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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 9.3.7
Chapter 9, Problem 9.3.7

"In Exercises 7-10, 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?
7. r =0.465"

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Identify the given correlation coefficient, which is r = 0.465.
Calculate the coefficient of determination by squaring the correlation coefficient: r2 = (0.465)^2. This value is denoted as r^2.
Interpret r^2 as the proportion of the total variation in the dependent variable that is explained by the independent variable through the regression line. It represents the explained variation.
Understand that the unexplained variation is the remaining proportion of the total variation, which can be found by subtracting r^2 from 1, i.e., 1 - r^2.
Summarize that a higher r^2 value indicates a better fit of the regression line to the data, meaning more variation is explained, while a lower r^2 means more variation remains 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 strong positive or negative linear relationships, respectively, and values near 0 indicate weak or no linear relationship.
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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 in a regression model. 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. Together, they sum to the total variation in the dependent variable.
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Related Practice
Textbook Question

Two variables have a bivariate normal distribution. Explain what this means.

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Textbook Question

3. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation: r =0.918 or r =- 0.932? Explain your reasoning.

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Textbook Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

24. Trees Construct a 90% prediction interval for the trunk diameter of a tree in Exercise 14 when the height is 80 feet."

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Textbook Question

"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.

10. y-intercept

a. \(\hat{y}\)_i

b. y_i

c. b

d. (\(\bar{x}\), \(\bar{y}\))

e. m

f. \(\bar{y}\)"

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Textbook Question

"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.

7. The y-value of a data point corresponding to x;

a. \(\hat{y}\)_i

b. y_i

c. b

d. (\(\bar{x}\), \(\bar{y}\))

e. m

f. \(\bar{y}\)"

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Textbook Question

In Exercise 24, remove the data for the student who is 57 inches tall and scored 128 on the IQ test. Describe how this affects the correlation coefficient r.

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