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Chapter 13: Problem 14
Briefly explain the assumptions of the population regression model.
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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.
Explain each of the following concepts. You may use graphs to illustrate each concept. a. Perfect positive linear correlation b. Perfect negative linear correlation c. Strong positive linear correlation d. Strong negative linear correlation e. Weak positive linear correlation f. Weak negative linear correlation g. No linear correlation
The following table provides information on the speed at takeoff (in meters per second) and distance traveled (in meters) by a random sample of 10 world- class long jumpers. $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Speed } & 8.5 & 8.8 & 9.3 & 8.9 & 8.2 & 8.6 & 8.7 & 9.0 & 8.7 & 9.1 \\ \hline \text { Distance } & 7.72 & 7.91 & 8.33 & 7.93 & 7.39 & 7.65 & 7.95 & 8.28 & 7.86 & 8.14 \\ \hline \end{array} $$ With distance traveled as the dependent variable and speed at takeoff as the independent variable, find the following: a. \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Standard deviation of errors c. SST, SSE, and SSR d. Coefficient of determination
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.
Why is the random error term included in a regression model?
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