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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
The value of \(r\) will be negative.

Step by step solution

01

Understanding Linear Correlation Coefficient

The linear correlation coefficient, also called Pearson's correlation coefficient, measures the strength and direction of a linear relationship between two variables. It is calculated as the covariance of the variables divided by the product of their standard deviations. The value of the linear correlation coefficient (\(r\)) is always between -1 and 1, inclusive. A positive \(r\) indicates a positive relationship, where both variables increase or decrease together. A negative \(r\) indicates a negative relationship, where one variable increases as the other decreases.
02

Relation between Regression Line Slope and Linear Correlation Coefficient

In linear regression, the slope of the line (\(b\)) quantifies the strength of the link between the two variables. It can be positive (indicating that the dependencies increase together) or negative (implying that when one factor increases, the other decreases). The sign of the slope corresponds to the sign of the correlation coefficient. Thus, if the slope is negative, the correlation coefficient is also likely to be negative.
03

Answer

Given that the slope of the regression line in the dataset is negative, implying that an increase in one variable corresponds to a decrease in the other, the linear correlation coefficient (\(r\)) should also be negative as the sign of the slope and the correlation coefficient always match.

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Most popular questions from this chapter

While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data are given in the following table. $$ \begin{array}{l|rrrrrrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the percentage of a magazine's space containing ads and the price of the magazine? b. Find the estimated regression equation of price on the percentage containing ads. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\underline{b}\) d. Plot the estimated regression line on the scatter diagram of part a, and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Predict the price of a magazine with \(50 \%\) of its space containing ads. f. Estimate the price of a magazine with \(99 \%\) of its space containing ads. Comment on this finding.

The following table gives information on the number of megapixels and the prices of nine randomly selected point-and-shoot digital cameras that were available on BestBuy.com on July 22, 2009 . $$ \begin{array}{l|rrrrrrrrr} \hline \text { Megapixels } & 10.3 & 10.2 & 7.0 & 9.1 & 10.0 & 12.1 & 8.0 & 5.0 & 14.7 \\ \hline \text { Price (\$) } & 130 & 150 & 62 & 160 & 200 & 280 & 125 & 60 & 400 \\ \hline \end{array} $$ Compute the following. a. \(\mathrm{SS}_{\mathrm{xr}} \mathrm{SS}_{y,}\) and \(\mathrm{SS}_{x y}\) b. Standard deviation of errors c. SST, SSE, and SSR d. Coefficient of determination

A population data set produced the following information. $$ N=460, \quad \Sigma x=3920, \quad \Sigma y=2650, \quad \Sigma x y=26,570, \quad \Sigma x^{2}=48,530 $$ Find the population regression line.

The following table provides information on the 10 NASDAQ companies with the largest percentage of their stocks traded on July \(6,2009 .\) Specifically, the table gives the information on the percentage of stocks traded and the change (in dollars per share) in each stock's price. $$ \begin{array}{lcc} \hline \text { Stock } & \text { Percentage Traded } & \text { Change (\$) } \\\ \hline \text { Matrixx } & 19.6 & -0.43 \\ \text { SpectPh } & 14.7 & -1.10 \\ \text { DataDom } & 12.4 & 0.85 \\ \text { CardioNet } & 9.3 & -0.11 \\ \text { DryShips } & 9.0 & -0.42 \\ \text { DynMatl } & 8.0 & -1.16 \\ \text { EvrgrSlr } & 7.9 & -0.04 \\ \text { EagleBulk } & 7.9 & -0.11 \\ \text { Palm } & 7.8 & -0.02 \\ \text { CentAl } & 7.4 & -0.57 \\ \hline \end{array} $$ a. With percentage traded as an independent variable and the change in the stock's price as a dependent variable, compute \(S S_{x x}, S S_{y y}\), and \(S S_{x y}\) b. Construct a scatter diagram for these data. Does the scatter diagram exhibit a negative linear relationship between the percentage of stock traded and the change in the stock's price? c. Find the regression equation \(\hat{y}=a+b x\). d. Give a brief interpretation of the values \(a\) and \(b\) calculated in part \(\mathrm{c}\). e. Compute the correlation coefficient \(r\). f. Predict the change in a stock's price if \(8.6 \%\) of the stock's shares are traded on a day, Using part b, how reliable do you think this prediction will be? Explain.

Will you expect a positive, zero, or negative linear correlation between the two variables for each of the following examples? a. SAT scores and GPAs of students b. Stress level and blood pressure of individuals c. Amount of fertilizer used and yield of corn per acre d. Ages and prices of houses e. Heights of husbands and incomes of their wives
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