Question

Ohio Swiss Milk Products manufactures and distributes ice cream in Ohio, Kentucky, and West Virginia. The company wants to expand operations by locating another plant in northern Ohio. The size of the new plant will be a function of the expected demand for ice cream within the area served by the plant. A mar-ket survey is currently under way to determine that demand. Ohio Swiss wants to estimate the relationship between the manufacturing cost per gallon and the number of gallons sold in a year to determine the demand for ice cream and, thus, the size of the new plant. The following data have been collected.
Plant Cost per Thousand Gallons (Y) Thousands of Gallons Sold (X)
1 $1,015 416.9
2 973 472.5
3 1,046 250
4 1,006 372.1
5 1,058 238.1
6 1,068 258,6
7 967 597
8 997 414
9 1,044 263.2
10 1,008 372
Total $10,182 3,654.40

Required:
a. Develop a regression equation to forecast the cost per gallon as a function of the number of gallons produced.
b. What are the correlation coefficient and the coefficient of determination? Comment on your regression equation in light of these measures.
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 peryear.

Answer 1

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.

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