; 1 if quality rating is very good, 0 otherwise

Develop an estimated regression equation that could be used to estimate the overall score giving the price and the quality rating.

d.) For the estimated regression equation developed in part c test for overall significance using a=.10

e.) For the estimated regression equation developed in part c use the t test to determine the significance for each independent variable. Use a =.10

f.) Develop a standardized residual plot. Does the pattern of the residual plot appear to be reasonable?

g.) Do the data contain any outliers or influential observations?

f.) Estimate the overall score for a treadmill with a price of $2000 and a good quality rating. How much would the estimated change if the quality rating were very good? Explain.

Answer 1

(Note - The first four questions have been solved)

a.

Step 1 - Put the data in excel as shown and arrange the variables as shown

Step 2 - Since we need to predict a score

The dependent variable is the score

The independent variable is the price

Step 3 - Select the regression option from the data analysis tab

Step 4 - Input the values as shown below.

Step 5 - The output is generated as follows.

We find the regression equation from the regression output coefficient(highlighted in blue)

Score = 67.6762   + 0.0046 Price

b.) Use a=.05 to test for overall significance

Ho: All the beta coefficient of equation is equal to zero
H1: At least one beta coefficient is not equal to zero.

From the regression output we see that pvalue = 0.000373703, is less than 0.05, hence we reject the null hypothesis and conclude that the regression equation is significant.

c.
Step 1 - Put the data in excel as shown and arrange the variables as shown. Also, create the dummy variables as shown.

Step 2 - Since we need to predict the score

The dependent variable is the score

The independent variable is price, dummy variables

Step 3 - Select the regression option from the data analysis tab

Step 4 - Input the values as shown below.

Step 5 - The output is generated as follows.

We find the regression equation from the regression output coefficient(highlighted in blue)

Score = 65.6597   + 10.2097 Quality-E + 5.9246 Quality-VG + 0.0023 Price

d.) d.) For the estimated regression equation developed in part c test for overall significance using a=.10

Ho: All the beta coefficient of the equation is equal to zero
H1: At least one beta coefficient is not equal to zero.

From the regression output, we see that pvalue = 0.00020494, is less than 0.10, hence we reject the null hypothesis and conclude that the regression equation is significant.