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What is the symbol used for the population correlation coefficient?

Let \(x\) be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let \(y\) be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of \(n=6\) professional basketball players gave the following information (Reference: The Official NBA Basketball Encyclopedia, Villard Books). $$ \begin{array}{c|cccccc} \hline x & 67 & 65 & 75 & 86 & 73 & 73 \\ \hline y & 44 & 42 & 48 & 51 & 44 & 51 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=439, \quad \Sigma y=280, \quad \Sigma x^{2}=32,393, \quad \Sigma y^{2}=13,142\), \(\Sigma x y=20,599\), and \(r \approx 0.784 .\) (b) Use a \(5 \%\) level of significance to test the claim that \(\rho>0\). (c) Verify that \(S_{e} \approx 2.6964, a \approx 16.542, b \approx 0.4117\), and \(\bar{x} \approx 73.167\). (d) Find the predicted percentage \(\hat{y}\) of successful field goals for a player with \(x=70 \%\) successful free throws. (e) Find a \(90 \%\) confidence interval for \(y\) when \(x=70\). (f) Use a \(5 \%\) level of significance to test the claim that \(\beta>0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and its meaning.

Over the past 50 years, there has been a strong negative correlation between average annual income and the record time to run 1 mile. In other words, average annual incomes have been rising while the record time to run 1 mile has been decreasing. (a) Do you think increasing incomes cause decreasing times to run the mile? Explain. (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

What is the optimal amount of time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let \(x=\) depth of dive in meters, and let \(y=\) optimal time in hours. A random sample of divers gave the following data (based on information taken from Medical Physiology by A. C. Guyton, M.D.). $$ \begin{array}{c|ccccccc} \hline x & 14.1 & 24.3 & 30.2 & 38.3 & 51.3 & 20.5 & 22.7 \\ \hline y & 2.58 & 2.08 & 1.58 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=201.4, \quad \Sigma y=12.6, \quad \Sigma x^{2}=6734.46, \quad \Sigma y^{2}=25.607\), \(\Sigma x y=311.292\), and \(r \approx-0.976\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho<0\). (c) Verify that \(S_{e} \approx 0.1660, a \approx 3.366\), and \(b \approx-0.0544\). (d) Find the predicted optimal time in hours for a dive depth of \(x=18\) meters. (e) Find an \(80 \%\) confidence interval for \(y\) when \(x=18\) meters. (f) Use a \(1 \%\) level of significance to test the claim that \(\beta<0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and its meaning.

It is not obvious from the formulas, but the values of the sample test statistic \(t\) for the correlation coefficient and for the slope of the least- squares line are equal for the same data set. This fact is based on the relation $$ b=r \frac{s_{y}}{s_{x}} $$ where \(s_{y}\) and \(s_{x}\) are the sample standard deviations of the \(x\) and \(y\) values, respectively. (a) Many computer software packages give the \(t\) value and corresponding \(P\) -value for \(b\). If \(\beta\) is significant, is \(\rho\) significant? (b) When doing statistical tests "by hand," it is easier to compute the sample test statistic \(t\) for the sample correlation coefficient \(r\) than it is to compute the sample test statistic \(t\) for the slope \(b\) of the sample least- squares line. Compare the results of parts (b) and (f) for Problems \(7-12\) of this problem set. Is the sample test statistic \(t\) for \(r\) the same as the corresponding test statistic for \(b\) ? If you conclude that \(\rho\) is positive, can you conclude that \(\beta\) is positive at the same level of significance? If you conclude that \(\rho\) is not significant, is \(\beta\) also not significant at the same level of significance?

The following data are based on information from Domestic Affairs. Let \(x\) be the average number of employees in a group health insurance plan, and let \(y\) be the average administrative cost as a percentage of claims. $$ \begin{array}{l|rrrrr} \hline x & 3 & 7 & 15 & 35 & 75 \\ \hline y & 40 & 35 & 30 & 25 & 18 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=135, \Sigma x^{2}=7133, \quad \Sigma y=148\), \(\Sigma y^{2}=4674\), and \(\Sigma x y=3040\). Compute \(r\). As \(x\) increases from 3 to 75 , does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.