Bicycling, the world's leading cycling magazine, reviews hundreds of bicycles throughout the year. Their "Road-Race" category contains reviews of bikes used by riders primarily interested in racing. One of the most important factors in selecting a bike for racing is the weight of the bike. The following data show the weight (pounds) and price ($) for 10 racing bikes reviewed by the magazine.†
Brand	Weight	Price ($)
FELT F5	17.8	2,100
PINARELLO Paris	16.1	6,250
ORBEA Orca GDR	14.9	8,370
EDDY MERCKX EMX-7	15.9	6,200
BH RC1 Ultegra	17.2	4,000
BH Ultralight 386	13.1	8,600
CERVELO S5 Team	16.2	6,000
GIANT TCR Advanced 2	17.1	2,580
WILIER TRIESTINA Gran Turismo	17.6	3,400
SPECIALIZED S-works Amira SL4	14.1	8,000
(a)
Use the data to develop an estimated regression equation that could be used to estimate the price for a bike given the weight. (Round your numerical values to the nearest integer).
ŷ = 
  
Incorrect: Your answer is incorrect.
(b)
Compute 
r2.
 (Round your answer to three decimal places.)
r2
 = 

To develop the estimated regression equation and compute $ r^2 $, we need to perform linear regression analysis on the given data. The general form of the regression equation is:

\[ \hat{y} = \beta_0 + \beta_1 x \]

where $ x $ is the weight of the bike, $ y $ is the price, and $ \beta_0 $ and $ \beta_1 $ are the coefficients we need to determine.

### Steps to Solve:

1. **Prepare the Data:**
   - List the weight $ x $ and price $ y $ for each bike.

2. **Calculate the Regression Coefficients:**
   - Use the formulas for the slope $ \beta_1 $ and intercept $ \beta_0 $:
     \[
     \beta_1 = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}
     \]
     \[
     \beta_0 = \frac{\sum y - \beta_1 \sum x}{n}
     \]

3. **Compute the Coefficient of Determination $ r^2 $:**
   \[
   r^2 = \frac{\sum (y_i - \bar{y})^2 - \sum (y_i - \hat{y})^2}{\sum (y_i - \bar{y})^2}
   \]
   where $ \bar{y} $ is the mean of the observed prices.

### Data Provided:
- Brand, Weight (x in pounds), Price ($)

Let’s proceed with the calculations:

1. **Weight and Price Data:**

\[
\begin{array}{|c|c|}
\hline
\text{Brand} & \text{Weight (x)} & \text{Price (y)} \\
\hline
\text{FELT F5} & 17.8 & 2100 \\
\text{PINARELLO Paris} & 16.1 & 6250 \\
\text{ORBEA Orca GDR} & 14.9 & 8370 \\
\text{EDDY MERCKX EMX-7} & 15.9 & 6200 \\
\text{BH RC1 Ultegra} & 17.2 & 4000 \\
\text{BH Ultralight 386} & 13.1 & 8600 \\
\text{CERVELO S5 Team} & 16.2 & 6000 \\
\text{GIANT TCR Advanced 2} & 17.1 & 2580 \\
\text{WILIER TRIESTINA Gran Turismo} & 17.6 & 3400 \\
\text{SPECIALIZED S-works Amira SL4} & 14.1 & 8000 \\
\hline
\end{array}
\]

### Perform the Regression Analysis:

Let's calculate the necessary values to find the regression equation:

1. **Calculate the Sums:**

\[
\sum x = 17.8 + 16.1 + 14.9 + 15.9 + 17.2 + 13.1 + 16.2 + 17.1 + 17.6 + 14.1 = 163.0
\]

\[
\sum y = 2100 + 6250 + 8370 + 6200 + 4000 + 8600 + 6000 + 2580 + 3400 + 8000 = 55300
\]

\[
\sum xy = (17.8 \cdot 2100) + (16.1 \cdot 6250) + (14.9 \cdot 8370) + (15.9 \cdot 6200) + (17.2 \cdot 4000) + (13.1 \cdot 8600) + (16.2 \cdot 6000) +(a) The estimated regression equation is:

\[ \hat{y} = 28,574 - 1,439x \]

(b) The coefficient of determination $ r^2 $ is approximately:

\[ r^2 = 0.864 \]

This means that about 86.4% of the variability in the price of the bikes can be explained by their weight.

Do you need further details or have any questions? Here are some questions you might find helpful:

1. How can we interpret the slope of the regression equation?
2. What assumptions are necessary for linear regression analysis to be valid?
3. Can we visualize this data and regression line on a scatter plot?
4. How does the coefficient of determination help us understand the fit of the model?
5. What other factors, besides weight, could affect the price of a racing bike?

**Tip:** When interpreting the slope of the regression equation, remember that it represents the change in the price for each additional pound of weight. A negative slope indicates that as the weight increases, the price tends to decrease.

Learn how to estimate the price of racing bikes using linear regression with weight data from Bicycling magazine. Explore the regression equation and coefficient of determination (r²) calculated from a dataset of 10 racing bikes. Perfect for statistical analysis enthusiasts and data-driven bike enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation