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Chapter 13: Problem 26

Explain the meaning of coefficient of determination.

Short Answer

Expert verified
The coefficient of determination (R^2) represents how much of the total variability in a dependent variable can be explained by a regression model. It is calculated as 1 - (sum of squares of residuals / total sum of squares), and ranges from 0 to 1. A higher R^2 represents a better fit of the model.

Step by step solution

01

Define Coefficient of Determination

The coefficient of determination, often denoted as R^2, is a statistical measurement that attempts to explain how much variation in a dependent variable can be explained by variation in independent variables in a regression model. In general terms, it represents the proportion of the total variation in the dependent variable that is associated with the variation in the independent variable(s).
02

Calculation of Coefficient of Determination

The calculation of the coefficient of determination starts by computing the total sum of squares (TSS), which measures the total variance in the response Y. Then, we compute the sum of squares of residuals (SSR), which measures the amount of Y's variance that is not explained by the model. The coefficient of determination, R^2, is calculated as 1 - (SSR / TSS). This means R^2 measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
03

Interpretation of Coefficient of Determination

The coefficient of determination ranges between 0 and 1. An R^2 of 100% indicates that all changes in the dependent variable are completely explained by changes in the independent variable(s). An R^2 of 0% indicates that the dependent variable cannot be predicted from the independent variable(s). A higher R^2 indicates a better fit of the model, meaning the regression model accounts for a great amount of the variation in the dependent variable.

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Most popular questions from this chapter

The following information is obtained from a sample data set. $$ \begin{aligned} &n=12, \quad \Sigma x=66, \quad \Sigma y=588, \quad \sum x y=2244, \\ &\sum x^{2}=396, \text { and } \Sigma y^{2}=58,734 \end{aligned} $$ Find the values of \(s_{e}\) and \(r^{2}\).

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 . 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 psi, and performed the bounce test mentioned above. The data obtained are given in the following table. $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Pressure (psi) } & 7.8 & 8.1 & 8.3 & 7.4 & 8.9 & 7.2 & 8.6 & 7.5 & 8.1 & 8.5 \\ \hline \begin{array}{l} \text { Bounce height } \\ \text { (inches) } \end{array} & 54.154 .3 & 55.2 & 53.3 & 55.4 & 52.2 & 55.7 & 54.6 & 54.8 & 55.3 \\ \hline \end{array} $$ a. With the pressure as an independent variable and bounce height as a dependent variable, compute \(\mathrm{SS}_{x}, \mathrm{SS}_{y y}\), 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 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 \(a=.05\), can you conclude that \(\rho\) is different from zero?

Construct a \(95 \%\) confidence interval for the mean value of \(y\) and a \(95 \%\) prediction interval for the predicted value of \(y\) for the following. a. \(\hat{y}=13.40+2.58 x\) for \(x=8\) given \(s_{e}=1.29, \bar{x}=11.30, \mathrm{SS}_{x x}=\) \(210.45\), and \(n=12\) b. \(\hat{y}=-8.6+3.72 x\) for \(x=24\) given \(s_{e}=1.89, \bar{x}=19.70, \mathrm{SS}_{x x}=\) \(315.40\), and \(n=10\)

The CTO Corporation has a large number of chain restaurants throughout the United States. The research department at the company wanted to find if the restaurants' sales depend on the mean income of households in the related areas. The company collected information on these two variables for 10 restaurants randomly selected from different areas. The following table gives information on the weekly sales (in thousands of dollars) of these restaurants and the mean annual incomes (in thousands of dollars) of the households for those areas. $$ \begin{array}{l|llllllllll} \hline \text { Sales } & 26 & 38 & 23 & 30 & 22 & 40 & 44 & 32 & 28 & 47 \\ \hline \text { Income } & 46 & 63 & 48 & 52 & 32 & 55 & 58 & 49 & 41 & 72 \\ \hline \end{array} $$ a. Taking income as an independent variable and sales 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. Briefly explain 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 \(2.5 \%\) significance level whether \(B\) is positive. h. Using a \(2.5 \%\) significance level, test whether \(\rho\) is positive.

The following data give the ages (in years) of husbands and wives for six couples. $$ \begin{array}{l|cccccc} \hline \text { Husband's age } & 43 & 57 & 28 & 19 & 35 & 39 \\ \hline \text { Wife's age } & 37 & 51 & 32 & 20 & 33 & 38 \\ \hline \end{array} $$ a. Do you expect the ages of husbands and wives 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 b? d. Using a \(5 \%\) significance level, test whether the correlation coefficient is different from zero.
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