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.

In a contest of National Football League behemoths, teams from the Rams and the $49$ers engage in a tug-of-war, pulling in opposite directions on a strong rope. The Rams exert a force of $9 200\mathrm{~N}$ and they are winning, making the center of the light rope move steadily toward themselves. Is it possible to know the tension in the rope from the information stated? Is it larger or smaller than $9 200\mathrm{~N}$? How hard are the $49$ers pulling on the rope? Would it change your answer if the $49$ers were winning or if the contest were even? The stronger team wins by exerting a larger force, on what? Explain your answers.

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

An economist wants to test if the average hourly wage is less than $\$ 22$. Assume that the population standard deviation is $\$ 6$.

a. State the null and the alternative hypotheses for the test.

b. The data accompanying this exercise show hourly wages. Find the value of the test statistic and the $p$-value.

c. At $\alpha=0.05$, what is the conclusion to the test? Is the average hourly wage less than $\$ 22 ?$

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?

American football is the highest paying sport on a per- game basis. Given that the quarterback is considered the most important player on an NFL team, he is typically well-compensated. Consider a portion of the following quarterback salary data (in $millions) in$2009

$$\begin{array}{|l|c|} \hline \text { Name } & \text { Salary } \\ \hline \text { Philip Rivers } & 25.5566 \\ \hline \text { Jay Cutler } & 22.0441 \\ \hline \vdots & \vdots \\ \hline \text { Tony Romo } & 0.6260 \\ \hline \end{array}$$

$

a. Compute and interpret the mean and the median salary for a quarterback.

b. Compute and interpret the range and the standard deviation for quarterback salaries.

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.

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.

A social scientist would like to analyze the relationship between educational attainment (in years of higher education) and annual salary (in $\$ 1,000$ s). He collects data on 20 individuals. A portion of the data is as follows:

$$\begin{array}{|c|c|} \hline \text { Salary } & \text { Education } \\ \hline 40 & 3 \\ \hline 53 & 4 \\ \hline \vdots & \vdots \\ \hline 38 & 0 \\ \hline \end{array}$$

What is the predicted salary for an individual who completed $7$ years of higher education?

A researcher wonders whether males get paid more, on average, than females at a large firm. She interviews 50 employees and collects data on each employee's hourly wage (Wage in $), years of higher education (EDUC), years of experience (EXPER), age (Age), and a Male dummy variable that equals 1 if male, 0 otherwise. A portion of the data is shown in the accompanying table.$

$$\begin{array}{|c|c|c|c|c|} \hline \text { Wage } & \text { EDUC } & \text { EXPER } & \text { Age } & \text { Male } \\ \hline 37.85 & 11 & 2 & 40 & 1 \\ \hline 21.72 & 4 & 1 & 39 & 0 \\ \hline \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline 24.18 & 8 & 11 & 64 & 0 \\ \hline \end{array}$$

$a. Estimate: Wage$=\beta_0+\beta_1$EDUC$+\beta_2$EXPER$+\beta_3$Age$+\beta_4$Male$+\varepsilon$.

b. Predict the hourly wage of a 40 -year-old male employee with 10 years of higher education and 5 years experience. Predict the hourly wage of a 40-year-old female employee with the same qualifications.

c. Interpret the estimated coefficient for Male. Is the variable Male significant at the$5 \%$ level? Do the data suggest that sex discrimination exists at this firm?

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

b. Find the standard error of the estimate.

The accompanying table shows a portion of data consisting of the selling price, the age, and the mileage for $20$ used sedans.

$$\begin{array}{|c|c|c|} \hline \text { Selling Price } & \text { Age } & \text { Mileage } \\ \hline 13590 & 6 & 61485 \\ \hline 13775 & 6 & 54344 \\ \hline \vdots & \vdots & \vdots \\ \hline 11988 & 8 & 42408 \\ \hline \end{array}$$

Predict the selling price of a five-year-old sedan with $65,000$ miles.

A researcher wonders whether males get paid more, on average, than females at a large firm. She interviews 50 employees and collects data on each employee's hourly wage (Wage), years of higher education (EDUC), experience (EXPER), age (AGE), and gender. The GENDER dummy variable equals 1 if male, 0 if female. A portion of the data is shown in the accompanying table.

$$\begin{array}{|c|c|c|c|c|} \hline \text { Wage } & \text { EDUC } & \text { EXPER } & \text { AGE } & \text { GENDER } \\ \hline \$ 37.85 & 11 & 2 & 40 & 1 \\ \hline 21.72 & 4 & 1 & 39 & 0 \\ \hline \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline 24.18 & 8 & 11 & 64 & 0 \\ \hline \end{array}$$

a. Estimate: Wage $=\beta_0+\beta_1$ EDUC $+\beta_2$ EXPER $+\beta_3 \mathrm{AGE}+$ $\beta_4$ GENDER $+\varepsilon$.

b. Predict the hourly wage of a 40-year-old male employee with 10 years of higher education and 5 years experience. Predict the hourly wage of a 40 -year-old female employee with the same qualifications.

c. Interpret the estimated coefficient for GENDER. Is the variable GENDER significant at the 5% level? Do the data suggest that gender discrimination exists at this firm?

It is advertised that the average braking distance for a small car traveling at 65 miles per hour equals 120 feet. A transportation researcher wants to determine if the statement made in the advertisement is false. She randomly test drives $36$ small cars at $65$ miles per hour and records the braking distance. The sample average braking distance is computed as $114$ feet. Assume that th population standard deviation is $22$ feet.

a. State the null and the alternative hypotheses for the test.

b. Calculate the value of the test statistic and the $p$-value.

c. Use $\alpha=0.01$ to determine if the average breaking distance differs from 120 feet.

d. Repeat the test with the critical value approach.