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Chapter 13: Problem 32

Explain the meaning of coefficient of determination.

Short Answer

Expert verified
The coefficient of determination \( R^2 \) is a measure in regression analysis that explains the variability of the dependent variable that is predictable from the independent variable(s). On a scale of 0 to 1, a high \( R^2 \) value close to 1 indicates a strong relationship, while a low value indicates a weak relationship. It's widely used in several scientific fields both to assess prediction accuracy and to predict future behaviors or phenomena.

Step by step solution

01

- Understanding the Definition

The coefficient of determination, known as \( R^2 \), is a statistic measure in regression analysis. It indicates the proportion of the variance in the dependent variable that is predictable from the independent variables.
02

- Interpreting the Coefficient

The scale for \( R^2 \) ranges from 0 to 1, where 1 means the independent variable perfectly predicts the dependent variable and 0 means the independent variable does not predict the dependent variable at all.
03

- Applications of Coefficient of Determination

In practical applications, \( R^2 \) is used to understand how strong the relationship is between two variables. It's often used in economics, business, psychology, social science and natural science to analyze and predict certain behaviors or phenomena.
04

- Calculating the Coefficient of Determination

To calculate \( R^2 \), square the correlation coefficient (also known as Pearson's correlation coefficient). This can be done using certain mathematical formulas or using statistical software like R or Python.

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Most popular questions from this chapter

The recommended air pressure in a basketball is between 7 and 9 pounds per square inch (psi). When dropped from a height of 6 feet, a properly inflated basketball should bounce upward between 52 and 56 inches (http://www.bestsoccerbuys.com/balls-basketball.html). The basketball coach at a local high school purchased 10 new basketballs for the upcoming season, inflated the balls to pressures between 7 and \(9 \mathrm{psi}\), and performed the bounce test mentioned above. The data obtained are given in the following table. $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Pressure (psi) } & 7.8 & 8.1 & 8.3 & 7.4 & 8.9 & 7.2 & 8.6 & 7.5 & 8.1 & 8.5 \\ \hline \text { Bounce height (inches) } & 54.1 & 54.3 & 55.2 & 53.3 & 55.4 & 52.2 & 55.7 & 54.6 & 54.8 & 55.3 \\ \hline \end{array} $$ a. With the pressure as an independent variable and bounce height as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the least squares regression line. c. Interpret the meaning of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Compute the standard deviation of errors. f. Predict the bounce height of a basketball for \(x=8.0\). g. Construct a \(98 \%\) confidence interval for \(B\). h. Test at a \(5 \%\) significance level whether \(B\) is different from zero. i. Using \(\alpha=.05\), can you conclude that \(\rho\) is different from zero?

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.

A population data set produced the following information. $$ \begin{aligned} &N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \\ &\Sigma x^{2}=485,870, \text { and } \Sigma y^{2}=135,675 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).

Explain the difference between \(y\) and \(\hat{y}\)

The following table gives information on the limited tread warranties (in thousands of miles) and the prices of 12 randomly selected tires at a national tire retailer as of July 2012 . $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Warranty (thousands of miles) } & 60 & 70 & 75 & 50 & 80 & 55 & 65 & 65 & 70 & 65 & 60 & 65 \\ \hline \text { Price per tire (\$) } & 95 & 135 & 94 & 90 & 121 & 70 & 140 & 80 & 92 & 125 & 160 & 155 \\ \hline \end{array} $$ a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y y}\) a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the regression of price per tire on warranty length. c. Briefly explain the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Plot the scatter diagram and the regression line. f. Predict the price of a tire with a warranty length of 73,000 miles. g. Compute the standard deviation of errors. h. Construct a \(95 \%\) confidence interval for \(B\). i. Test at a \(5 \%\) significance level if \(B\) is positive. j. Using \(\alpha=.025\), can you conclude that the linear correlation coefficient is positive?
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