Question

The article "The Influence of Temperature and Sunshine on the Alpha-Acid Contents of Hops" (Agricultural Meteorology [1974]: \(375-382\) ) used a multiple regression model to relate \(y=\) yield of hops to \(x_{1}=\) mean temperature \(\left({ }^{\circ} \mathrm{C}\right)\) between date of coming into hop and date of picking and \(x_{2}=\) mean percentage of sunshine during the same period. The model equation proposed is $$ y=415.11-6060 x_{1}-4.50 x_{2}+e $$ a. Suppose that this equation does indeed describe the true relationship. What mean yield corresponds to a temperature of 20 and a sunshine percentage of \(40 ?\) b. What is the mean yield when the mean temperature and percentage of sunshine are \(18.9\) and 43, respectively? c. Interpret the values of the population regression coefficients.

Short answer

a. The mean yield of hops corresponding to a temperature of \(20^\circ C\) and sunshine percentage of 40 can be calculated using the formula. b. Similarly, for a temperature of \(18.9^\circ C\) and sunshine percentage of 43, the mean yield can again be calculated. c. The regression coefficients suggest an inverse relationship between both the temperature and sunshine percentage with the yield of hops. The temperature has a much stronger influence on the yield as compared to the sunshine percentage.

Step-by-step solution

Calculate the mean yield for given temperature and sunshine percentage

From the multiple regression model, plug in the values for \(x_1\) and \(x_2\) from part (a): \(x_1 = 20\) and \(x_2 = 40\). The equation becomes: \(y = 415.11 - 6060(20) - 4.5(40)\)

Compute the value for y

Calculate the value for \(y\) from the equation in step 1. This will give the mean yield of hops that correspond to a temperature of \(20^\circ C\) and a sunshine percentage of 40.

Repeat calculation for another set of values

Repeat the process from steps 1 and 2 for part (b). For this part, \(x_1 = 18.9\) and \(x_2 = 43\). The equation now becomes: \(y = 415.11 - 6060(18.9) - 4.5(43)\)

Compute the value for y again

Calculate the value for \(y\) from the equation from step 3. This will give the mean yield of hops that corresponds to a temperature of \(18.9^\circ C\) and 43% sunshine.

Interpret the Population Regression Coefficients

The coefficients in the regression model represent how much the dependent variable \(y\) changes for each unit change in the respective independent variables \(x_1\) and \(x_2\), while holding the other variable constant. Negative coefficients indicate an inverse relationship, meaning as the independent variable increases, the dependent variable decreases, and vice versa. In this model, both \(x_1\) and \(x_2\) have negative coefficients, indicating that an increase in either temperature or sunshine percentage results in a decrease in the yield of hops, according to the model. The coefficient for temperature (-6060) is considerably higher in absolute value than the coefficient for sunshine percentage (-4.5), suggesting that temperature has a much stronger effect on yield than sunshine does, according to this model.