Question 1

Which of the following best describes a residual in linear regression?

Accepted Answer

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

Question 2

Given the regression line equation $$\hat{y} = mx + b$$, what does $$b$$ represent?

Accepted Answer

The y-intercept, or the value of y when x equals zero.

Question 3

If the correlation coefficient $$r$$ between two variables is close to zero, what does this imply about the regression line?

Accepted Answer

The regression line does not provide meaningful predictions because the variables are not strongly related.

Question 4

Which formula is used to calculate the slope $$m of $$the regression line?

Accepted Answer

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

Question 5

A regression line is only meaningful for predictions when:

Accepted Answer

The x-value is within the range of the original data and the correlation is strong.

Question 6

What is the main purpose of the regression line in statistics?

Accepted Answer

To model the relationship between the response variable y and the explanatory variable x and make predictions.

Question 7

If the slope $$m of a $$regression line is negative, what does this indicate about the relationship between x and y?

Accepted Answer

There is a negative linear correlation; as x increases, y decreases.

Question 8

Given the following sums for a data set: $$\sum x = 54$$, $$\sum y = 908$$, $$\sum xy = 3724$$, $$\sum x^2$ = 332$$, and $$n = 12$$, what is the slope $$m of $$the regression line?

Accepted Answer

$$m = \frac{(12 \ast 3724) - (54 \ast 908)}{(12 \ast 332) - (54)^2$} \approx -4.067$$

Question 9

Why is extrapolation often unreliable when using a regression line?

Accepted Answer

Because the regression line models the trend only within the range of the given data, and the trend may not continue beyond that range.

Question 10

If you are given the regression equation $$\hat{y} = 0.065x + 0.465$$ and the range of x-values is 486 to 869, is it meaningful to predict y for $$x = 400$$?

Accepted Answer

No, because $$x = 400 is $$outside the range of the original data.

Question 11

What is the formula for calculating the y-intercept $$b of $$the regression line?

Accepted Answer

$$b = \left(\frac{\sum y}{n}\right) - \left(m \ast \frac{\sum x}{n}\right)$$

Question 12

A student uses the regression equation $$\hat{y} = -4.067x + 93.970 to $$predict the test score for $$x = 13$$ hours spent online, but the original data range for x is 0 to 10. What should the student conclude?

Accepted Answer

It is not meaningful to predict the value of y for $$x = 13$$, because $$x = 13 is $$outside the range of the original data.

Question 13

Which of the following statements about residuals is TRUE?

Accepted Answer

The smaller the residual, the more accurate the prediction.

Question 14

In the context of linear regression, what does the slope $$m$$ indicate?

Accepted Answer

The rate at which y changes for each unit increase in x.

Question 15

If a regression line is used to predict y for an x-value that is very close to, but not within, the original data range, what is this process called?

Accepted Answer

Extrapolation

Question 16

Which of the following is NOT a step in finding the regression line for a data set?

Accepted Answer

Calculating $$\sum y^2$ for the y-values.

Question 17

A researcher finds a regression line with equation $$\hat{y} = 12.45x + 9$$ and a correlation coefficient $$r = 0.8912. $$What can be concluded?

Accepted Answer

The regression line can produce meaningful predictions because the correlation is strong.

Question 18

Which equation represents the predicted y-value for a given x-value using the regression line?

Accepted Answer

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

Question 19

In a scatterplot, what does it mean if the regression line goes up from left to right?

Accepted Answer

There is a positive linear correlation between x and y.

Section 9.2 – Linear Regression: Regression Lines, Residuals, and Prediction

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