91Ó°ÊÓ

In the least squares line \(\hat{y}=5+3 x\), what is the marginal change in \(\hat{y}\) for each unit change in \(x ?\)

Suppose two variables are positively correlated. Does the response variable increase or decrease as the explanatory variable increases?

Aviation and high-altitude physiology is a specialty in the study of medicine. Let \(x=\) partial pressure of oxygen in the alveoli (air cells in the lungs) when breathing naturally available air. Let \(y=\) partial pressure when breathing pure oxygen. The \((x, y)\) data pairs correspond to elevations from 10,000 feet to 30,000 feet in 5000 -foot intervals for a random sample of volunteers. Although the medical data were collected using airplanes, they apply equally well to Mt. Everest climbers (summit 29,028 feet). $$ \begin{array}{l|rrrrr} \hline x & 6.7 & 5.1 & 4.2 & 3.3 & 2.1(\text { units: } \mathrm{mm} \mathrm{Hg} / 10) \\ \hline y & 43.6 & 32.9 & 26.2 & 6.2 & 13.9(\text { units: } \mathrm{mm} \mathrm{Hg} / 10) \\ \hline \end{array} $$ (Based on information taken from Medical Physiology by A. C. Guyton, M.D.) (a) Verify that \(\Sigma x=21.4, \Sigma y=132.8, \Sigma x^{2}=103.84, \Sigma y^{2}=4125.46, \Sigma x y=\) 652\. 6 , and \(r \approx 0.984\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho>0\). (c) Verify that \(S_{e} \approx 2.5319, a \approx-2.869\), and \(b \approx 6.876\). (d) Find the predicted pressure when breathing pure oxygen if the pressure from breathing available air is \(x=4.0\). (e) Find a \(90 \%\) confidence interval for \(y\) when \(x=4.0\). (f) Use a \(1 \%\) level of significance to test the claim that \(\beta>0\). (g) Find a \(95 \%\) confidence interval for \(\beta\) and interpret its meaning.

The following data are based on information from the book Life in America's Small Cities (by G. S. Thomas, Prometheus Books). Let \(x\) be the percentage of 16 - to 19 -year-olds not in school and not high school graduates. Let \(y\) be the reported violent crimes per 1000 residents. Six small cities in Arkansas (Blytheville, El Dorado, Hot Springs, Jonesboro, Rogers, and Russellville) reported the following information about \(x\) and \(y\) : $$ \begin{array}{r|rrrrrr} \hline x & 24.2 & 19.0 & 18.2 & 14.9 & 19.0 & 17.5 \\ \hline y & 13.0 & 4.4 & 9.3 & 1.3 & 0.8 & 3.6 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=112.8, \Sigma y=32.4, \Sigma x^{2}=2167.14\), \(\Sigma y^{2}=290.14, \Sigma x y=665.03\), and \(r \approx 0.764\). (f) If the percentage of 16 - to 19 -year-olds not in school and not graduates reaches \(24 \%\) in a similar city, what is the predicted rate of violent crimes per 1000 residents?