Q: Which formula gives 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 main criterion for the 'best' least-squares regression line?

Accepted Answer

It minimizes the sum of the squared vertical distances (errors) between the observed values and the 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

$$3.7$$

Question 4

Which of the following statements about the correlation coefficient $$r$$ and the regression line is TRUE?

Accepted Answer

Switching the explanatory and response variables does not change $$r$$, but it does change the regression line.

Question 5

Given summary statistics $$s_x = 2$$, $$s_y = 4$$, $$\bar{x} = 5$$, $$\bar{y} = 7$$, and $$r = 0.5$$, what is the equation of the least-squares regression line for predicting $$y$$ from $$x$$?

Accepted Answer

$$\hat{y} = 2x - 3$$

Question 6

What is the correct interpretation of the slope in the regression equation $$\hat{y} = 3.1661x - 55.7966$$ for golf club speed ($$x) $$and distance ($$y$$)?

Accepted Answer

For every 1 mph increase in club-head speed, the distance increases by 3.1661 yards, on average.

Question 7

Why should we not interpret the $$y-$$intercept in the regression equation for golf club speed and distance?

Accepted Answer

Because a club-head speed of 0 mph is not a reasonable or observed value in the data.

Question 8

Which formula gives the slope $$b_1 of $$the least-squares regression line?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Which of the following best defines a residual in the context of least-squares regression?

Accepted Answer

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

Question 11

Mathematically, the least-squares regression line minimizes which of the following quantities?

Accepted Answer

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

Question 12

If a regression line is used to predict $$y$$ for values of $$x$$ much larger or smaller than those observed, this is called:

Accepted Answer

Extrapolation, which is generally unreliable.

Question 13

Which of the following is NOT a requirement for interpreting the $$y-$$intercept of a regression line?

Accepted Answer

The $$y-$$intercept must be positive.

Question 14

A student claims that the regression line is the same regardless of which variable is chosen as explanatory or response. Is this correct?

Accepted Answer

No, because the regression line minimizes vertical distances for the response variable, and switching variables changes the line.

Question 15

In the context of least-squares regression, what does the slope $$b_1$$ represent?

Accepted Answer

The predicted change in $$y$$ for a one-unit increase in $$x.$$

Question 16

A scatterplot of data shows a strong linear relationship. Which of the following is the most appropriate next step?

Accepted Answer

Fit a least-squares regression line to summarize and predict the relationship.

Question 17

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 is the best possible linear fit to the data in the least-squares sense.

Question 18

Which of the following is a correct formula for the $$y-$$intercept $$b_0 of $$the least-squares regression line?

Accepted Answer

$$b_0$$ = \bar{y} - $$b_1$$ \bar{x}$$

Question 19

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

Accepted Answer

$$r$$ will become more negative (further from 0).

Question 20

Which of the following is a real-world example where using a regression model outside the observed data range could lead to misleading conclusions?

Accepted Answer

Predicting a child's height at age 50 using a regression model built from data on children aged 5-15.

Least Squares Regression and Correlation: Study Notes

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