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Chapter 13: Problem 63

For a sample data set, the linear correlation coefficient \(r\) has a positive value. Which of the following is true about the slope \(b\) of the regression line estimated for the same sample data? a. The value of \(b\) will be positive. b. The value of \(b\) will be negative. c. The value of \(b\) can be positive or negative.

Short Answer

Expert verified
The value of \(b\) will be positive.

Step by step solution

01

Understanding the linear correlation coefficient \(r\)

The linear correlation coefficient, also known as Pearson's \(r\), describes the strength and direction of the relationship between two variables. A positive \(r\) indicates a positive linear relationship, meaning as one variable increases, so does the other. Alternatively, as one variable decreases, the other also decreases.
02

Relating the linear correlation coefficient to the slope of the regression line

In a linear regression model, the slope \(b\) of the regression line represents the rate at which the dependent variable changes for every unit change in the independent variable. If the linear correlation coefficient \(r\) is positive, this means that as one variable increases, the other also increases. Correspondingly, the slope of the regression line would also be positive.
03

Concluding the relationship

Hence, if the correlation coefficient is positive, the slope of the regression line would also be positive. This is because, the slope of the regression line and the correlation coefficient both indicate the direction of the relationship between two variables.

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