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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 9 - Correlation and RegressionProblem 9.RE.15
Chapter 9, Problem 9.RE.15

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
15. r = 0.642"

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1
Step 1: Recall the formula for the coefficient of determination, r², which is the square of the correlation coefficient r. Mathematically, r² = r × r.
Step 2: Substitute the given value of r (0.642) into the formula. This means you will calculate r² = (0.642)².
Step 3: Interpret the coefficient of determination, r². It represents the proportion of the variation in the dependent variable (y) that is explained by the independent variable (x) through the regression line.
Step 4: To find the explained variation, multiply r² by 100 to express it as a percentage. This percentage indicates how much of the total variation in the data is explained by the regression model.
Step 5: To find the unexplained variation, subtract the explained variation percentage from 100%. This represents the proportion of the variation in the data that is not explained by the regression model.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Correlation Coefficient (r)

The correlation coefficient, denoted as r, measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values around 0 suggest no linear correlation. Understanding r is crucial for interpreting the relationship between the variables in a dataset.
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Coefficient of Determination (r^2)

The coefficient of determination, represented as r^2, quantifies the proportion of variance in the dependent variable that can be explained by the independent variable in a regression model. It is calculated by squaring the correlation coefficient (r). An r^2 value closer to 1 indicates that a large proportion of the variance is explained by the model, while a value closer to 0 suggests that the model explains little of the variance.
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Explained vs. Unexplained Variation

Explained variation refers to the portion of the total variation in the dependent variable that is accounted for by the regression model, as indicated by r^2. Conversely, unexplained variation is the portion that remains after accounting for the model, representing the variability that cannot be predicted by the independent variable. Understanding these concepts helps in assessing the effectiveness of the regression model in capturing the underlying data patterns.
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Related Practice
Textbook Question

"In Exercises 27 and 28, use the multiple regression equation to predict the y-values for the values of the independent variables.

28. Use the regression equation found in Exercise 25.

a. x_1 = 9.0, x_2 = 0.70

b. x_1 = 3.0, x_2 = 0.25

c. x_1 = 8.0, x_2 = 0.60

d. x_1 = 5.2, x_2 = 0.46"

Textbook Question

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?

13. r =- 0.450"

Textbook Question

"[APPLET] For Exercises 2–9, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years. (Source: U.S. Bureau of Labor Statistics)

8. Find the standard error of estimate Se and interpret the result."

Textbook Question

"The U.S. Food and Drug Administration (FDA) requires nutrition labeling for most foods. Un FDA regulations, manufacturers are required to list the amounts of certain nutrients in their foods, such as calories, sugar, fat, and carbohydrates. This nutritional information is displayed in the ""Nutrition Facts"" panel on the food's package.

The table shows the nutritional content below for one cup of each of 21 different breakfast

cereals.

C = calories

S = sugar in grams

F = fat in grams

R = carbohydrates in grams

7. Use the equations from Exercise 6 to predict the calories in 1 cup of cereal that has 7 grams of sugar, 0.5 gram of fat, and 31 grams of carbohydrates."

Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."

Textbook Question

"1. Net Sales The equation used to predict the net sales (in millions of dollars) for a fiscal

year for a clothing retailer is y=23,769 + 9.18x_1 - 8.41x_2

where x_1 is the number of stores open at the end of the fiscal year and x_2 is the average

square footage per store. Use the multiple regression equation to predict the y-values for

the values of the independent variables.

a. x_1 = 1057, x_2 = 3698

b. x_1 = 1012, x_2 = 3659

c. x_1 = 952, x_2 = 3601

d. x_1 = 914, x_2 = 3594"