Q: Which of the following is the correct formula for the coefficient of determination?

$$r^2$$ = \frac{\text{Explained variation}}{\text{Total variation}}$$

Q: Which of the following is true about the relationship between total, explained, and unexplained variation?

$$\text{Total variation} = \text{Explained variation} + \text{Unexplained variation}$$

Question 1

Which of the following best describes the total variation in a regression analysis?

Accepted Answer

The sum of the squares of the differences between each observed y-value and the mean of y.

Question 2

Given the regression equation $$\hat{y} = 545.290x - 827.355$$, and the sum $$\sum (y_i - \hat{y}_i)^2$ = 312625.8$$ for $$n = 11$$ data points, what is the standard error of estimate?

Accepted Answer

$$s_e = \sqrt{\frac{312625.8}{11-2}} = 186.38$$

Question 3

What does the coefficient of determination ($$r^2) $$represent in the context of regression analysis?

Accepted Answer

The proportion of the total variation in y that is explained by the regression line.

Question 4

If the coefficient of determination $$r^2 = 0.905$, what percentage of the variation in the dependent variable is unexplained by the regression model?

Accepted Answer

9.5%

Question 5

Which formula correctly calculates the unexplained variation in a regression analysis?

Accepted Answer

$$\sum (y_i - \hat{y}_i)^2$

Question 6

A researcher finds $$r^2 = 0.964$ for the relationship between children’s age and vocabulary size. What does this imply?

Accepted Answer

96.4% of the variation in vocabulary size can be explained by the variation in age.

Question 7

Which of the following is the correct formula for the coefficient of determination?

Accepted Answer

$$r^2$$ = \frac{	ext{Explained variation}}{	ext{Total variation}}$$

Question 8

In a study of building heights and number of stories, the regression output gives a standard error of 2.138. What does this value represent?

Accepted Answer

The average distance of the observed number of stories from the regression line is about 2.138 stories.

Question 9

Which of the following steps is NOT required when calculating the standard error of estimate?

Accepted Answer

Calculate the sum of the squares of the differences between each observed y-value and the mean of y.

Question 10

A regression analysis yields $$r^2 = 0.973$ for the relationship between temperature and ice cream cones sold per hour. What is the best interpretation?

Accepted Answer

97.3% of the variation in ice cream cones sold per hour can be explained by temperature.

Question 11

Which of the following is true about the relationship between total, explained, and unexplained variation?

Accepted Answer

$$	ext{Total variation} = 	ext{Explained variation} + 	ext{Unexplained variation}$$

Question 12

Suppose you are using Excel to perform a regression analysis. Which output value directly gives you the coefficient of determination?

Accepted Answer

R Square

Question 13

If the standard error of estimate is close to zero, what does this indicate about the regression model?

Accepted Answer

The observed y-values are very close to the predicted y-values, indicating a good fit.

Question 14

Which of the following best describes explained variation in regression?

Accepted Answer

The sum of the squares of the differences between each predicted y-value and the mean of y.

Question 15

A city planner uses regression to predict the number of stories in a building based on its height. The regression output gives $$r^2 = 0.905$. What does this mean in practical terms?

Accepted Answer

About 90.5% of the variation in the number of stories can be explained by the building's height.

Measures of Regression and Prediction Intervals: Variation, Coefficient of Determination, and Standard Error of Estimate

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