The following equation estimates the dependence of CANS (the weekly number of cans of brand A tuna sold in thousands) on the price of brand \(\mathrm{A}\) in dollars (PRA) and the prices of two competing brands \(\mathrm{B}\) and \(\mathrm{C}(P R B\) and \(P R C)\). The equation was estimated using 52 weekly observations.

a. When \(P R B\) and \(P R C\) are omitted from the equation, the sum of squared errors increases to 1513.6. Using a \(10 \%\) significance level, test whether the prices of the competing brands should be included in the equation. \(\left(F_{(0.9,2,48)}=2.417\right)\)

b. Consider the following two estimated equations: \(\widehat{E}(P R B \mid P R A)=0.5403+0.3395 P R A\) and \(\widehat{E}(P R C \mid P R A)=0.7708+0.0292 P R A\). If \(P R B\) and \(P R C\) are omitted from the original equation for CANS, by how much will the coefficient estimate for PRA change? By how much will the intercept estimate change?

c. Find point and \(95 \%\) interval estimates of \(E(C A N S \mid P R A=0.91, P R B=0.91, P R C=0.90)\) using the original equation. The required standard error is 1.58.

d. Find a point estimate for \(E(C A N S \mid P R A=0.91)\) using the equation you constructed in part (b). Can you suggest why the point estimates in (c) and (d) are different? Are there values for \(P R B\) and \(P R C\) for which they would be identical?

e. Find a 95\% prediction interval for \(C A N S\) when \(P R A=0.91, P R B=0.91\) and \(P R C=0.90\). If you were a statistical consultant to the supermarket selling the tuna, how would you report this interval?

f. When \(\widehat{C A N S}^{2}\) is added to the original equation as a regressor the sum of squared errors decreases to 1198.9. Is there any evidence that the equation is misspecified?

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E(CANS PRA, PRB, PRC) = 22.96-47.08PRA + 9.30PRB + 16.51PRC SSE = 1358.7