The following table shows a portion of the monthly returns data (in percent) for 2010-2016 for two of Vanguard's mutual funds: the Vanguard Energy Fund and the Vanguard Healthcare Fund.

$$\begin{array}{|c|c|c|} \hline \text { Date } & \text { Energy } & \text { Healthcare } \\ \hline \text { Jan-10 } & -4.86 & -0.13 \\ \hline \text { Feb-10 } & 1.50 & 0.58 \\ \hline \vdots & \vdots & \vdots \\ \hline \text { Dec-16 } & -0.30 & -5.26 \\ \hline \end{array}$$

b. Specify the competing hypotheses in order to determine whether the population correlation coefficient is different from zero.

Is it defense or offense that wins football games? Consider the following portion of data, which includes a team's winning record (Win in %), the average number of yards gained, and the average number of yards allowed during the $2009$ NFL season.

$$\begin{array}{|l|c|c|c|} \hline \text { Team } & \text { Win } & \text { Yards Gained } & \text { Yards Allowed } \\ \hline \text { Arizona Cardinals } & 62.50 & 344.40 & 346.40 \\ \hline \text { Atlanta Falcons } & 56.30 & 340.40 & 348.90 \\ \hline \vdots & \vdots & \vdots & \vdots \\ \hline \text { Washington Redskins } & 25.00 & 312.50 & 319.70 \\ \hline \end{array}$$

Estimate a multiple linear regression model, Model 3, that applies both Yards Gained and Yards Allowed to forecast the winning percentage. Is this model an improvement over the other two models? Explain.

Is it defense or offense that wins football games? Consider the following portion of data, which includes a team's winning record (Win in %), the average number of yards gained, and the average number of yards allowed during the $2009$ NFL season.

$$\begin{array}{|l|c|c|c|} \hline \text { Team } & \text { Win } & \text { Yards Gained } & \text { Yards Allowed } \\ \hline \text { Arizona Cardinals } & 62.50 & 344.40 & 346.40 \\ \hline \text { Atlanta Falcons } & 56.30 & 340.40 & 348.90 \\ \hline \vdots & \vdots & \vdots & \vdots \\ \hline \text { Washington Redskins } & 25.00 & 312.50 & 319.70 \\ \hline \end{array}$$

a. Compare two simple linear regression models, where Model 1 predicts the winning percentage based on Yards Gained and Model 2 uses Yards Allowed.

b. Estimate a multiple linear regression model, Model 3, that applies both Yards Gained and Yards Allowed to forecast the winning percentage. Is this model an improvement over the other two models? Explain.

A financial analyst uses the following model to estimate a firm's stock return: Return $=\beta_0{ }^{+}$ $\beta_1 P / E+\beta_2 P / S+\varepsilon$, where $P / E$ is a firm's price-to-earnings ratio and $P / S$ is a firm's price-to-sales ratio. For a sample of 30 firms, she finds that $S S E=4,402.786$ and $S S T=5,321.532$.

c. Calculate the adjusted $\mathrm{R}^2$.

A financial analyst uses the following model to estimate a firm's stock return: Return $=\beta_0{ }^{+}$ $\beta_1 P / E+\beta_2 P / S+\varepsilon$, where $P / E$ is a firm's price-to-earnings ratio and $P / S$ is a firm's price-to-sales ratio. For a sample of 30 firms, she finds that $S S E=4,402.786$ and $S S T=5,321.532$.

b. Calculate and interpret the coefficient of determination.

A financial analyst uses the following model to estimate a firm's stock retum: Return $=\beta_0{ }^{+}$ $\beta_1 \mathrm{P} E+\beta_2 \mathrm{P} / \mathrm{S}+\varepsilon$, where $\mathrm{P} / \mathrm{E}$ is a firm's price-to-earnings ratio and $\mathrm{P} / \mathrm{S}$ is a firm's price-tosales ratio. For a sample of 30 firms, she finds that SSE $=4,402.786$ and $S S T=5,321.532$.

a. Calculate the standard error of the estimate.

b. Calculate and interpret the coefficient of determination.

c. Calculate the adjusted $\mathrm{R}^2$.

For a sample of $41 \mathrm{New}$ England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $\$ 1,000$ s). He finds that $S S E=4,182,663$ and $S S T=7,732,451$.

b. What proportion of the sample variation in crime rate is explained by the variability in the explanatory variables? What proportion is unexplained?

For a sample of 41 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $\$ 1,000$ s). He finds that $S S E=4,182,663$ and $S S T=7,732,451$.

a. Calculate the standard error of the estimate.

b. What proportion of the sample variation in crime rate is explained by the variability in the explanatory variables? What proportion is unexplained?

The accompanying data file shows the selling price of a used sedan, its age, and its mileage. Estimate two models:

$$\begin{aligned} &\text { Model 1: Price }=\beta_0+\beta_1 x_1 \text { Age }+\varepsilon \\ &\text { Model 2: Price }=\beta_0+\beta_1 \text { Age }+\beta_2 \text { Mileage }+\varepsilon \end{aligned}$$

Which model provides a better fit for $y$ ? Justify your response with two goodness-of-fit measures.

The accompanying data file shows the square footage and associated property taxes for 20 homes in an affluent suburb 30 miles outside New York City.

c. What proportion of the sample variation in property taxes is unexplained by the home's size?

The accompanying data file shows the square footage and associated property taxes for 20 homes in an affluent suburb 30 miles outside New York City.

b. What proportion of the sample variation in property taxes is explained by the home's size?

The accompanying data file shows the square footage and associated property taxes for 20 homes in an affluent suburb 30 miles outside New York City.

a. Estimate a home's property taxes as a linear function of the size of the home (measured by its square footage).

b. What proportion of the sample variation in property taxes is explained by the home's size?

c. What proportion of the sample variation in property taxes is unexplained by the home's size?

The director of college admissions at a local university is trying to determine whether a student's high school GPA or SAT score is a better predictor of the student's subsequent college GPA. She formulates two models:

Model 1. College GPA $=\beta_0+\beta_1$ High School GPA $+\varepsilon$

Model 2. College GPA $=\beta_0+\beta_1$ SAT Score $+\varepsilon$

She estimates these models and obtains the following goodness-of-fit measures.

$$\begin{array}{|l|c|c|} \hline & \text { Model } 1 & \text { Model } 2 \\ \hline R^2 & 0.5595 & 0.5322 \\ \hline \text { Adjusted } R^2 & 0.5573 & 0.5298 \\ \hline s_e & 40.3684 & 41.6007 \\ \hline \end{array}$$

Which model provides a better fit for y? Justify your response with two goodness-of-fit measures.

The accompanying data file shows the midterm and final scores for 32 students in a statistics course.

c. Find and interpret the coefficient of determination.