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.
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 9.3.8
"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?
8.r =- 0.328"
Verified step by step guidance1
Step 1: Recall the formula for the coefficient of determination, which is r². This is calculated by squaring the value of the correlation coefficient r. In this case, r = -0.328, so r² = (-0.328)².
Step 2: Compute r² by squaring the value of r. Note that squaring a negative number results in a positive value, so the sign of r does not affect r².
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. For example, if r² = 0.107, this means 10.7% of the variation in y is explained by x.
Step 4: Calculate the unexplained variation. This is the remaining proportion of the variation in y that is not explained by x. It is given by 1 - r². For instance, if r² = 0.107, then the unexplained variation is 1 - 0.107 = 0.893, or 89.3%.
Step 5: Summarize the findings. The coefficient of determination (r²) quantifies how well the regression line fits the data. A higher r² value indicates a better fit, meaning more of the variation in y is explained by x. Conversely, a lower r² value indicates a weaker fit, with more unexplained variation.
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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 -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. In this context, a negative value of r suggests that as one variable increases, the other tends to decrease.
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Correlation Coefficient
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. An r^2 value close to 1 indicates that a large proportion of the variance is explained, while a value close to 0 suggests that little variance is explained.
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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 factors not captured by the independent variable. Understanding these concepts helps in assessing the effectiveness of the regression model in predicting outcomes.
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Related Practice
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."
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.
Textbook Question
Graphical Analysis In Exercises 11–14, determine whether there is a perfect positive linear correlation, a strong positive linear correlation, a perfect negative linear correlation, a strong negative linear correlation, or no linear correlation between the variables.
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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.
22. Mean Hourly Wage Construct a 95% prediction interval for the mean hourly wage in Exercise 12 when the median hourly wage is \$21.50."
Textbook Question
"Graphical Analysis In Exercises 1–3, use the figure.
1. Describe the total variation about a regression line in words and in symbols."