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

Firm Return PE PS

$14\mathrm{.}514\mathrm{.}352\mathrm{.}43$

$24\mathrm{.}610\mathrm{.}990\mathrm{.}80$

$$

$3016\mathrm{.}613\mathrm{.}981\mathrm{.}93$

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a$-1.$ Estimate: Return $=\backslash$beta $0+\backslash$beta $1$PE $+\backslash$beta $2$PS $+\backslash$epsi $.($Negative values should be indicated by a minus sign. Round your answers to $2$ decimal places.$)$

a$-2.$ Are the signs on the coefficients as expected?

multiple choice $1$

Yes

No

b$.$ Interpret the slope coefficient of the PS ratio.

multiple choice $2$

As the PS ratio decreases by $1$ unit, the predicted return of the firm decreases by $33\mathrm{.}01\%,$ holding PE constant.

As the PS ratio increases by $1$ unit, the predicted return of the firm decreases by $33\mathrm{.}01\%,$ holding PE constant.

As the PS ratio increases by $1$ unit, the predicted return of the firm decreases by $3\mathrm{.}43\%,$ holding PE constant.

As the PS ratio increases by $1$ unit, the predicted return of the firm increases by $3\mathrm{.}43\%,$ holding PE constant.

c$.$ What is the predicted return for a firm with a PE ratio of $10$ and a PS ratio of $2?($Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round final answer to $2$ decimal places.$)$

d$.$ What is the standard error of the estimate? $($Round your answer to $2$ decimal places.$)$

e$.$ Interpret R$2.$

multiple choice $3$

$40\mathrm{.}24\%$ of the sample variation in y is explained by the sample regression equation.

$40\mathrm{.}24\%$ of the sample variation in x is explained by the sample regression equation.

$63\mathrm{.}44\%$ of the sample variation in x is explained by the sample regression equation.

$35\mathrm{.}82\%$ of the sample variation in y is explained by the sample regression equation