Q: Which formula is used to calculate the slope $$b_1 of $$the least-squares regression line?

$$b_1 = r$$ \times \frac{s_y}{s_x}$$

Question 1

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

Accepted Answer

It is the line that minimizes the sum of the squared vertical distances (errors) between observed and predicted values.

Question 2

Given the regression equation $$\hat{y} = 3.1661x - 55.7966$$, what is the predicted value of $$y$$ when $$x = 103$$?

Accepted Answer

$$\hat{y} = 270.3$$

Question 3

Suppose the observed value of $$y$$ for $$x = 103 is 274. $$What is the residual for this observation using the regression equation $$\hat{y} = 3.1661x - 55.7966$$?

Accepted Answer

Residual $$= 274 - 270.3 = 3.7$$

Question 4

Which formula is used to calculate the slope $$b_1 of $$the least-squares regression line?

Accepted Answer

$$b_1 = r$$ 	imes \frac{s_y}{s_x}$$

Question 5

Why should you not interpret the y-intercept of a regression equation if the value $$x = 0 is $$not reasonable or not observed in the data?

Accepted Answer

Because the regression model may not be valid outside the range of observed $$x$$ values.

Question 6

If the correlation coefficient $$r is 0.5$$, $$s_x = 2$$, $$s_y = 4$$, $$\bar{x} = 5$$, and $$\bar{y} = 7$$, what is the equation of the least-squares regression line?

Accepted Answer

$$\hat{y} = x + 5$$

Question 7

What does the slope of a least-squares regression line represent in the context of club-head speed and golf ball distance?

Accepted Answer

The expected increase in distance for each 1 mph increase in club-head speed.

Question 8

Which of the following is true when switching the explanatory and response variables in regression analysis?

Accepted Answer

The correlation coefficient $$r$$ remains the same, but the regression line changes.

Question 9

Mathematically, the least-squares regression line minimizes which quantity?

Accepted Answer

$$\sum_{i=1}^n (y_i - (b_0$ + b_1$ x_i))^2$

Question 10

A researcher uses a regression model to predict golf ball distance for a club-head speed of 140 mph, even though the highest observed speed is 105 mph. What is the main issue with this prediction?

Accepted Answer

It is outside the scope of the model and may not be reliable.

Question 11

Which statement about residuals is correct?

Accepted Answer

Residuals are the difference between observed and predicted values: $$	ext{Residual} = y - \hat{y}.$$

Question 12

In the context of regression, what is the main difference between correlation and regression?

Accepted Answer

Regression specifies which variable is explanatory and which is response, while correlation does not.

Question 13

If a regression line is fitted to a scatterplot, what criterion is used to determine the 'best' line?

Accepted Answer

The line that minimizes the sum of the squares of the vertical distances from the points to the line.

Question 14

Which of the following is a valid interpretation of the slope in a regression equation relating club-head speed ($$x) to $$golf ball distance ($$y$$)?

Accepted Answer

For each 1 mph increase in club-head speed, the expected distance the golf ball will travel increases by the value of the slope.

Question 15

Why is it important not to use a regression model to make predictions for values of the explanatory variable that are much larger or smaller than those observed?

Accepted Answer

Because the linear relationship may not hold outside the observed range.

Question 16

If you switch the explanatory and response variables in a regression analysis, what happens to the sum of squared errors minimized by the regression line?

Accepted Answer

It changes from minimizing vertical distances to minimizing horizontal distances.

Question 17

Given the regression equation $$y = 2.8333x - 22.4967$$, what is the predicted distance for a club-head speed of 104 mph?

Accepted Answer

$$272.2$$ yards

Question 18

Which of the following is NOT a valid reason to avoid interpreting the y-intercept in a regression equation?

Accepted Answer

The correlation coefficient is positive.

Question 19

A dataset has a correlation coefficient $$r = -0.85. If a $$point far below the existing data is added, what happens to the correlation coefficient?

Accepted Answer

It becomes less than $$-0.85.$$

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