Question

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.

Answer 1

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.