Answer:
a. The regression equation required is Y = 915.18 – 0.2819X.
b. b-1. Correlation coefficient (r) = –0.9423
b-2. Coefficient of determination = r^2 = 88.80%
b-3. The negative correlation coefficient of -0.9423 implies that increase in X mostly causes a decrease in Y. The coefficient of determination implies that 88.80% variation in Y is explained by X.
c. The manufacturing cost per gallon is $823.56.
Explanation:
Note: See the attached excel file for the calculation of Mean of X and Y and other values.
a. Develop a regression equation to forecast the cost per gallon as a function of the number of gallons produced.
The regression can be written as follows:
Y = bo + b1X ………………… (1)
b1 = (Sum of (Y - Mean of Y) * (X - Mean of X)) / (Sum of (X - Mean of X)^2) = –34,273.08 / 121,585.14 = –0.2819
b0 = Mean of Y – (b1 * Mean of X) = 1,018.20 - (365.44 * 0.2819) = 915.18
Substituting b) and b1 values into equation (1), regression equation to forecast the cost per gallon as a function of the number of gallons produced can be written as follows:
Y = 915.18 – 0.2819X ……………………….. (2)
Equation (2) is the regression equation required.
b. What are the correlation coefficient and the coefficient of determination? Comment on your regression equation in light of these measures.
b-1. Correlation coefficient (r) can be calculated using the following formula:
r = (Sum of (Y - Mean of Y) * (X - Mean of X)) / ((Sum of (Y - Mean of Y)^2) * (Sum of (X - Mean of X)^2))^0.5 = –34,273.08 / (10,879.60 * 121,585.14)^0.5 = –0.9423
b-2. Coefficient of determination = r^2 = –0.94^2 = 0.8880, or 88.80%
b-3. The negative correlation coefficient of -0.9423 implies that increase in X mostly causes a decrease in Y. The coefficient of determination implies that 88.80% variation in Y is explained by X.
c. Suppose that the market survey indicates a demand of 325,000 gallons in the Bucyrus Ohio, area. Estimate the manufacturing cost per gallon for a plant producing 325,000 gallons per year.
Since X and Y are in thousands, 325,000 gallons implies we have:
X = 325
Substitute X = 325 into equation (2), we have:
Y = 915.18 - (0.2819 * 325)
Expressing in full form, we have:
Y = $823
Therefore, the manufacturing cost per gallon is $823.56.
