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Chapter 13: Problem 19
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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A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x $$ a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?
For a sample data set on two variables, the value of the linear correlation coefficient is (close to) zero. Does this mean that these variables are not related? Explain.
The following table, reproduced from Exercise \(13.28\), lists the percentages of space for eight magazines that contain advertisements and the prices of these magazines. $$ \begin{array}{l|rrrrrrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 & 70 & 28 & 65 & 32 \\ \hline \text { Price }(\$) & 5.50 & 6.95 & 4.95 & 5.75 & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?
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 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?
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