Where Q is the number of cans of tennis balls sold quarterly, P is the wholesale price Wilpen charges for a can of tennis balls, M is the consumers average household income, and Pr is the average price of tennis rackets. The regression results are as follows

Dependant Variable: Q R-Square F-Ratio P-Value on F
Observations: 20 0.8435 28.75 0.001
Variable Parameter Standard
Estimate Error T-Ratio P-Value
Intercept 425120.0 220300.0 1.93 0.0716
P -37260.6 12587 -22.96 0.0093
M 1.49 0.3651 4.08 0.0009
PR -1456.0 460.75 -3.16 0.0060

a. Discuss the statistical significance of the parameter estimates a, b, c, and d using the p- values. Are the signs of b, c, and d consistent with the theory of demand?
b. What is the estimated number of cans of tennis ball demanded?
c. At the values of P, M, and Pr given, what are the estimated values of the price (E), income (Em), and cross-price elasticities (Exr) of demand?
d. What will happen, in percentage terms to the number of cans of tennis balls demanded if the price of tennis balls decreases by 15 percent?
e. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the average household income increases by 20 percent?
f. What will happen, in percentage terms to the number of cans of tennis balls demanded if the average price of tennis rackets increases 25 percent?

a. To discuss the statistical significance of the parameter estimates (a, b, c, and d) using the p-values, we need to compare the p-values to the commonly used significance level, such as 0.05. If the p-value is less than 0.05, we can reject the null hypothesis and conclude that there is a statistically significant relationship between the independent variable and the dependent variable.

For the given regression results:
- Parameter a (intercept) has a p-value of 0.0716, which is greater than 0.05. This means that the intercept is not statistically significant.
- Parameter b (P) has a p-value of 0.0093, which is less than 0.05. Therefore, there is a statistically significant relationship between the wholesale price of tennis balls (P) and the number of cans of tennis balls sold (Q).
- Parameter c (M) has a p-value of 0.0009, which is less than 0.05. This indicates a statistically significant relationship between the average household income (M) and the number of cans of tennis balls sold (Q).
- Parameter d (PR) has a p-value of 0.0060, which is less than 0.05. This implies a statistically significant relationship between the average price of tennis rackets (PR) and the number of cans of tennis balls sold (Q).

Regarding the signs of b, c, and d, we can see that:
- The sign of b (-37,260.6) is negative, which is consistent with the theory of demand. When the price of tennis balls increases, we expect the quantity demanded to decrease.
- The sign of c (1.49) is positive, which is also consistent with the theory of demand. As household income increases, we expect the quantity demanded to increase.
- The sign of d (-1,456.0) is negative, which again aligns with the theory of demand. When the price of tennis rackets increases, we expect the quantity demanded of tennis balls to decrease.

b. The estimated number of cans of tennis balls demanded can be calculated using the regression equation. From the regression results, the estimated equation is:
Q = 425,120.0 - 37,260.6P + 1.49M - 1,456.0PR

You can substitute the values of P, M, and PR into this equation to find the estimated number of cans of tennis balls demanded.

c. To find the estimated values of the price elasticity of demand (E), income elasticity of demand (Em), and cross-price elasticity of demand (Exr) at the given values of P, M, and PR, we need to use the estimated coefficients from the regression results.

- Price elasticity of demand (E): The coefficient of P (-37,260.6) can be used to calculate the price elasticity of demand. The formula for price elasticity is E = (P/Q) * (dQ/dP). You can substitute the values of Q and P, and the coefficient of P into this formula to calculate the price elasticity.

- Income elasticity of demand (Em): The coefficient of M (1.49) can be used to calculate the income elasticity of demand. The formula for income elasticity is Em = (M/Q) * (dQ/dM). You can substitute the values of Q and M, and the coefficient of M into this formula to calculate the income elasticity.

- Cross-price elasticity of demand (Exr): The coefficient of PR (-1,456.0) can be used to calculate the cross-price elasticity of demand. The formula for cross-price elasticity is Exr = (PR/Q) * (dQ/dPR). You can substitute the values of Q and PR, and the coefficient of PR into this formula to calculate the cross-price elasticity.

d. To calculate the percentage change in the number of cans of tennis balls demanded if the price of tennis balls decreases by 15 percent, we need to use the price elasticity of demand (E) calculated in part c. The percentage change in the quantity demanded is given by the formula: %∆Q = E * %∆P, where %∆P is -15% (decrease in price).

e. To calculate the percentage change in the number of cans of tennis balls demanded if the average household income increases by 20 percent, we need to use the income elasticity of demand (Em) calculated in part c. The percentage change in the quantity demanded is given by the formula: %∆Q = Em * %∆M, where %∆M is 20% (increase in income).

f. To calculate the percentage change in the number of cans of tennis balls demanded if the average price of tennis rackets increases by 25 percent, we need to use the cross-price elasticity of demand (Exr) calculated in part c. The percentage change in the quantity demanded is given by the formula: %∆Q = Exr * %∆PR, where %∆PR is 25% (increase in price of tennis rackets).