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Chapter 13: Problem 47
Explain the difference between \(r\) and \(\rho\).
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An investigation of the relationship between \(x=\) traffic flow (thousands of cars per 24 hours) and \(y=\) lead content of bark on trees near the highway (mg/g dry weight) yielded the accompanying data. A simple linear regression model was fit, and the resulting estimated regression line was \(\hat{y}=28.7+33.3 x .\) Both residuals and standardized residuals are also given. \(\begin{array}{lrrrrr}x & 8.3 & 8.3 & 12.1 & 12.1 & 17.0 \\ y & 227 & 312 & 362 & 521 & 640 \\ \text { Residual } & -78.1 & 6.9 & -69.6 & 89.4 & 45.3 \\ \text { St. resid. } & -0.99 & 0.09 & -0.81 & 1.04 & 0.51 \\ x & 17.0 & 17.0 & 24.3 & 24.3 & 24.3 \\ y & 539 & 728 & 945 & 738 & 759 \\ \text { Residual } & -55.7 & 133.3 & 107.2 & -99.8 & -78.8 \\ \text { St. resid. } & -0.63 & 1.51 & 1.35 & -1.25 & -0.99\end{array}\) a. Plot the \((x\), residual \()\) pairs. Does the resulting plot suggest that a simple linear regression model is an appropriate choice? Explain your reasoning. b. Construct a standardized residual plot. Does the plot differ significantly in general appearance from the plot in Part (a)?
Hormone replacement therapy (HRT) is thought to increase the risk of breast cancer. The accompanying data on \(x=\) percent of women using HRT and \(y=\) breast cancer incidence (cases per 100,000 women) for a region in Germany for 5 years appeared in the paper. The authors of the paper used a simple linear regression model to describe the relationship between HRT use and breast cancer incidence. \begin{tabular}{cc} & Breast Cancer \\ HRT Use & Incidence \\ \hline \(46.30\) & \(103.30\) \\ \(40.60\) & \(105.00\) \\ \(39.50\) & \(100.00\) \\ \(36.60\) & \(93.80\) \\ \(30.00\) & \(83.50\) \\ \hline \end{tabular} a. What is the equation of the estimated regression line? b. What is the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use? c. What would you predict the breast cancer incidence to be in a year when HRT use was \(40 \%\) ? d. Should you use this regression model to predict breast cancer incidence for a year when HRT use was \(20 \%\) ? Explain. e. Calculate and interpret the value of \(r^{2}\). f. Calculate and interpret the value of \(s_{e}\).
If the sample correlation coefficient is equal to 1, is it necessarily true that \(\rho=1\) ? If \(\rho=1\), is it necessarily true that \(r=1 ?\)
The paper suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length \((\mathrm{mm})\) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation \(0.005\). a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m}\) ? c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) has a shell thickness of greater than .175? Less than . 178 ?
Television is regarded by many as a prime culprit for the difficulty many students have in performing well in school. The article reported that for a random sample of \(n=528\) college students, the sample correlation coefficient between time spent watching television \((x)\) and grade point average \((y)\) was \(r=-.26\). a. Does this suggest that there is a negative correlation between these two variables in the population from which the 528 students were selected? Use a test with significance level \(.01\). b. Would the simple linear regression model explain a substantial percentage of the observed variation in grade point average? Explain your reasoning.
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