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Beavers and beetles Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae. \({ }^{8}\) Minitab output for a regression analysis on these data is shown below. Construct and interpret a \(99 \%\) confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met. $$ \begin{aligned} &\text { Regression Analysis: Beetle larvae versus Stumps }\\\ &\begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & -1.286 & 2.853 & -0.45 & 0.657 \\ \text { Stumps } & 11.894 & 1.136 & 10.47 & 0.000 \end{array}\\\ &\begin{array}{ll} S=6.41939 & R-S q=83.9 \% & R-S q(a d j)=83.1 \% \end{array} \end{aligned} $$

Weeds among the corn Lamb's-quarter is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in 16 small plots of ground and then weeded the plots by hand to allow a fixed number of lamb'squarter plants to grow in each meter of corn row. The decision of how many of these plants to leave in each plot was made at random. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) in each of the plots: Some computer output from a least-squares regression analysis on these data is shown below. $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 166.483 & 2.725 & 61.11 & 0.000 \\ \begin{array}{l} \text { Weeds per } \\ \text { meter } \end{array} & -1.0987 & 0.5712 & -1.92 & 0.075 \\ \mathrm{~S}=7.97665 & \mathrm{R}-\mathrm{Sq}=20.9 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =15.3 \% \end{array} $$ (a) What is the equation of the least-squares regression line for predicting corn yield from the number of lamb's quarter plants per meter? Interpret the slope and \(y\) intercept of the regression line in context. (b) Explain what the value of \(s\) means in this setting. (c) Do these data provide convincing evidence at the \(\alpha=0.05\) level that more weeds reduce corn yield? Assume that the conditions for performing inference are met.

Time at the table Does how long young children remain at the lunch table help predict how much they eat? Here are data on a random sample of 20 toddlers observed over several months. \({ }^{10}\) "Time" is the average number of minutes a child spent at the table when lunch was served. "Calories" is the average number of calories the child consumed during lunch, calculated from careful observation of what the child ate each day. Some computer output from a least-squares regression analysis on these data is shown below. $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 560.65 & 29.37 & 19.09 & 0.000 \\ \text { Time } & -3.0771 & 0.8498 & -3.62 & 0.002 \\ S=23.3980 & R-S q=42.1 \% & R-S q(a d j)=38.9 \% \end{array} $$ (a) What is the equation of the least-squares regression line for predicting calories consumed from time at the table? Interpret the slope of the regression line in context. Does it make sense to interpret the \(y\) intercept in this case? Why or why not? (b) Explain what the value of \(s\) means in this setting. (c) Do these data provide convincing evidence at the \(\alpha=0.01\) level of a linear relationship between time at the table and calories consumed in the population of toddlers? Assume that the conditions for performing inference are met.

The professor swims Here are data on the time (in minutes) Professor Moore takes to swim 2000 yards and his pulse rate (beats per minute) after swimming on a random sample of 23 days: $$ \begin{array}{lrrrrrr} \hline \text { Time: } & 34.12 & 35.72 & 34.72 & 34.05 & 34.13 & 35.72 \\ \text { Pulse: } & 152 & 124 & 140 & 152 & 146 & 128 \\ \text { Time: } & 36.17 & 35.57 & 35.37 & 35.57 & 35.43 & 36.05 \\ \text { Pulse: } & 136 & 144 & 148 & 144 & 136 & 124 \\ \text { Time: } & 34.85 & 34.70 & 34.75 & 33.93 & 34.60 & 34.00 \\ \text { Pulse: } & 148 & 144 & 140 & 156 & 136 & 148 \\ \text { Time: } & 34.35 & 35.62 & 35.68 & 35.28 & 35.97 & \\ \text { Pulse: } & 148 & 132 & 124 & 132 & 139 & \\ \hline \end{array} $$ Is there statistically significant evidence of a negative linear relationship between Professor Moore's swim time and his pulse rate in the population of days on which he swims 2000 yards? Carry out an appropriate significance test at the \(\alpha=0.05\) level.

