Q: A car dealership wants to predict the number of cars sold per week ($$y) $$based on the number of TV ads run per week ($$x). $$The regression equation is $$\hat{y} = 4.47 + 3.89x. $$What is the predicted number of cars sold if 5 TV ads are run in a week?

$$\hat{y} = 4.47 + 3.89 \times 5 = 24.92$$

Question 1

Which of the following statements best describes the difference between the independent variable and the dependent variable in a simple linear regression model?

Accepted Answer

The independent variable explains the variation in the dependent variable, but not the reverse.

Question 2

Suppose a correlation coefficient $$r is $$calculated to be $$-0.85$$ between two variables. Which of the following best interprets this value?

Accepted Answer

There is a strong negative linear relationship between the two variables.

Question 3

A car dealership wants to predict the number of cars sold per week ($$y) $$based on the number of TV ads run per week ($$x). $$The regression equation is $$\hat{y} = 4.47 + 3.89x. $$What is the predicted number of cars sold if 5 TV ads are run in a week?

Accepted Answer

$$\hat{y} = 4.47 + 3.89 	imes 5 = 24.92$$

Question 4

Which of the following is NOT an assumption required for simple linear regression analysis?

Accepted Answer

The variance of the residuals increases as the independent variable increases.

Question 5

If the coefficient of determination $$R^2$$ for a regression model is $$0.70$$, what does this mean?

Accepted Answer

70% of the variation in the dependent variable is explained by the independent variable.

Question 6

A hypothesis test for the population correlation coefficient $$ho is $$conducted with $$H_0$$: ho = 0$$ and H_1$: ho 
eq 0. If $$the calculated $$t-$$statistic is $$4.2$$ and the critical value is $$2.776$$, what is the correct conclusion at $$\alpha = 0.05$$?

Accepted Answer

Reject $$H_0$$; there is evidence of a linear relationship.

Question 7

Which formula is used to calculate the slope $$b_1 of $$the regression line in simple linear regression?

Accepted Answer

$$b_1$$ = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x}$$)^2$$}$$

Question 8

In the context of financial markets, why is a correlation close to zero between assets desirable for investors?

Accepted Answer

It means the assets move independently, which helps diversify risk.

Question 9

Given the regression output: $$R^2 = 0.69954$, F = 9.313$, p$-value = 0.03796$, and F$$_{\alpha} = 7.709$$ for $$\alpha = 0.05$$, what is the correct interpretation?

Accepted Answer

Reject the null hypothesis; the independent variable explains a significant portion of the variation in the dependent variable.

Question 10

Which of the following best describes the difference between a confidence interval and a prediction interval in regression analysis?

Accepted Answer

A confidence interval estimates the range for the average value of $$y$$, while a prediction interval estimates the range for an individual value of $$y.$$

Question 11

Suppose the regression equation for predicting gas mileage (MPG) from speed (MPH) is $$\hat{y} = 40.7 - 0.26x. $$What is the predicted MPG at a speed of 60 MPH?

Accepted Answer

$$\hat{y} = 40.7 - 0.26 	imes 60 = 25.7$$

Question 12

Which of the following is a potential pitfall when using regression analysis for prediction?

Accepted Answer

Assuming correlation implies causation.

Question 13

If the residuals from a regression analysis show a clear pattern when plotted against the independent variable, what does this indicate?

Accepted Answer

The relationship between the variables is likely non-linear or another assumption is violated.

Question 14

Which formula is used to calculate the correlation coefficient $$r$$ for a sample?

Accepted Answer

$$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2$ - (\sum x)^2$][n\sum y^2$ - (\sum y)^2$]}}$$

Question 15

In the context of the law of supply and demand, what type of correlation would you expect between price and demand for a product?

Accepted Answer

A strong negative correlation.

Question 16

Which of the following best describes homoscedasticity in regression analysis?

Accepted Answer

The variance of the residuals is constant across all values of the independent variable.

Question 17

A regression analysis yields a slope estimate $$b_1 = 3.89$ with a standard error s_b = 1.276$. For n = 6$ and $$\alpha = 0.05$$, the critical t$-value is 2.776$. What is the 95% confidence interval for the slope?

Accepted Answer

$$3.89 \pm 2.776 	imes 1.276$$

Question 18

Which of the following Excel functions can be used to calculate the correlation coefficient between two variables?

Accepted Answer

=CORREL(array1, array2)

Correlation and Simple Linear Regression: Chapter 14 Study Guide

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