Question

day 3: practice problem 7.32d classwork
at a conference for high school teachers, some data were collected on the age and current mileage of each teachers primary vehicle. the data were entered into a ti - 84. here is some output from running two - variable statistics on the data.
the correlation between age and mileage for these data is 0.877.

1. calculate the equation of the least - squares line for predicting a high school teachers vehicle mileage from its age. show your work.
2. interpret the slope of the regression line in context.
3. here is a residual plot from this linear regression analysis. describe what the residual plot tells you about how well the linear model fits the data.
4. what is the actual vehicle mileage for the teacher whose point is highlighted in the plot?
5. state and interpret the value of ( r^{2} ) in the context of this problem.

Explanation:

Step1: Calculate the slope \(b_1\)

The formula for the slope of the least - squares line is \(b_1=r\frac{s_y}{s_x}\). Given \(r = 0.877\), \(s_y=45738.95716\), and \(s_x = 2.683281573\).

$$b_1=0.877\times\frac{45738.95716}{2.683281573}$$

$$b_1=0.877\times17046.613$$

$$b_1 = 14949.88$$

Step2: Calculate the y - intercept \(b_0\)

The formula for the y - intercept is \(b_0=\bar{y}-b_1\bar{x}\). Given \(\bar{x} = 5\) and \(\bar{y}=60939.83333\)

$$b_0=60939.83333-14949.88\times5$$

$$b_0=60939.83333 - 74749.4$$

$$b_0=-13809.57$$

Answer:

The equation of the least - squares line is \(\hat{y}=-13809.57 + 14949.88x\)