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Chapter 13: Problem 64

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

Short Answer

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

Step by step solution

01

Understand the terms

The slope of the regression line (or just 'slope') describes the steepness, incline, gradient, or declination of the line. It can be presumed from its sign whether the correlation between two variables is positive or negative. A negative slope means that as one variable increases, the other decreases. The correlation coefficient, \(r\), is a statistical measure that describes the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 1 a perfect positive linear relationship, and 0 no linear relationship.
02

Interpreting the relationship

If the slope \(b\) of the regression line has a negative value, it means that as one variable increases, the other decreases. According to the definition of the correlation coefficient \(r\), it will also be negative. The sign of \(r\) always matches the sign of \(b\).
03

Conclusion

Based on the analysed information, if the slope is negative, it implies the correlation coefficient will also be negative.

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Most popular questions from this chapter

For a sample data set on two variables, the value of the linear correlation coefficient is (close to) zero. Does this mean that these variables are not related? Explain.

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y^{\circ}}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is positive. h. Using a \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

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