Q: Given $$r = 0.8$$, $$s_x = 2$$, $$s_y = 5$$, what is the slope $$b_1 of $$the least-squares regression line?

$$b_1 = 0.8$$ \times \frac{5}{2} = 2$$

Q: Suppose the least-squares regression line for predicting distance (yards) from club-head speed (mph) is $$\hat{y} = 3.1661x - 55.7966. $$What is the predicted distance for a club-head speed of 103 mph?

$$\hat{y} = 3.1661 \times 103 - 55.7966 = 270.3$$ yards

Question 1

Which of the following best describes the response variable in a scatter diagram?

Accepted Answer

The variable whose value can be explained by the value of the explanatory or predictor variable.

Question 2

A scatter diagram of club-head speed (x) and golf ball distance (y) shows that as club-head speed increases, the distance also increases. What type of association is this?

Accepted Answer

Positive association

Question 3

Which of the following is NOT a property of the linear correlation coefficient $$r$$?

Accepted Answer

$$r is $$resistant to outliers in the data.

Question 4

Given the following data for $$x$$ and $$y$$: $$\bar{x} = 4$$, $$s_x = 2.44949$$, $$\bar{y} = 10$$, $$s_y = 5.612486$$, and $$\sum \frac{(x_i - \bar{x})}{s_x} \frac{(y_i - \bar{y})}{s_y} = -3.7824$$ for $$n = 5$$, what is the sample correlation coefficient $$r ($$rounded to three decimal places)?

Accepted Answer

$$r = -0.946$$

Question 5

Suppose a study finds a correlation coefficient of $$r = 0.939$$ between club-head speed and driving distance for $$n = 8$$ samples. The critical value for $$n = 8 is$$ $$0.707. $$What can you conclude?

Accepted Answer

There is a positive linear relation between club-head speed and distance.

Question 6

Which of the following statements about correlation and causation is correct?

Accepted Answer

A lurking variable can explain a correlation between two variables without a direct causal relationship.

Question 7

In a study, both air-conditioning bills and crime rates increase as temperature rises. What is the lurking variable in this scenario?

Accepted Answer

Air temperature

Question 8

Which formula correctly represents the sample linear correlation coefficient $$r$$?

Accepted Answer

$$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$$

Question 9

A scatter diagram shows points forming a pattern that curves upward. Which of the following best describes the relationship between the variables?

Accepted Answer

There is a nonlinear relationship.

Question 10

If the data points in a scatter diagram are equally dispersed in all four quadrants, what is the likely value of the correlation coefficient $$r$$?

Accepted Answer

$$r is $$close to $$0$$

Question 11

Which of the following is the equation of the least-squares regression line?

Accepted Answer

$$\hat{y} = b_1$ x + b_0$

Question 12

Given $$r = 0.8$$, $$s_x = 2$$, $$s_y = 5$$, what is the slope $$b_1 of $$the least-squares regression line?

Accepted Answer

$$b_1 = 0.8$$ 	imes \frac{5}{2} = 2$$

Question 13

Suppose the least-squares regression line for predicting distance (yards) from club-head speed (mph) is $$\hat{y} = 3.1661x - 55.7966. $$What is the predicted distance for a club-head speed of 103 mph?

Accepted Answer

$$\hat{y} = 3.1661 	imes 103 - 55.7966 = 270.3$$ yards

Question 14

What does the residual represent in the context of least-squares regression?

Accepted Answer

The difference between the observed value of $$y$$ and the predicted value of $$y.$$

Question 15

Why is it generally inappropriate to interpret the y-intercept of a regression line if $$x = 0 is $$not a reasonable value for the explanatory variable?

Accepted Answer

Because the regression model is only valid for values of $$x$$ within the observed range.

Question 16

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

Accepted Answer

It minimizes the sum of the squared residuals.

Question 17

A researcher finds a correlation of $$r = -0.806$$ between number of colas consumed per week and bone mineral density, with a critical value of $$0.514$$ for $$n = 15. $$What is the correct interpretation?

Accepted Answer

There is a significant negative linear relation between cola consumption and bone mineral density.

Question 18

Which of the following is a correct interpretation of the slope in the regression equation $$\hat{y} = 3.1661x - 55.7966$$ for club-head speed and distance?

Accepted Answer

For each 1 mph increase in club-head speed, the expected distance increases by 3.1661 yards.

Question 19

If the sum of squared residuals for a line is 56.91, and for the least-squares regression line it is 49.46, what does this indicate?

Accepted Answer

The least-squares regression line fits the data better than the other line.

Chapter 4: Describing the Relation between Two Variables – Scatter Diagrams and Correlation

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