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Chapter 13: Problem 49

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 answer is (a); the value of \(b\) will be positive.

Step by step solution

01

Identify the meaning of correlation coefficient

The linear correlation coefficient, \(r\), measures the strength and direction of a linear relationship between two variables. The value of \(r\) is always between -1 and 1. If \(r > 0\), it means that as one variable increases, the other also increases, indicating a positive relationship. If \(r < 0\), it means that as one variable increases, the other decreases, indicating a negative relationship.
02

Identify the meaning of slope in regression line

The slope of the regression line, also called the coefficient of the predictor or independent variable, determines how much the outcome variable changes for each one-unit increase in the predictor variable. If the slope is positive, the regression line rises; if it is negative, it falls.
03

Link the correlation coefficient and slope of regression line

The sign of the correlation coefficient \(r\) determines the direction (positive or negative) of the regression line. If \(r\) is positive, the slope of the regression line will also be positive. Conversely, if \(r\) is negative, the slope of the regression line will be negative. Since given \(r\) is positive, the slope of the regression line will also be positive.

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Most popular questions from this chapter

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 different varieties of cereal. $$ \begin{array}{l|lllllllllllll} \hline \begin{array}{l} \text { Sugar } \\ \text { (grams) } \end{array} & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Find the correlation coefficient. b. Test at a \(1 \%\) significance level whether the linear correlation coefficient between the two variables listed in the table is positive.

The following data give information on the ages (in years) and the number of breakdowns during the last month for a sample of seven machines at a large company. $$ \begin{array}{l|rrrrrrr} \hline \text { Age (years) } & 12 & 7 & 2 & 8 & 13 & 9 & 4 \\ \hline \begin{array}{l} \text { Number of } \\ \text { breakdowns } \end{array} & 10 & 5 & 1 & 4 & 12 & 7 & 2 \\ \hline \end{array} $$ a. Taking age as an independent variable and number of breakdowns as a dependent variable, what is your hypothesis about the sign of \(B\) in the regression line? (In other words, do you expect \(B\) to be positive or negative?) b. Find the least squares regression line. Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at a \(2.5 \%\) significance level whether \(B\) is positive. h. At a \(2.5 \%\) significance level, can you conclude that \(\rho\) is positive? Is your conclusion the same as in part g?

An auto manufacturing company wanted to investigate how the price of one of its car models depreciates with age. The research department at the company took a sample of eight cars of this model and collected the following information on the ages (in years) and prices (in hundreds of dollars) of these cars. $$ \begin{array}{l|rrrrrrrr} \hline \text { Age } & 8 & 3 & 6 & 9 & 2 & 5 & 6 & 3 \\ \hline \text { Price } & 45 & 210 & 100 & 33 & 267 & 134 & 109 & 235 \\ \hline \end{array} $$ a. Calculate the standard deviation of errors. b. Compute the coefficient of determination and give a brief interpretation of it.

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

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact? $$
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