Consider the following estimated area response model for sugar cane (area of sugar cane planted in thousands of hectares in a region of Bangladesh), as a function of relative price (100 times the price of sugar cane divided by the price of jute, which is an alternative crop to sugar cane, planted by Bangladesh farmers), \(\widehat{A R E A}_{t}=-0.24+0.50 R P R I C E_{t}\) using 34 annual observations.

a. The sample average of RPRICE is 114.03 , with a minimum of 74.9 and a maximum of 182.2. RPRICE is the price of sugar cane taken as a percentage of the price of jute. What do these sample statistics tell us about the relative price of sugar cane?

b. Interpret the intercept and slope of the estimated relation.

c. The \(t\)-statistic is -0.01 for the hypothesis that the intercept parameter is zero. What do you conclude? Is this an economically surprising result? Explain.

d. The sample mean area planted is 56.83 thousand hectares, and the sample mean for relative price is 114.03. Taking these values as given, test at the 5\% level of significance the hypothesis that the elasticity of area response to price at the means is 1.0. The estimated variance of the coefficient of RPRICE is 0.020346 .

e. The model is re-estimated in log-linear form, obtaining \(\widehat{\ln \left(A R E A_{t}\right)}=3.21+0.0068 R P R I C E_{t}\). Interpret the coefficient of RPRICE. The standard error of the slope estimate is 0.00229 . What does that tell us about the estimated relationship?

f. Using the model in (e), test the null hypothesis that a \(1 \%\) increase in the price of sugar cane relative to the price of jute increases the area planted in sugar cane by \(1 \%\). Use the \(5 \%\) level of significance and a two-tail test. Include (i) the test statistic and its distribution if the null hypothesis is true, (ii) a sketch of the rejection region, (iii) show the location of the test statistic value, (iv) state your conclusion, and (v) show on the sketch, the region that would represent the \(p\)-value.