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 research analyst is trying to determine whether a firm's price-earnings (P/E) and price-sales (P/S) ratios can explain the firm's stock performance over the past year. A P/E ratio is calculated as a firm's share price compared to the income or profit earned by the firm per share. Generally, a high P/E ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E ratio. The $\mathrm{P} / \mathrm{S}$ ratio is calculated by dividing a firm's share price by the firm's revenue per share for the trailing 12 months. In short, investors can use the $\mathrm{P} / \mathrm{S}$ ratio to determine how much they are paying for a dollar of the firm's sales rather than a dollar of its earnings (P/E ratio). In general, the lower the $\mathrm{P} / \mathrm{S}$ ratio, the more attractive the investment. The accompanying table shows the year-to-date returns (Return in %) and the P/E and P/S ratios for a portion of the 30 firms included in the Dow Jones Industrial Average.

$$\begin{array}{|c|c|c|} \hline \text { Return } & \text { P/E } & \text { P/S } \\ \hline 4.4 & 14.37 & 2.41 \\ \hline-4.5 & 11.01 & 0.78 \\ \hline \vdots & \vdots & \vdots \\ \hline 16.3 & 13.94 & 1.94 \\ \hline \end{array}$$

Interpret $R^2$.

A research analyst is trying to determine whether a firm's price-earnings (P/E) and price-sales (P/S) ratios can explain the firm's stock performance over the past year. A P/E ratio is calculated as a firm's share price compared to the income or profit earned by the firm per share. Generally, a high P/E ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E ratio. The $\mathrm{P} / \mathrm{S}$ ratio is calculated by dividing a firm's share price by the firm's revenue per share for the trailing 12 months. In short, investors can use the $\mathrm{P} / \mathrm{S}$ ratio to determine how much they are paying for a dollar of the firm's sales rather than a dollar of its earnings (P/E ratio). In general, the lower the $\mathrm{P} / \mathrm{S}$ ratio, the more attractive the investment. The accompanying table shows the year-to-date returns (Return in %) and the P/E and P/S ratios for a portion of the 30 firms included in the Dow Jones Industrial Average.

$$\begin{array}{|c|c|c|} \hline \text { Return } & \text { P/E } & \text { P/S } \\ \hline 4.4 & 14.37 & 2.41 \\ \hline-4.5 & 11.01 & 0.78 \\ \hline \vdots & \vdots & \vdots \\ \hline 16.3 & 13.94 & 1.94 \\ \hline \end{array}$$

What is the predicted return for a firm with a $\mathrm{P} / \mathrm{E}$ ratio of 10 and a $\mathrm{P} / \mathrm{S}$ ratio of 2 ?

A research analyst is trying to determine whether a firm's price-earnings (P/E) and price-sales (P/S) ratios can explain the firm's stock performance over the past year. A P/E ratio is calculated as a firm's share price compared to the income or profit earned by the firm per share. Generally, a high P/E ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E ratio. The $\mathrm{P} / \mathrm{S}$ ratio is calculated by dividing a firm's share price by the firm's revenue per share for the trailing 12 months. In short, investors can use the $\mathrm{P} / \mathrm{S}$ ratio to determine how much they are paying for a dollar of the firm's sales rather than a dollar of its earnings (P/E ratio). In general, the lower the $\mathrm{P} / \mathrm{S}$ ratio, the more attractive the investment. The accompanying table shows the year-to-date returns (Return in %) and the P/E and P/S ratios for a portion of the 30 firms included in the Dow Jones Industrial Average.

$$\begin{array}{|c|c|c|} \hline \text { Return } & \text { P/E } & \text { P/S } \\ \hline 4.4 & 14.37 & 2.41 \\ \hline-4.5 & 11.01 & 0.78 \\ \hline \vdots & \vdots & \vdots \\ \hline 16.3 & 13.94 & 1.94 \\ \hline \end{array}$$

What is the standard error of the estimate?

A research analyst is trying to determine whether a firm's price-earnings (P/E) and price-sales (P/S) ratios can explain the firm's stock performance over the past year. A P/E ratio is calculated as a firm's share price compared to the income or profit earned by the firm per share. Generally, a high P/E ratio suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E ratio. The $\mathrm{P} / \mathrm{S}$ ratio is calculated by dividing a firm's share price by the firm's revenue per share for the trailing 12 months. In short, investors can use the $\mathrm{P} / \mathrm{S}$ ratio to determine how much they are paying for a dollar of the firm's sales rather than a dollar of its earnings (P/E ratio). In general, the lower the $\mathrm{P} / \mathrm{S}$ ratio, the more attractive the investment. The accompanying table shows the year-to-date returns (Return in %) and the P/E and P/S ratios for a portion of the 30 firms included in the Dow Jones Industrial Average.

$$\begin{array}{|c|c|c|} \hline \text { Return } & \text { P/E } & \text { P/S } \\ \hline 4.4 & 14.37 & 2.41 \\ \hline-4.5 & 11.01 & 0.78 \\ \hline \vdots & \vdots & \vdots \\ \hline 16.3 & 13.94 & 1.94 \\ \hline \end{array}$$

Interpret the slope coefficient of the $\mathrm{P} / \mathrm{S}$ ratio.

Two boxes with given letters are shown. Choose a box and draw three letters at random, one-by-one, without replacement. If the outcome is SOS, you win a prize.
1.$\boxed{S O S}$

2.$\boxed{S O S S O S}$

a. Which box should you choose?
b. Which box would you choose if the letters are to be drawn with replacement?

The position of an object as a function of time is given as $x=A t^3+B t^2+C t+D$. The constants are $A=2.1 \mathrm{~m} / \mathrm{s}^3$, $B=1.0 \mathrm{~m} / \mathrm{s}^2, C=-4.1 \mathrm{~m} / \mathrm{s}$, and $D=3 \mathrm{~m}$. a) What is the velocity of the object at $t=10.0 \mathrm{~s}$ ? b) At what time(s) is the object at rest? c) What is the acceleration of the object at $t=0.50 \mathrm{~s}$ ? d) Plot the acceleration as a function of time for the time interval from $t=-10.0 \mathrm{~s}$ to $t=10.0 \mathrm{~s}$.

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.