Question

An automobile company is running a new television commercial in five cities with approximately the same population. The following table shows the number of times the commercial is run on TV in each city and the number of car sales (in hundreds). Find the linear regression line for the data given in the table. Round any intermediate calculations to no less than six decimal places, and round the coefficients to two decimal places.

Number of TV commercials, x 4
6
12
14
17
Car sales, y (in hundreds) 2
5
2
8
9

Answer 1

the linear regression line for the given data is: y = 0.24x + 2.49

Explanation:

To find the linear regression line for the given data, we need to first calculate the mean of x, the mean of y, the sum of squares of x, the sum of squares of y, and the sum of cross-products of x and y.

Using these values, we can then calculate the slope and y-intercept of the regression line.

First, we calculate the mean of x and y:

mean of x = (4 + 6 + 12 + 14 + 17) / 5 = 10.6

mean of y = (2 + 5 + 2 + 8 + 9) / 5 = 5.2

Next, we calculate the sum of squares of x and y:

sum of squares of x = (4 - 10.6)^2 + (6 - 10.6)^2 + (12 - 10.6)^2 + (14 - 10.6)^2 + (17 - 10.6)^2 = 240.4

sum of squares of y = (2 - 5.2)^2 + (5 - 5.2)^2 + (2 - 5.2)^2 + (8 - 5.2)^2 + (9 - 5.2)^2 = 44.8

Finally, we calculate the sum of cross-products of x and y:

sum of cross-products of x and y = (4 - 10.6)(2 - 5.2) + (6 - 10.6)(5 - 5.2) + (12 - 10.6)(2 - 5.2) + (14 - 10.6)(8 - 5.2) + (17 - 10.6)(9 - 5.2) = 56.8

Using these values, we can calculate the slope of the regression line:

slope = sum of cross-products of x and y / sum of squares of x = 56.8 / 240.4 = 0.236

Next, we can calculate the y-intercept of the regression line:

y-intercept = mean of y - slope * mean of x = 5.2 - 0.236 * 10.6 = 2.488

Therefore, the linear regression line for the given data is:

y = 0.24x + 2.49

Note: The slope and y-intercept have been rounded to two decimal places, as per the instructions.