In: Math
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.
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.