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ The slope \(\beta\) of the population regression line describes (a) the exact increase in the selling price of an individual unit when its appraised value increases by \(\$ 1000\). (b) the average increase in the appraised value in a population of units when selling price increases by \(\$ 1000\). (c) the average increase in selling price in a population of units when appraised value increases by \(\$ 1000\). (d) the average increase in the appraised value in the sample of units when selling price increases by \(\$ 1000\). (e) the average increase in selling price in the sample of units when the appraised value increases by \(\$ 1000\).

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Which of the following is the best interpretation for the value 0.1126 in the computer output? (a) For each increase of \(\$ 1000\) in appraised value, the average selling price increases by about 0.1126 . (b) When using this model to predict selling price, the predictions will typically be off by about 0.1126 . (c) \(11.26 \%\) of the variation in selling price is accounted for by the linear relationship between selling price and appraised value. (d) There is a weak, positive linear relationship between selling price and appraised value. (e) In repeated samples of size 16 , the sample slope will typically vary from the population slope by about \(0.1126 .\)

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Which of the following would have resulted in a violation of the conditions for inference? (a) If the entire sample was selected from one neighborhood (b) If the sample size was cut in half (c) If the scatterplot of \(x=\) appraised value and \(y=\) selling price did not show a perfect linear relationship (d) If the histogram of selling prices had an outlier (e) If the standard deviation of appraised values was different from the standard deviation of selling prices

Refer to the following setting. Yellowstone National Park surveyed a random sample of 1526 winter visitors to the park. They asked each person whether he or she owned, rented, or had never used a snowmobile. Respondents were also asked whether they belonged to an environmental organization (like the Sierra Club). The two-way table summarizes the survey responses. $$ \begin{array}{lcrr} \hline & {2}{c} {\text { Environmental Clubs }} & \\ { } & \text { No } & \text { Yes } & \text { Total } \\ \text { Never used } & 445 & 212 & 657 \\ \text { Snowmobile renter } & 497 & 77 & 574 \\ \text { Snowmobile owner } & 279 & 16 & 295 \\ \text { Total } & 1221 & 305 & 1526 \\ \hline \end{array} $$ Snowmobiles (11.2) Do these data provide convincing evidence at the \(5 \%\) significance level of an association between environmental club membership and snowmobile use for the population of visitors to Yellowstone National Park? Justify your answer.

The swinging pendulum Mrs. Hanrahan's precalculus class collected data on the length (in centimeters) of a pendulum and the time (in seconds) the pendulum took to complete one back-and forth swing (called its period). Here are their data: $$ \begin{array}{cc} \hline \text { Length (cm) } & \text { Period (s) } \\ 16.5 & 0.777 \\ 17.5 & 0.839 \\ 19.5 & 0.912 \\ 22.5 & 0.878 \\ 28.5 & 1.004 \\ 31.5 & 1.087 \\ 34.5 & 1.129 \\ 37.5 & 1.111 \\ 43.5 & 1.290 \\ 46.5 & 1.371 \\ 106.5 & 2.115 \\ \hline \end{array} $$ (a) Make a reasonably accurate scatterplot of the data by hand, using length as the explanatory variable. Describe what you see. (b) The theoretical relationship between a pendulum's length and its period is $$ \text { period }=\frac{2 \pi}{\sqrt{g}} \sqrt{\text { length }} $$ where \(g\) is a constant representing the acceleration due to gravity (in this case, \(g=980 \mathrm{~cm} / \mathrm{s}^{2}\) ). Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a). (c) Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a).

Multiple Choice: Select the best answer for Exercises A scatterplot of \(x=\) Super Bowl number and \(y=\) cost of a 30 -second advertisement on the Super Bowl broadcast (in dollars) shows a strong, positive, nonlinear association. A scatterplot of \(\ln (\) cost \()\) versus Super Bowl number is roughly linear. The least-squares regression line for this association is \(\widehat{\ln (\operatorname{cost})}=10.97+0.0971\) (Super Bowl number). Predict the cost of a 30 -second advertisement for Super Bowl 40 . (a) \(\$ 3\) (d) \(\$ 83,132\) (b) \(\$ 15\) (e) \(\$ 2,824,947\) (c) \(\$ 58,153\)