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Chapter 5: Problem 34

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

Short Answer

Expert verified
The sign of the slope \(b\) of the least-squares line is always the same as the sign of the sample correlation coefficient \(r\) because both measure the direction of the linear relationship between two variables. A positive sign indicates y generally increases as x increases, while a negative sign indicates y generally decreases as x increases.

Step by step solution

01

- Definition of Correlation Coefficient

The correlation coefficient (\(r\)) measures the strength and direction of a linear relationship between two variables. The value of the correlation coefficient varies between -1 and 1, where -1 indicates a perfect negative linear relationship, 1 a perfect positive linear relationship, and 0 no linear relationship. The sign of the correlation coefficient indicates the direction of the relationship.
02

- Explanation of the Least-Squares Line

The least-squares line, also known as the regression line, represents the best-fit line that minimizes the sum of the squared differences (or errors) between the observed and predicted values. The slope of this line, denoted here as \(b\), will be positive if, in general, y increases as x increases and negative if y decreases as x increases.
03

- Relation Between the Signs of b and r

Since both the correlation coefficient and the slope of the least-squares line indicate the direction of a linear relationship, their signs will always be the same. If the correlation coefficient is positive, indicating a positive linear relationship, the slope of the least-squares line will also be positive. Conversely, if the correlation coefficient is negative, indicating a negative linear relationship, the slope of the least-squares line will also be negative.

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Most popular questions from this chapter

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.

Both \(r^{2}\) and \(s_{e}\) are used to assess the fit of a line. a. Is it possible that both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set? Explain. (A picture might be helpful.) b. Is it possible that a bivariate data set could yield values of \(r^{2}\) and \(s_{e}\) that are both small? Explain. (Again, a picture might be helpful.) c. Explain why it is desirable to have \(r^{2}\) large and \(s_{e}\) small if the relationship between two variables \(x\) and \(y\) is to be described using a straight line.

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have fulltime jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17,2001\(]: 86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at \(15.4\) percent, and SAT I was last at \(13.3\) percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would the value of \(r^{2}\) have been? b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would have been very accurate? Explain why or why not.

The paper "Root Dentine Transparency: Age Determination of Human Teeth Using Computerized Densitometric Analysis" (American Journal of Physical Anthropology \([1991]: 25-30\) ) reported on an investigation of methods for age determination based on tooth characteristics. With \(y=\) age (in years) and \(x=\) percentage of root with transparent dentine, a regression analysis for premolars gave \(n=36\), SSResid \(=5987.16\), and \(\mathrm{SSTo}=\) \(17,409.60 .\) Calculate and interpret the values of \(r^{2}\) and \(s_{e}\)
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