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Chapter 13: Problem 88

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 \(n=10\)

Short Answer

Expert verified
By using the provided formulas and the given values, it's possible to construct a 95% confidence interval and a 95% prediction interval for \(\hat{y}\) at the given x-values for both part a and part b.

Step by step solution

01

Calculate regression estimate

For part a, substitute \(x=8\) into the model to get regression estimate, \(\hat{y}\), i.e., \(\hat{y}=13.40+2.58*8 \). You do the same for part b with \(x=24\).
02

Calculate the confidence interval

The formula for the confidence interval for \(y\) at a given value of \(x\) is given by \[ \hat{y} \pm t_{\alpha/2} s_{e} \sqrt{ \frac{1}{n} + \frac{(x-\bar{x})^{2}}{SS_{xx}} } \], where \( t_{\alpha/2} \) is the t statistic corresponding to the desired level of confidence (95%), \(s_e\) is the standard error of estimate, \(n\) is the number of observations, \(x\) is the given x-value, \(\bar{x}\) is the mean of x-values, and \(SS_{xx}\) is the sum of squares of x. Substitute the given values into this formula for part a and part b.
03

Calculate the prediction interval

The formula for the prediction interval for \(y\) at a given value of \(x\) is given by \[ \hat{y} \pm t_{\alpha/2} s_{e} \sqrt{1+\frac{1}{n} + \frac{(x-\bar{x})^{2}}{SS_{xx}} } \] Substitute the given values into this formula for part a and part b.

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Most popular questions from this chapter

A sample data set produced the following information. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680, \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r\). b. Using a \(2 \%\) significance level, can you conclude that \(\rho\) is different from zero?

Briefly explain the difference between estimating the mean value of \(y\) and predicting a particular value of \(y\) using a regression model.

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. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between ages and prices of cars? b. Find the regression line with price as a dependent variable and age as an independent variable. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Plot the regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the regression line. e. Predict the price of a 7 -year-old car of this model. \(\mathbf{f}\). Estimate the price of an 18 -year-old car of this model. Comment on this finding.

The following table gives the total payroll (in millions of dollars) on the opening day of the 2011 season and the percentage of games won during the 2011 season by each of the National League baseball teams. $$ \begin{array}{lrc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Games Won } \end{array} \\ \hline \text { Arizona Diamondbacks } & 53.60 & 58.0 \\ \text { Atlanta Braves } & 87.00 & 54.9 \\ \text { Chicago Cubs } & 125.50 & 43.8 \\ \text { Cincinnati Reds } & 76.20 & 48.8 \\ \text { Colorado Rockies } & 88.00 & 45.1 \\ \text { Houston Astros } & 70.70 & 34.6 \\ \text { Los Angeles Dodgers } & 103.80 & 50.9 \\ \text { Miami Marlins } & 56.90 & 44.4 \\ \text { Milwaukee Brewers } & 85.50 & 59.3 \\ \text { New York Mets } & 120.10 & 47.5 \\ \text { Philadelphia Phillies } & 173.00 & 63.0 \\ \text { Pittsburgh Pirates } & 46.00 & 44.4 \\ \text { San Diego Padres } & 45.90 & 43.8 \\ \text { San Francisco Giants } & 118.20 & 53.1 \\ \text { St. Louis Cardinals } & 105.40 & 55.6 \\ \text { Washington Nationals } & 63.70 & 49.7 \\ \hline \end{array} $$ a. Find the least squares regression line with total payroll as the independent variable and percentage of games won as the dependent variable. b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the \(y\) -intercept and the slope of the regression line give \(A\) and \(B\) or \(a\) and \(b\) ? c. Give a brief interpretation of the values of the \(y\) -intercept and the slope obtained in part a. d. Predict the percentage of games won by a team with a total payroll of \(\$ 100\) million.

The following information is obtained from a sample data set. $$ n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680, \quad \Sigma x^{2}=1140 $$ Find the estimated regression line.
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