The students in Mr. Shenk’s class measured the arm spans and heights (in inches) of a random sample of $18$ students from their large high school. Here is computer output from a least-squares regression analysis of these data. Construct and interpret a $90\%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met.

$$\begin{gathered} \mathrm{Predictor}\mathrm{Coef}\mathrm{Stdev}\mathrm{t}-\mathrm{ratio}\mathrm{P} \\ \mathrm{Constant}11.5475.6002.060.056 \\ \mathrm{Armspan}0.840420.0809110.390.000 \\ \mathrm{S}=1.613\mathrm{R}-\mathrm{Sq}=87.1\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=86.3\% \end{gathered}$$

Exercises T12.4–T12.8 refer to the following setting. An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for
inference about the slope are met. Here is computer output from the regression analysis:

T12.6 The P -value for the test in Exercise T12.5 is 0.0087. Which of the following is a correct interpretation of this result?a. The probability there is no linear relationship between average number of putts per hole and total winnings for these 69 players is 0.0087.b. The probability there is no linear relationship between average number of putts per hole and total winnings for all players on the PGA Tour’s world money list is 0.0087.c. If there is no linear relationship between average number of putts per hole and total winnings for the players in the sample, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of −4,139,198 or less is 0.0087.d. If there is no linear relationship between average number of putts per hole and total winnings for the players on the PGA Tour’s world money list, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of −4,139,198 or less is 0.0087.e. The probability of making a Type I error is 0.0087.

After a name-brand drug has been sold for several years, the Food and Drug Administration (FDA) will allow other companies to produce a generic equivalent. The FDA will permit the generic drug to be sold as long as there isn't convincing evidence that it is less effective than the name-brand drug. For a proposed generic drug intended to lower blood pressure, the following hypotheses will be us

where

$\mu \mathrm{G}=$ true mean reduction in blood pressureusing the generic drug $\mu \mathrm{G}=$ true mean reduction in blood pressureusing the name@brand drug

${\mu }_G=$ true mean reduction in blood pressure

using the generic drug

${\mu }_G=$ true mean reduction in blood pressure

using the name@brand drug

In the context of this situation, which of the following describes a Type I error?

a. The FDA finds convincing evidence that the generic drug is less effective, when in reality it is less effective.

b. The FDA finds convincing evidence that the generic drug is less effective, when in reality it is equally effective.

c. The FDA finds convincing evidence that the generic drug is equally effective, when in reality it is less effective.

d. The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is less effective.

e. The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is equally effective.

Counting carnivores Ecologists look at data to learn about nature’s patterns. One pattern they have identified relates the size of a carnivore (body mass in kilograms) to how many of those carnivores exist in an area. A good measure of “how many” is to count carnivores per 10,000 kg of their prey in the area. The scatterplot shows this relationship between body mass and abundance for 25 carnivore species.

The following graphs show the results of two different transformations of the data. The first graph plots the logarithm (base 10) of abundance against body mass. The second graph plots the logarithm (base 10) of abundance against the logarithm (base 10) of body mass.

a. Based on the scatterplots, would an exponential model or a power model provide a better description of the relationship between abundance and body mass? Justify your answer.

b. Here is a computer output from a linear regression analysis of log(abundance) and log(body mass). Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Use your model from part (b) to predict the abundance of black bears, which have a body mass of 92.5 kg.

d. Here is a residual plot for the linear regression in part (b). Do you expect your prediction in part (c) to be too large, too small, or about right? Justify your answer.

Boyle’s law Refers to Exercise 34. We took the logarithm (base $10$) of the values for both volume and pressure. Here is some computer output from a linear regression analysis of the transformed data.

a. Based on the output, explain why it would be reasonable to use a power model to describe the relationship between pressure and volume.

b. Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Use the model from part (b) to predict the pressure in the syringe when the volume is $17$ cubic centimeters.

Exercises T12.4–T12.8 refer to the following setting. An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions forinference about the slope are met. Here is computer output from the regression analysis:

T12.5 Suppose that the researchers test the hypotheses $H0:{\beta }_1=0$ versus

$Ha:{\beta }_1<0$. Which of the following is the value of the t statistic for this