Q: Suppose the regression line for predicting home price (in thousands) from size (in square feet) is $$y = 0.15x + 50. $$What is the predicted price for a 2000 square foot home?

$$y = 0.15 \times 2000 + 50 = 350$$

Q: If the regression line for predicting y from x is $$y = 2x + 5$$, and the mean of x is 10, what is the mean of y?

$$\bar{y} = 2 \times 10 + 5 = 25$$

Question 1

Which of the following best describes the purpose of a regression line in statistics?

Accepted Answer

It summarizes the overall trend between two variables and helps make predictions.

Question 2

What is the main criterion used to determine the 'best fit' regression line for a set of data points?

Accepted Answer

Minimizing the sum of the squared residuals.

Question 3

Given the regression equation $$y = mx + b$$, what does the slope $$m$$ represent in the context of predicting home prices from home size?

Accepted Answer

The average increase in price for each additional unit of home size.

Question 4

Suppose the regression line for predicting home price (in thousands) from size (in square feet) is $$y = 0.15x + 50. $$What is the predicted price for a 2000 square foot home?

Accepted Answer

$$y = 0.15 	imes 2000 + 50 = 350$$

Question 5

What is a residual in the context of regression analysis?

Accepted Answer

The difference between the observed y-value and the predicted y-value for a given x.

Question 6

Why do we square the residuals when calculating the regression line using the least squares method?

Accepted Answer

To make all residuals positive so they do not cancel each other out.

Question 7

Which of the following statements about $$R^2 (R-$$squared) is correct?

Accepted Answer

$$R^2$$ measures the proportion of variation in y explained by x.

Question 8

A residual plot for a regression model shows a random scatter of points around zero with no clear pattern. What does this indicate about the model?

Accepted Answer

The linear model is appropriate for the data.

Question 9

If the regression line for predicting y from x is $$y = 2x + 5$$, and the mean of x is 10, what is the mean of y?

Accepted Answer

$$\bar{y} = 2 	imes 10 + 5 = 25$$

Question 10

Which of the following is NOT a valid interpretation of the regression line in a real estate context?

Accepted Answer

It guarantees the exact price for every home of a given size.

Question 11

Which formula gives the predicted y-value ($$\hat{y}) $$for a given x-value in a regression equation?

Accepted Answer

$$\hat{y} = mx + b$$

Question 12

A real estate analyst finds that $$R^2 = 0.85$ for the regression of home price on home size. What does this mean?

Accepted Answer

85% of the variation in home price is explained by home size.

Question 13

If a regression line is calculated using the least squares method, which point does it always pass through?

Accepted Answer

The point $$(\bar{x}, \bar{y})$$, where $$\bar{x}$$ and $$\bar{y}$$ are the means of x and y.

Question 14

Which of the following scenarios would suggest that a linear regression model is NOT appropriate for the data?

Accepted Answer

The residual plot shows a clear curve or pattern.

Question 15

Suppose you use a regression equation to predict the price of a home much larger than any in your data set. What is this practice called, and why is it risky?

Accepted Answer

Extrapolation; because the relationship may not hold outside the observed range.

Chapter 9: Correlation and Regression – Linear Regression and Model Assessment

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