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Chapter 13: Problem 59

What does a linear correlation coefficient tell about the relationship between two variables? Within what range can a correlation coefficient assume a value?

Short Answer

Expert verified
The Linear Correlation Coefficient measures the strength and direction of the linear relationship between two variables. It can assume a value between -1 and 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and values near zero suggest no correlation.

Step by step solution

01

Understand the concept of a correlation coefficient

The linear correlation coefficient, also known as Pearson's correlation coefficient, measures the strength and direction of association between two continuous variables. If the correlation coefficient is close to +1, it means there's a strong positive linear relationship, meaning as one variable increases, the other also increases. If it鈥檚 close to -1, there's a strong negative linear relationship, which means as one variable increases, the other decreases. A value close to zero suggests there's no linear correlation between the variables.
02

Understand the range of values a correlation coefficient can assume

The correlation coefficient is always between -1 and 1, inclusive. This means the lowest possible value it can assume is -1 (indicating a perfect negative correlation), the highest possible value is 1 (indicating a perfect positive correlation) and it could be anywhere between these values.

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Most popular questions from this chapter

Bob's Pest Removal Service specializes in removing wild creatures (skunks, bats, reptiles, etc.) from private homes. He charges $$\$ 70$$ to go to a house plus $$\$ 20$$ per hour for his services. Let \(y\) be the total amount (in dollars) paid by a household using Bob's services and \(x\) the number of hours Bob spends capturing and removing the animal(s). The equation for the relationship between \(x\) and \(y\) is $$ y=70+20 x $$ a. Bob spent 3 hours removing a coyote from under Alice's house. How much will he be paid? b. Suppose nine persons called Bob for assistance during a week. Strangely enough, each of these jobs required exactly 3 hours. Will each of these clients pay Bob the same amount, or do you expect each one to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

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?

Explain the meaning of the words simple and linear as used in simple linear regression.

The following table gives information on GPAs and starting salaries (rounded to the nearest thousand dollars) of seven recent college graduates. $$ \begin{array}{l|rrrrrrr} \hline \text { GPA } & 2.90 & 3.81 & 3.20 & 2.42 & 3.94 & 2.05 & 2.25 \\ \hline \text { Starting salary } & 48 & 53 & 50 & 37 & 65 & 32 & 37 \\ \hline \end{array} $$ a. With GPA as an independent variable and starting salary as a dependent variable, compute \(\mathrm{SS}_{x x}\), \(\mathrm{SS}_{\mathrm{yv}}\), and \(\mathrm{SS}_{x v}\) b. Find the least squares regression line. c. Interpret the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(95 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is different from zero. h. Test at a \(1 \%\) significance level whether \(\rho\) is positive.

The following table lists the midterm and final exam scores for seven students in a statistics class. $$ \begin{array}{l|ccccccc} \hline \text { Midterm score } & 79 & 95 & 81 & 66 & 87 & 94 & 59 \\ \hline \text { Final exam score } & 85 & 97 & 78 & 76 & 94 & 84 & 67 \\ \hline \end{array} $$ a. Do you expect the midterm and final exam scores to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1, or \(-1\) ? c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and \(\mathrm{b}\) ? d. Using a \(1 \%\) significance level, test whether the linear correlation coefficient is positive.
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