Question

Really do not understand any of the concepts behind the following problem:

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 (Ib)	Price ($)
Fierro 7B	18	2,200
HX 5000	16	6,300
Durbin Ultralight	15	8,470
Schmidt	16	6,300
WSilton Advanced	17	4,100
Bicyclette velo	13	8,700
Supremo Team	16	6,100
XTC Racer	17	2,680
D’Onofrio Pro	18	3,500
Americana #6	14	8,100

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). Fit a regression line into the data. What is the estimated regression model represented by the regression line?

c). Create a regression model output in Excel at 0.05 level of significance. Are the regression parameters b₀ and b₁ equal to zero? Is there a significant relationship between Weight and Price for the bicycles? Explain your answer.

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 15.5 pounds D’Onofrio Elite bicycle later this year. Use the regression model you estimated to predict the price of the D’Onofrio Elite bicycle. Show your working.

f). Create a scatter plot of residual versus x-variable. From this scatter plot, could the estimated regression model obtained from the bicycle sample data be used to make a statistical inference? Explain how your plot of residual versus x-variable indicate whether or not the regression assumptions for statistical inference are met.

Answer 1

a)

From the scatter we can assume that the price of bicycles shows a decline as the weight increases.

b) The regression model that fits the data is

The data with the regression values ( Calculated for each weight with the above equation) is as follows

The regression line is

C) The regression output in Excel is as follows

The regression coefficients b₀ = 27371.67 ?and b?₁ = -1357.92 hence both are not equal to zero.

Here the R²? = 0.922 indicating that 92% change in the price can be explained by the independent variable weight. There exist a strong relationship between weight of the cycle and price. Also the p values for the coefficient (p values for b₀? 2.94 E-05 and b₁? 0.000147) are much lower than the significance levelof 0.05 indicating that both the coefficients are significant in the equation.

d) The value of R²?? indicates the percentage of variation in the values of the dependent variable that can be explained by the variation in the independent variable. Here R²? = 0.922? indicating that the model can explain 92.2% of the variation.

e) The estimated price of the bicycle can be found by using the equation

Weight is given as 15.5 lb

The price of D’Onofrio Elite bicycle planned to be introduced later this year with 15.5 lb weight is $6,324

f) A residual is the difference between the measured value and the predicted value of a regression model. They show how accurate a mathematical function, such as a line, is in representing a set of data. So lets plot the residual the residual with X values

From the scatter we find two values (indicated by red) deviated much from the regression line. But the residual line in a similar pattern above and below the line hence the line can be used for further statistical inference. As we discussed earlier there would be a 10% deviation in the value with the estimated.