Data on weekly sales of a major brand of canned tuna by a supermarket chain in a large midwestern U.S. city during a mid-1990s calendar year are contained in the data file tuna. There are 52 observations for each of the variables. The variable \(S A L 1=\) unit sales of brand no. 1 canned tuna, and \(A P R 1=\) price per can of brand no. 1 tuna (in dollars).

a. Calculate the summary statistics for \(S A L 1\) and \(A P R 1\). What are the sample means, minimum and maximum values, and their standard deviations. Plot each of these variables versus WEEK. How much variation in sales and price is there from week to week?

b. Plot the variable \(S A L 1\) (y-axis) against APR1 ( \(x\)-axis). Is there a positive or inverse relationship? Is that what you expected, or not? Why?

c. Create the variable \(P R I C E 1=100 A P R 1\). Estimate the linear regression \(S A L 1=\beta_{1}+\) \(\beta_{2}\) PRICE \(1+e\). What is the point estimate for the effect of a one cent increase in the price of brand no. 1 on the sales of brand no. 1? What is a \(95 \%\) interval estimate for the effect of a one cent increase in the price of brand no. 1 on the sales of brand no. 1 ?

d. Construct a \(90 \%\) interval estimate for the expected number of cans sold in a week when the price per can is 70 cents.

e. Construct a 95\% interval estimate of the elasticity of sales of brand no. 1 with respect to the price of brand no. 1 "at the means." Treat the sample means as constants and not random variables. Do you find the sales are fairly elastic, or fairly inelastic, with respect to price? Does this make economic sense? Why?

f. Test the hypothesis that elasticity of sales of brand no. 1 with respect to the price of brand no. 1 from part (e) is minus three against the alternative that the elasticity is not minus three. Use the \(10 \%\) level of significance. Clearly, state the null and alternative hypotheses in terms of the model parameters, give the rejection region, and the \(p\)-value for the test. What is your conclusion?