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Chapter 13: Problem 31
What are the degrees of freedom for a simple linear regression model?
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Explain the least squares method and least squares regression line. Why are they called by these names?
Explain the difference between exact and nonexact relationships between two variables.
The following table provides information on the high temperature for each day and the number of crimes committed in Chicago, Illinois, during the period July 1, 2009 to July 14, 2009. $$ \begin{array}{l|rrrrrrr} \hline \text { High temperature }\left({ }^{\circ} \mathrm{F}\right) & 65 & 73 & 79 & 69 & 81 & 86 & 77 \\ \hline \text { Number of crimes } & 1110 & 1134 & 1117 & 1044 & 1014 & 1105 & 1152 \\ \hline \text { High temperature }\left({ }^{\circ} \mathrm{F}\right) & 65 & 79 & 82 & 85 & 82 & 79 & 80 \\ \hline \text { Number of crimes } & 1046 & 1127 & 1160 & 1065 & 1126 & 1041 & 1038 \\ \hline \end{array} $$ a. Find the least squares regression line \(\hat{y}=a+b x\). Take high temperature as an independent variable and number of crimes committed as a dependent variable. b. Give a brief interpretation of the values of \(a\) and \(b\). c. Compute \(r\) and \(r^{2}\) and explain what they mean. d. Predict the number of crimes committed on a day with a high temperature of \(83^{\circ} \mathrm{F}\). e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Testing at the \(1 \%\) significance level, can you conclude that \(B\) is different from zero? h. Using \(\alpha=.01\), can you conclude that the correlation coefficient is different from zero?
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 \(\mathrm{SS}_{x x}\) \(\mathrm{SS}_{y y}\), 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 the \(1 \%\) significance level whether \(B\) is different from zero. h. Test at the \(1 \%\) significance level whether \(\rho\) is positive.
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?
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