Question 1

Which of the following best describes the meaning of $$R^2 ($$coefficient of determination) in a regression analysis?

Accepted Answer

$$R^2 is $$the proportion of the variation in the dependent variable that is accounted for by the regression model.

Question 2

Given the regression equation $$\hat{Y} = 4.72 + 0.70X$$, what is the predicted value of $$Y$$ when $$X = 10$$?

Accepted Answer

$$\hat{Y} = 11.72$$

Question 3

Which of the following is NOT a required condition for performing a linear regression analysis?

Accepted Answer

The residuals must all be positive.

Question 4

If the correlation coefficient $$r = 0.76$$ for a dataset, what is the value of $$R^2$$?

Accepted Answer

$$R^2 = 0.5776$

Question 5

A regression model for calories ($$Y) on $$sugar ($$X) in $$breakfast cereals is $$\hat{Y} = 89.6 + 2.49X. $$What is the predicted calorie content for a cereal with 10g of sugar?

Accepted Answer

$$\hat{Y} = 114.5$$

Question 6

Which of the following statements about extrapolation is correct?

Accepted Answer

Extrapolation can be dangerous because it assumes future changes will continue at the same rate as observed in the past.

Question 7

In the context of regression, what does it mean if the residual plot shows a clear pattern (e.g., a curve)?

Accepted Answer

The straight enough condition is violated, and a nonlinear model may be more suitable.

Question 8

Suppose you fit a regression model to predict fat content from protein content in fast food items. The residual plot shows increasing spread for higher protein values. Which regression assumption is violated?

Accepted Answer

Equal variance

Question 9

Why is it important to report $$R^2$$ along with the regression equation?

Accepted Answer

$$R^2$$ allows readers to judge how well the regression model fits the data.

Question 10

Which of the following is a common mistake when interpreting regression results?

Accepted Answer

Claiming that a change in $$X$$ causes a change in $$Y$$ based solely on the regression.

Question 11

Given $$r = 0.564$$ for the relationship between sugar and calories in cereals, what percent of the variation in calories is explained by sugar?

Accepted Answer

31.8%

Question 12

Which scenario best illustrates the danger of extrapolation?

Accepted Answer

Using a regression model for crude oil prices from 1970-1990 to predict prices in 2025.

Question 13

If a regression model has $$R^2 = 0.45$, what does this mean?

Accepted Answer

45% of the variation in $$Y is $$explained by the model.

Question 14

Which of the following is NOT a recommended practice when fitting a regression model?

Accepted Answer

Fitting a straight line to a nonlinear relationship.

Question 15

A student fits a regression line to predict Exam 2 scores from Exam 1 scores and finds $$r = 0.67. $$What is the $$R^2$$ value, and what does it represent?

Accepted Answer

$$R^2 = 0.45$, representing 45% of the variation in Exam 2 scores explained by Exam 1 scores.

Question 16

Which of the following is a sign that the 'straight enough' condition for regression may not be satisfied?

Accepted Answer

The scatterplot of the data is curved.

Question 17

Why is it problematic to invert the regression equation (predict $$X$$ from $$Y$$ using the regression of $$Y on$$ $$X$$)?

Accepted Answer

The regression of $$Y on$$ $$X$$ minimizes vertical distances, not horizontal distances.

Question 18

Which of the following best describes the 'equal variance' condition in regression?

Accepted Answer

The spread of the data around the regression line is similar for all values of $$X.$$

Question 19

In a regression analysis, what is the main purpose of examining the residual plot?

Accepted Answer

To check for patterns that indicate violations of regression assumptions.

Regression, $R^2$, and Extrapolation: Key Concepts and Pitfalls

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