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Bicycling World, a magazine devoted to cycling, reviews hundreds of bicycles throughout the year. Its Road-Race...

Bicycling World, a magazine devoted to cycling, reviews hundreds of bicycles throughout the year. Its Road-Race category contains reviews of bicycles used by riders primarily interested in racing. One of the most important factors in selecting a bicycle for racing is its weight. The following data show the weight (pounds) and price ($) for ten racing bicycles reviewed by the magazine:

Model

Weight (lbs)

Price ($)

Fierro 7B

17.9

2200

HX 5000

16.2

6350

Durbin Ultralight

15.0

8470

Schmidt

16.0

6300

WSilton Advanced

17.3

4100

bicyclette vélo

13.2

8700

Supremo Team

16.3

6100

XTC Racer

17.2

2680

D’Onofrio Pro

17.7

3500

Americana #6

14.2

8100

A. Develop a scatter chart with weight as the independent variable. What does the scatter chart indicate about the relationship between the weight and price of these bicycles?

B. Use the data to develop an estimated regression equation that could be used to estimate the price for a bicycle, given its weight. What is the estimated regression model?

C. Test whether each of the regression parameters  and  is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?

D. How much of the variation in the prices of the bicycles in the sample does the regression model you estimated in part b explain?

E. The manufacturers of the D’Onofrio Pro plan to introduce the 15-pound D’Onofrio Elite bicycle later this year. Use the regression model you estimated in part a to predict the price of the D’Ononfrio Elite.

Solutions

Expert Solution

A.

Following is the scatter plot of the data:

Scatter plot shows a negative linear relationship between the variables.

B.

Following table shows the calculaitons:

Model Weight (lbs) (X) Price ($) (Y) X^2 Y^2 XY
Fierro 7B 17.9 2200 320.41 4840000 39380
HX 5000 16.2 6350 262.44 40322500 102870
Durbin Ultralight 15 8470 225 71740900 127050
Schmidt 16 6300 256 39690000 100800
WSilton Advanced 17.3 4100 299.29 16810000 70930
bicyclette vélo 13.2 8700 174.24 75690000 114840
Supremo Team 16.3 6100 265.69 37210000 99430
XTC Racer 17.2 2680 295.84 7182400 46096
D’Onofrio Pro 17.7 3500 313.29 12250000 61950
Americana #6 14.2 8100 201.64 65610000 115020
Total 161 56500 2613.84 371345800 878366

Now we have

Slope of the regression equation is

and intercept of the equation will be

So the regression equation will be

y'=28818.00-1439.01x

C.

Following is the output of the regression analysis generated by excel:

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.929366408
R Square 0.86372192
Adjusted R Square 0.84668716
Standard Error 942.2660545
Observations 10
ANOVA
df SS MS F Significance F
Regression 1 45017877.46 45017877.46 50.70349813 9.99374E-05
Residual 8 7102922.539 887865.3174
Total 9 52120800
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 28818.00368 3267.256839 8.820244351 2.14888E-05 21283.6959 36352.31146
X Variable 1 -1439.00644 202.0895123 -7.120638885 9.99374E-05 -1905.02569 -972.987189

Intercept:

Hypotheses are:

t-value of intercept is 8.820

P-value is 0.0000

Since p-value is less than 0.05 so intercept is significant to the model.

Slope:

Hypotheses are:

t-value of intercept is -7.12

P-value is 0.0000

Since p-value is less than 0.05 so slope is significant to the model.

D.

Since from regression output R-square is 0.864 so 86.4% of the variation  in the prices of the bicycles is accounted for by the weight of bicycles.

E.

For X= 15 estimated y value is

y'=28818.00-1439.01*15 = 7232.85

hence, required predicted price is $7232.85.


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