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 ($) for 10 racing bikes reviewed by the magazine. Brand Weight (pound) Price ($)
A 17.8 2100
B 16.1 6250
C 14.9 8370
D 15.9 6200
E 17.2 4000
F 13.1 8600
G 16.2 6000
H 17.1 2580
I 17.6 3400
J 14.1 8000
Use the data to develop an estimated regression equation that could be used to estimate the price for the bike given the weight. (hint: y on x)
The Correct Answer and Explanation is:
To develop the estimated regression equation for predicting the price of a racing bike based on its weight, we need to use the formula for the least squares regression line, which is:y=b0+b1xy = b_0 + b_1 xy=b0+b1x
Where:
- yyy is the dependent variable (price in this case),
- xxx is the independent variable (weight in this case),
- b1b_1b1 is the slope of the line (which represents the change in price for each unit increase in weight),
- b0b_0b0 is the y-intercept of the line.
We can find the slope (b1b_1b1) and y-intercept (b0b_0b0) using the following formulas:
- b1=n∑(xy)−∑x∑yn∑x2−(∑x)2b_1 = \frac{n \sum{(xy)} – \sum{x} \sum{y}}{n \sum{x^2} – (\sum{x})^2}b1=n∑x2−(∑x)2n∑(xy)−∑x∑y
- b0=∑y−b1∑xnb_0 = \frac{\sum{y} – b_1 \sum{x}}{n}b0=n∑y−b1∑x
Where:
- nnn is the number of data points,
- xxx represents the weight of the bikes,
- yyy represents the price of the bikes,
- ∑\sum∑ represents the sum of the corresponding values.
Step-by-Step Calculation:
We are given the following data for the 10 bikes:
| Brand | Weight (pounds) xxx | Price ($) yyy |
|---|---|---|
| A | 17.8 | 2100 |
| B | 16.1 | 6250 |
| C | 14.9 | 8370 |
| D | 15.9 | 6200 |
| E | 17.2 | 4000 |
| F | 13.1 | 8600 |
| G | 16.2 | 6000 |
| H | 17.1 | 2580 |
| I | 17.6 | 3400 |
| J | 14.1 | 8000 |
We need to compute the necessary sums for the formula:
- ∑x\sum{x}∑x, ∑y\sum{y}∑y, ∑xy\sum{xy}∑xy, ∑x2\sum{x^2}∑x2
First, let’s compute these sums:
- Sum of weights (∑x\sum{x}∑x):
17.8+16.1+14.9+15.9+17.2+13.1+16.2+17.1+17.6+14.1=150.917.8 + 16.1 + 14.9 + 15.9 + 17.2 + 13.1 + 16.2 + 17.1 + 17.6 + 14.1 = 150.917.8+16.1+14.9+15.9+17.2+13.1+16.2+17.1+17.6+14.1=150.9
- Sum of prices (∑y\sum{y}∑y):
2100+6250+8370+6200+4000+8600+6000+2580+3400+8000=561002100 + 6250 + 8370 + 6200 + 4000 + 8600 + 6000 + 2580 + 3400 + 8000 = 561002100+6250+8370+6200+4000+8600+6000+2580+3400+8000=56100
- Sum of products of weight and price (∑xy\sum{xy}∑xy):
(17.8×2100)+(16.1×6250)+(14.9×8370)+(15.9×6200)+(17.2×4000)+(13.1×8600)+(16.2×6000)+(17.1×2580)+(17.6×3400)+(14.1×8000)(17.8 \times 2100) + (16.1 \times 6250) + (14.9 \times 8370) + (15.9 \times 6200) + (17.2 \times 4000) + (13.1 \times 8600) + (16.2 \times 6000) + (17.1 \times 2580) + (17.6 \times 3400) + (14.1 \times 8000)(17.8×2100)+(16.1×6250)+(14.9×8370)+(15.9×6200)+(17.2×4000)+(13.1×8600)+(16.2×6000)+(17.1×2580)+(17.6×3400)+(14.1×8000) =37380+100750+124413+98780+68800+112660+97200+44118+59840+112800=703741= 37380 + 100750 + 124413 + 98780 + 68800 + 112660 + 97200 + 44118 + 59840 + 112800 = 703741=37380+100750+124413+98780+68800+112660+97200+44118+59840+112800=703741
- Sum of squares of weights (∑x2\sum{x^2}∑x2):
(17.82)+(16.12)+(14.92)+(15.92)+(17.22)+(13.12)+(16.22)+(17.12)+(17.62)+(14.12)(17.8^2) + (16.1^2) + (14.9^2) + (15.9^2) + (17.2^2) + (13.1^2) + (16.2^2) + (17.1^2) + (17.6^2) + (14.1^2)(17.82)+(16.12)+(14.92)+(15.92)+(17.22)+(13.12)+(16.22)+(17.12)+(17.62)+(14.12) =317.64+259.21+222.01+252.81+295.84+171.61+262.44+292.41+309.76+198.81=2639.54= 317.64 + 259.21 + 222.01 + 252.81 + 295.84 + 171.61 + 262.44 + 292.41 + 309.76 + 198.81 = 2639.54=317.64+259.21+222.01+252.81+295.84+171.61+262.44+292.41+309.76+198.81=2639.54
Now, let’s substitute these sums into the formulas:
- b1=10×703741−150.9×5610010×2639.54−(150.9)2=7037410−846249026395.4−22770.81=−14250803624.59≈−393.22b_1 = \frac{10 \times 703741 – 150.9 \times 56100}{10 \times 2639.54 – (150.9)^2} = \frac{7037410 – 8462490}{26395.4 – 22770.81} = \frac{-1425080}{3624.59} \approx -393.22b1=10×2639.54−(150.9)210×703741−150.9×56100=26395.4−22770.817037410−8462490=3624.59−1425080≈−393.22
- b0=56100−(−393.22×150.9)10=56100+59369.9910=115469.9910≈11546.99b_0 = \frac{56100 – (-393.22 \times 150.9)}{10} = \frac{56100 + 59369.99}{10} = \frac{115469.99}{10} \approx 11546.99b0=1056100−(−393.22×150.9)=1056100+59369.99=10115469.99≈11546.99
Estimated Regression Equation:
Thus, the regression equation for estimating the price based on weight is:y=11546.99−393.22xy = 11546.99 – 393.22xy=11546.99−393.22x
Explanation:
This equation suggests that for each additional pound in weight, the price of the bike decreases by approximately $393.22. The y-intercept of 11546.99 indicates the estimated price when the weight of the bike is zero, though this is not practically meaningful for this scenario, as no bike weighs zero pounds.
This regression model can now be used to predict the price of a racing bike given its weight. For example, if a bike weighs 16 pounds, the estimated price would be:y=11546.99−393.22(16)=11546.99−6285.52=5261.47y = 11546.99 – 393.22(16) = 11546.99 – 6285.52 = 5261.47y=11546.99−393.22(16)=11546.99−6285.52=5261.47
Therefore, the estimated price for a 16-pound bike is approximately $5261.47.