Question

suppose weve come up with a line that allows us to predict the top speed of a roller coaster, using its height.

1. for a certain roller coaster, the model predicts a top speed of 50 miles per hour, but it actually has a top speed of 52 miles per hour. whats the residual?
2. for another roller coaster, the model predicts a top speed of 40 miles per hour, and we see that the residual is -8 miles per hour. whats the actual top speed of this roller coaster?
3. there is one roller coaster that has a residual of 0. explain what that means.

a scatterplot of house price (in thousands of dollars) vs. house size (in thousands of square feet) for houses sold recently in saratoga, new york, shows a relationship that is straight, with only moderate scatter and no outliers. the equation of the linear model is \\( \widehat{price}=8.4 + 88.67size \\).

1. what does the slope of 88.67 mean?
2. what are the units of the slope?

Explanation:

Step1: Residual formula

Residual = Observed value - Predicted value.
For question 1: Observed value = 52, Predicted value = 50.
Residual = \(52 - 50\)

Step2: Rearranged residual formula

Observed value = Residual + Predicted value.
For question 2: Residual = - 8, Predicted value = 40.
Observed value = \(-8+40\)

Step3: Residual meaning

Residual = Observed - Predicted. If residual = 0, then Observed = Predicted.
For question 3: It means the actual top - speed of the roller - coaster is equal to the predicted top - speed.

Step4: Slope interpretation

The slope in a regression equation \(\widehat{y}=a + bx\) (here \(\widehat{price}=8.4 + 88.67Size\)) represents the change in \(y\) (price) for a one - unit change in \(x\) (size).
For question 4: For each additional 1 thousand square feet in house size, the predicted house price increases by 88.67 thousand dollars.

Step5: Slope unit calculation

The slope \(b=\frac{\Delta y}{\Delta x}\). \(y\) (price) is in thousands of dollars and \(x\) (size) is in thousands of square feet.
For question 5: The units of the slope are \(\frac{\text{thousands of dollars}}{\text{thousands of square feet}}=\text{dollars per square foot}\)

Answer:

1. \(2\) miles per hour
2. \(32\) miles per hour
3. The actual top - speed of the roller - coaster is equal to the predicted top - speed.
4. For each additional 1 thousand square feet in house size, the predicted house price increases by 88.67 thousand dollars.
5. Dollars per square foot