4. A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, $x$ years, and the mileage, $y$ thousand miles. The data collected from the cars are shown in the table below: | Age, x (years) | 2 | 2.5 | 3 | 4 | 4.5 | 5 | 6 | 6.5 | |------------------|---|-----|---|---|-----|---|---|-----| | Mileage, y (thousands) | 22 | 34 | 37 | 40 | 45 | 58 | 39 | 58 | [You may assume that $\sum x = 41, \sum y = 406, \sum x^2 = 188, \sum xy = 1818.5$] (a) Find $S_x$ and $S_y$. (b) Find the equation of the least squares regression line in the form $y = ax + b$. Give the values of $a$ and $b$ to 2 decimal places. (c) Give a practical interpretation of the slope $b$. (d) Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.

A-Level Edexcel Maths Statistics Correlation & Regression: 4. A second hand car dealer has

4. A second hand car dealer has 10 cars for sale - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 1

Question 4

4. A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, $x$ years, and the mileage, $y$ thousand miles. Th... show full transcript

Worked Solution & Example Answer:4. A second hand car dealer has 10 cars for sale - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 1

Step 1

Find $S_x$ and $S_y$

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Answer

$S_x = \frac{\sum x}{n} = \frac{41}{10} = 4.1, \\ S_y = \frac{\sum y}{n} = \frac{406}{10} = 40.6$ .

Step 2

Find the equation of the least squares regression line in the form $y = ax + b$. Give the values of $a$ and $b$ to 2 decimal places.

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Answer

To find the slope $b$ , we use the formula:

$b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$

Substituting the values:

$b = \frac{10(1818.5) - (41)(406)}{10(188) - (41)^2} = \frac{18185 - 16646}{1880 - 1681} = \frac{1248.5}{199} \approx 6.27$ .

Now for $a$ :

$a = \frac{\sum y - b(\sum x)}{n} = \frac{406 - 6.27(41)}{10} = \frac{406 - 257.07}{10} = 14.89$ .

Thus, the equation is:

$y = 6.27x + 14.89$ .

Step 3

Give a practical interpretation of the slope $b$.

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Answer

The slope $b = 6.27$ indicates that for each additional year in the age of a car, the mileage increases by approximately 6.27 thousand miles.

Step 4

Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.

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Answer

For a 5 year old car, substituting $x = 5$ into the regression equation:

$y = 6.27(5) + 14.89 = 31.35 + 14.89 = 46.24$ .

Thus, the predicted mileage is approximately 46.24 thousand miles.

4. A second hand car dealer has 10 cars for sale - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 1

Worked Solution & Example Answer:4. A second hand car dealer has 10 cars for sale - Edexcel - A-Level Maths Statistics - Question 4 - 2008 - Paper 1

Find $S_x$ and $S_y$

Find the equation of the least squares regression line in the form $y = ax + b$. Give the values of $a$ and $b$ to 2 decimal places.

Give a practical interpretation of the slope $b$.

Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.

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