Q: Suppose you fit a regression line to predict total fat (g) from protein (g) in Burger King menu items. If $$r = 0.827$$, $$\bar{x} = 17.1$$, $$\bar{y} = 23.1$$, $$s_x = 13.8$$, $$s_y = 16.3$$, what is the value of the slope $$b_1$$?

$$b_1$$ = \frac{r s_y}{s_x} = \frac{0.827 \times 16.3}{13.8} \approx 0.975$$

Question 1

Which of the following best describes the difference between correlation and simple linear regression?

Accepted Answer

Correlation treats both variables as equals, while regression distinguishes between an independent (predictor) and a dependent (response) variable.

Question 2

In the context of simple linear regression, what is the primary purpose of the least squares method?

Accepted Answer

To minimize the sum of squared residuals (errors) between observed and predicted values.

Question 3

Given the regression equation $$\hat{y} = b_0$ + b_1$ x$$, what does the slope $$b_1$$ represent?

Accepted Answer

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

Question 4

Suppose you fit a regression line to predict total fat (g) from protein (g) in Burger King menu items. If $$r = 0.827$$, $$\bar{x} = 17.1$$, $$\bar{y} = 23.1$$, $$s_x = 13.8$$, $$s_y = 16.3$$, what is the value of the slope $$b_1$$?

Accepted Answer

$$b_1$$ = \frac{r s_y}{s_x} = \frac{0.827 	imes 16.3}{13.8} \approx 0.975$$

Question 5

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

Accepted Answer

The predictor variable must be normally distributed.

Question 6

What is the definition of a residual in linear regression?

Accepted Answer

The difference between the observed value and the predicted value: $$e = y - \hat{y}$$

Question 7

If a residual plot shows a clear pattern or trend, what does this indicate about the linear regression model?

Accepted Answer

The linear model may not be appropriate; there may be non-linearity or violation of assumptions.

Question 8

Which statistic measures the proportion of variation in the response variable explained by the linear regression model?

Accepted Answer

The coefficient of determination $$R^2$$

Question 9

In a regression analysis, what is the effect of an influential data point?

Accepted Answer

It can unduly affect the estimated regression coefficients and predicted responses.

Question 10

Suppose you switch the roles of the explanatory and response variables in a regression. Is the new slope simply the reciprocal of the original slope?

Accepted Answer

No, the new slope is $$b_1$$^* = \frac{r s_x}{s_y}$$, not 1$$/$$b_1.$$

Question 11

Which R function would you use to obtain the predicted values from a fitted linear regression model?

Accepted Answer

predict(lm.obj)

Question 12

In multiple linear regression, what does the coefficient $$b_1$$ represent?

Accepted Answer

The change in the mean response of $$y$$ per unit increase in $$x_1$$, holding all other predictors constant.

Question 13

Which of the following best describes 'leverage' in regression analysis?

Accepted Answer

A data point with an extreme value of the predictor variable $$x.$$

Question 14

If the $$R^2$$ value of a regression model is 0.68, what does this mean?

Accepted Answer

68% of the variation in the response variable is explained by the regression model.

Question 15

Which diagnostic plot is most useful for checking the normality assumption of residuals in linear regression?

Accepted Answer

A Q-Q plot of the residuals

Question 16

In the context of regression to the mean, if the predictor variable is $$k$$ standard deviations from its mean, what is the expected number of standard deviations the response variable will be from its mean?

Accepted Answer

$$r 	imes k$$, where $$r is $$the correlation coefficient

Question 17

Which of the following is a potential issue when including many predictors in a multiple linear regression model?

Accepted Answer

Some predictors may be highly correlated, leading to inaccurate estimation and inefficiency.

Question 18

What is the formula for the total sum of squares (SST) in regression analysis?

Accepted Answer

$$SST = \sum_{i=1}^n (y_i - \bar{y})^2$

Question 19

Which of the following R commands fits a multiple linear regression model with three predictors $$x_1$$, $$x_2$$, and $$x_3$$ for response $$y$$?

Accepted Answer

lm(y ~ x1 + x2 + x3, data)

Question 20

Which of the following best describes the interpretation of the intercept $$b_0 in a $$simple linear regression equation?

Accepted Answer

The expected value of $$y$$ when $$x = 0$$

Linear Regression: Concepts, Methods, and Diagnostics

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