Q: Which of the following is the correct formula for a residual $$d_i in $$least-squares regression?

$$d_i = (\text{observed y-value}) - (\text{predicted y-value})$$

Question 1

Which of the following best describes the purpose of the least-squares regression line in statistics?

Accepted Answer

To minimize the sum of the squares of the residuals between observed and predicted values.

Question 2

Given the regression equation $$\hat{y} = 1.249x + 18.29$$, where $$x is $$the percentage of adults with at least a bachelor's degree and $$\hat{y} is $$the predicted median income (in thousands of dollars), what is the predicted median income for a state where 25% of adults have at least a bachelor's degree?

Accepted Answer

$$\49$$,565$

Question 3

If Virginia has 41.8% of adults with at least a bachelor's degree and a median income of $$\66$$,000$$, use the regression equation $$\hat{y} = 1.249x + 18.29$ to determine if Virginia's median income is higher or lower than predicted. What is the predicted value?

Accepted Answer

Predicted income is $$\70$$,400$, so Virginia's income is lower than predicted.

Question 4

What does the slope $$1.249 in $$the regression equation $$\hat{y} = 1.249x + 18.29$$ represent in the context of the problem?

Accepted Answer

For every 1% increase in adults with a bachelor's degree, the median income increases by $$\1$$,249$ (in thousands of dollars).

Question 5

Why does it not make sense to interpret the y-intercept in the regression equation $$\hat{y} = 1.249x + 18.29$$ for this scenario?

Accepted Answer

Because a state with 0% adults with a bachelor's degree is not realistic.

Question 6

Which of the following is the correct formula for a residual $$d_i in $$least-squares regression?

Accepted Answer

$$d_i = (	ext{observed y-value}) - (	ext{predicted y-value})$$

Question 7

Suppose the regression line for a dataset is $$\hat{y} = -0.5x + 10. $$What does the negative slope indicate about the relationship between $$x$$ and $$y$$?

Accepted Answer

As $$x$$ increases, $$y$$ decreases.

Question 8

Which of the following statements about the regression line is TRUE?

Accepted Answer

The regression line always passes through the mean of the x-values and the mean of the y-values.

Question 9

If the sum of the squared residuals for a regression line is minimized, what does this imply about the fit of the line?

Accepted Answer

The line fits the data as closely as possible in the least-squares sense.

Question 10

A researcher fits a regression line to data and finds the equation $$\hat{y} = 2x + 5. $$What is the predicted value of $$y$$ when $$x = 7$$?

Accepted Answer

$$\hat{y} = 19$$

Question 11

Which of the following is NOT a valid interpretation of a residual in regression analysis?

Accepted Answer

A residual is the horizontal distance between an observed data point and the regression line.

Question 12

If a regression line has the equation $$\hat{y} = 3x - 4$$, what is the y-intercept and what does it represent?

Accepted Answer

The y-intercept is $$-4$$, representing the predicted value of $$y$$ when $$x = 0.$$

Question 13

A regression analysis is performed on data relating weekly cola consumption ($$x) $$and bone density ($$y$$), resulting in the equation $$\hat{y} = -0.0287x + 0.9005. $$What is the predicted bone density for someone who consumes 5 colas per week?

Accepted Answer

$$0.7560$$

Question 14

Which of the following best explains why the method of least squares is used in regression analysis?

Accepted Answer

It minimizes the sum of the squared differences between observed and predicted values.

Least-Squares Regression: Interpreting and Applying the Regression Line

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