Textbook Question
"In Exercises 19-22, two variables are given that have been shown to have correlation but no cause-and-effect relationship. Describe at least one possible reason for the correlation.
19. Value of home and life span"
Ch. 9 - Correlation and Regression
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 9, Problem 9.1.16
"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?
16. r = 0.795"
Verified step by step guidance1
Identify the given correlation coefficient, which is .
Calculate the coefficient of determination by squaring the correlation coefficient: . This value is denoted as .
Interpret as the proportion of the total variation in the dependent variable that is explained by the variation in the independent variable through the regression line.
Understand that the explained variation corresponds to , while the unexplained variation corresponds to , representing the variation not accounted for by the regression model.
Summarize that a higher value indicates a better fit of the regression line to the data, meaning more of the variation is explained, and less is unexplained.
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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 or -1 indicate a strong linear relationship, and values near 0 indicate little to no linear association.
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Correlation Coefficient
Coefficient of Determination (r²)
The coefficient of determination, r², is the square of the correlation coefficient and represents the proportion of the variance in the dependent variable explained by the independent variable. It ranges from 0 to 1, with higher values indicating a better fit of the regression line to the data.
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Explained vs. Unexplained Variation
Explained variation refers to the portion of total variation in the data accounted for by the regression model, quantified by r². Unexplained variation is the remaining variation not captured by the model, representing random error or other factors outside the regression.
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Related Practice
Textbook Question
"Old Vehicles In Exercises 31–34, use the figure shown at the left.
Regression Line Find and draw the regression line."
Textbook Question
"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
4. Sorghum Yield The equation used to predict the annual sorghum yield (in bushels per
acre) is y = 80.1-20.2x_1 +21.2x_2
where x_1 is the number of acres planted (in millions) and x_2 is the number of acres harvested (in millions). (Adapted from United States Department of Agriculture)
a. x_1 = 5.5, x_2 = 3.9
b. x_1 = 8.3, x_2 = 7.3
c. x_1 = 6.5, x_2 = 5.7
d. x_1 = 9.4, x_2= 7.8"