91Ó°ÊÓ

Ninth-grade students at the Webb Schools go on a backpacking trip each fall. Students are divided into hiking groups of size 8 by selecting names from a hat. Before leaving, students and their backpacks are weighed. The data here are from one hiking group in a recent year. Make a scatterplot by hand that shows how backpack weight relates to body weight. $$\begin{array}{lrrrrrrrr}\hline \text { Body weight (?): } & 120 & 187 & 109 & 103 & 131 & 165 & 158 & 116 \\\\\text { Backpack weight (Ib): } & 26 & 30 & 26 & 24 & 29 & 35 & 31 & 28 \\ \hline\end{array}$$

Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. We will examine some data. If the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the lengths of a pair of bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones: \(^{10}\) $$\begin{array}{lccccc}\hline \text { Femur }(x): & 38 & 56 & 59 & 64 & 74 \\\\\text { Humerus }(y): & 41 & 63 & 70 & 72 & 84 \\\\\hline\end{array}$$ (a) Make a scatterplot. Do you think that all five specimens come from the same species? Explain. (b) Find the correlation \(r\) step by step, using the formula on page 154 . Explain how your value for \(r\) matches your graph in part (a).

We expect a car's highway gas mileage to be related to its city gas mileage. Data for all 1198 vehicles in the government's recent Fuel Economy Guide give the regression line: predicted highway \(\mathrm{mpg}=4.62+1.109(\mathrm{city} \mathrm{mpg})\) (a) What's the slope of this line? Interpret this value in context. (b) What's the \(y\) intercept? Explain why the value of the intercept is not statistically meaningful. (c) Find the predicted highway mileage for a car that gets 16 miles per gallon in the city.

We expect that a baseball player who has a high batting average in the first month of the season will also have a high batting average the rest of the season. Using 66 Major League Baseball players from the 2010 season, \({ }^{23}\) a least-squares regression line was calculated to predict rest-of- season batting average \(y\) from first-month batting average \(x .\) Note: \(\mathrm{A}\) player's batting average is the proportion of times at bat that he gets a hit. A batting average over 0.300 is considered very good in Major League Baseball. (a) State the equation of the least-squares regression line if each player had the same batting average the rest of the season as he did in the first month of the season. (b) The actual equation of the least-squares regression line is \(\hat{y}=0.245+0.109 x .\) Predict the rest-of-season batting average for a player who had a 0.200 batting average the first month of the season and for a player who had a 0.400 batting average the first month of the season. (c) Explain how your answers to part (b) illustrate regression to the mean.

What is the relationship between rushing yards and points scored in the 2011 National Football League? The table below gives the number of rushing yards and the number of points scored for each of the 16 games played by the 2011 Jacksonville Jaguars. $$\begin{array}{ccc}\hline \text { Game } & \text { Rushing yards } & \text { Points scored } \\\1 & 163 & 16 \\\2 & 112 & 3 \\\3 & 128 & 10 \\\4 & 104 & 10 \\\5 & 96 & 20 \\\6 & 133 & 13 \\\7 & 132 & 12 \\\8 & 84 & 14 \\\9 & 141 & 17 \\\10 & 108 & 10 \\\11 & 105 & 13 \\\12 & 129 & 14 \\\13 & 116 & 41 \\\14 & 116 & 14 \\ 15 & 113 & 17 \\\16 & 190 & 19 \\\\\hline\end{array}$$ (a) Make a scatterplot with rushing yards as the explanatory variable. Describe what you see. (b) The number of rushing yards in Game 16 is an outlier in the \(x\) direction. What effect do you think this game has on the correlation? On the equation of the leastsquares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers. (c) The number of points scored in Game 13 is an outlier in the \(y\) direction. What effect do you think this game has on the correlation? On the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers.

Each year, students in an elementary school take a standardized math test at the end of the school year. For a class of fourth-graders, the average score was 55.1 with a standard deviation of \(12.3 .\) In the third grade, these same students had an average score of 61.7 with a standard deviation of \(14.0 .\) The correlation between the two sets of scores is \(r=0.95\). Calculate the equation of the least-squares regression line for predicting a fourth-grade score from a third-grade score. (a) \(\hat{y}=3.60+0.835 x\) (b) \(\hat{y}=15.69+0.835 x\) (c) \(\hat{y}=2.19+1.08 x\) (d) \(\hat{y}=-11.54+1.08 x\) (e) Cannot be calculated without the data.

Measurements on young children in Mumbai, India, found this least-squares line for predicting height \(y\) from \(\operatorname{arm} \operatorname{span} x:\) $$\hat{y}=6.4+0.93 x$$ Measurements are in centimeters \((\mathrm{cm})\). Suppose that the measurements of arm span and height were converted from centimeters to meters by dividing each measurement by \(100 .\) How will this conversion affect the values of \(r^{2}\) and \(s ?\) (a) \(r^{2}\) will increase, \(s\) will increase. (b) \(r^{2}\) will increase, \(s\) will stay the same. (c) \(r^{2}\) will increase, \(s\) will decrease. (d) \(r^{2}\) will stay the same, \(s\) will stay the same. (e) \(r^{2}\) will stay the same, \(s\) will decrease.