Q: Given the regression equation $$\hat{y} = 2775.38 - 248.90x$$, what is the predicted number of beer packs sold when the price is $9.5?

$$\hat{y} = 2775.38 - 248.90 \times 9.5 = 1320.83$$

Question 1

Which of the following best describes the meaning of the slope in the regression equation $$\hat{y} = 2775.38 - 248.90x$$ for beer sales versus price?

Accepted Answer

For each $1 increase in price, the predicted sale decreases by 248.90 packs.

Question 2

Given the regression equation $$\hat{y} = 2775.38 - 248.90x$$, what is the predicted number of beer packs sold when the price is $9.5?

Accepted Answer

$$\hat{y} = 2775.38 - 248.90 	imes 9.5 = 1320.83$$

Question 3

What does the coefficient of determination $$R^2 = 0.737$ indicate about the regression model for beer sales and price?

Accepted Answer

About 73.7% of the variation in beer sales is explained by the price.

Question 4

If a store sells 449 packs at a price of $$8.4, what is the residual for this observation using the regression equation $$\hat{y} = 2775.38 - 248.90x$?

Accepted Answer

Residual = $$449 - 684.24 = -235.24$$

Question 5

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

Accepted Answer

The explanatory variable must be categorical.

Question 6

What is the correct interpretation of the y-intercept $$b_0 = 2775.38$ in the regression equation for beer sales?

Accepted Answer

It is the predicted sale when the price is $$0, but it may not have practical meaning since 0 is $$outside the observed range.

Question 7

Which value of the correlation coefficient $$r$$ indicates the strongest negative linear relationship?

Accepted Answer

$$r = -0.95$$

Question 8

In the hypothesis test for the slope $$\$$beta_1$$ in regression, what is the null hypothesis?

Accepted Answer

$$H_0$$: \$$beta_1 = 0$

Question 9

If the p-value for the slope in the regression is less than 0.05, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is a significant linear relationship between price and sales.

Question 10

What is the 95% confidence interval for the slope $$\$$beta_1$$ in the regression model for beer sales and price?

Accepted Answer

$$(-291.10, -206.70)$$

Question 11

Which plot is used to check the normality assumption of residuals in regression analysis?

Accepted Answer

Normal quantile plot (Q-Q plot) of residuals.

Question 12

What does the Root Mean Square Error (RMSE) represent in the Summary of Fit table?

Accepted Answer

The standard deviation of the residuals, measuring the average prediction error.

Question 13

Which formula gives the standard error for the estimated mean value of $$Y at$$ $$x_0$$?

Accepted Answer

$$SE(\hat{\mu}_0) = \sqrt{SE^2$(b_1$) \cdot (x_0$ - \bar{x})^2$ + \frac{S_e^2$}{n}}$$

Question 14

Which formula gives the standard error for the predicted individual value of $$Y at$$ $$x_0$$?

Accepted Answer

$$SE(\hat{y}_0) = \sqrt{SE^2$(b_1$) \cdot (x_0$ - \bar{x})^2$ + \frac{S_e^2$}{n} + S_e^2$}$$

Question 15

Which of the following best describes extrapolation in regression analysis?

Accepted Answer

Using the regression equation to predict values outside the range of observed data.

Question 16

If the scatterplot of residuals shows a 'fan' shape, which regression assumption is likely violated?

Accepted Answer

Equal variance assumption.

Question 17

What is the main purpose of constructing a confidence interval for the slope $$\$$beta_1$$ in regression?

Accepted Answer

To estimate the range of plausible values for the true population slope.

Question 18

Which of the following statements about the regression model for beer sales and price is correct?

Accepted Answer

The model predicts lower sales as price increases, and explains about 74% of the variation in sales.

Simple Linear Regression Analysis: Price and Sales Example

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