Let \(y=\) expenditure ( \(\$\) ) on food away from home per household member per month in the past quarter and \(x=\) monthly household income (in hundreds of dollars) during the past year.

a. Using 2013 data from three-person households \((N=2334)\), we obtain least squares estimates \(\hat{y}=13.77+0.52 x\). Interpret the estimated slope and intercept from this relation.

b. Predict the expenditures on food away from home for a household with \(\$ 2000\) a month income.

c. Calculate the elasticity of expenditure on food away from home with respect to income when household income is \(\$ 2000\) per month. [Hint: Elasticity must be calculated for a point on the fitted regression.]

d. We estimate the log-linear model to be \(\widehat{\ln (y)}=3.14+0.007 x\). What is the estimated elasticity of expenditure on food away from home with respect to income, if household income is \(\$ 2000\) per month?

e. For the log-linear model in part (d), calculate \(\hat{y}=\exp (3.14+0.007 x)\) when \(x=20\) and when \(x=30\). Evaluate the slope of the relation between \(y\) and \(x, d y / d x\), for each of these \(\hat{y}\) values. Based on these calculations for the log-linear model, is expenditure on food away from home increasing with respect to income at an increasing or decreasing rate?

f. When estimating the log-linear model in part (d), the number of observations used in the regression falls to \(N=2005\). How many households in the sample reported no expenditures on food away from home in the past quarter?