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He owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the following historical data What is the monthly rate of change (slope) of the least squares trend line for these data? Month #visits April 100

May 140
June 110
July 150
August 120
Sept 160
a. 320
b. 102
c. 8
d. -4
e. -8

User SPoage
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The owner of Darkest Tans Unlimited in a local mall is forecasting this month's (October's) demand for the one new tanning booth based on the given historical data. We need to find out the monthly rate of change (slope) of the least squares trend line for these data.How to find the monthly rate of change (slope) of the least squares trend line for the given data?First, we have to construct a table for X (Month) and Y (# visits). Then, we have to find the mean of X and Y. After that, we have to calculate the slope of the least squares trend line for these data. So, the solution is as follows:Table for X (Month) and Y (# visits)Month (X)Y (Number of visits)April (1)100May (2)140June (3)110July (4)150August (5)120September (6)160Here, X has values from 1 to 6. So, we have n = 6.X-mean = (-2 + -1 + 0 + 1 + 2 + 3) / 6 = -0.33 [mean of X = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5]Y-mean = (100 + 140 + 110 + 150 + 120 + 160) / 6 = 128.33We know that the formula for the slope of the least squares trend line is given by:b = Σ((Xi - X-mean)(Yi - Y-mean)) / Σ(Xi - X-mean)2Now, we will calculate the numerator of the above formula:Numerator = Σ((Xi - X-mean)(Yi - Y-mean))Here,X1 = 1, X2 = 2, X3 = 3, X4 = 4, X5 = 5, X6 = 6Y1 = 100, Y2 = 140, Y3 = 110, Y4 = 150, Y5 = 120, Y6 = 160Σ((Xi - X-mean)(Yi - Y-mean)) = (1 - 3.5)(100 - 128.33) + (2 - 3.5)(140 - 128.33) + (3 - 3.5)(110 - 128.33) + (4 - 3.5)(150 - 128.33) + (5 - 3.5)(120 - 128.33) + (6 - 3.5)(160 - 128.33)Σ((Xi - X-mean)(Yi - Y-mean)) = (-2.5)(-28.33) + (-1.5)(11.67) + (-0.5)(-18.33) + (0.5)(21.67) + (1.5)(-8.33) + (2.5)(31.67)Σ((Xi - X-mean)(Yi - Y-mean)) = 70.83Now, we will calculate the denominator of the above formula:Denominator = Σ(Xi - X-mean)2Here,X1 = 1, X2 = 2, X3 = 3, X4 = 4, X5 = 5, X6 = 6Σ(Xi - X-mean)2 = (1 - 3.5)2 + (2 - 3.5)2 + (3 - 3.5)2 + (4 - 3.5)2 + (5 - 3.5)2 + (6 - 3.5)2Σ(Xi - X-mean)2 = (-2.5)2 + (-1.5)2 + (-0.5)2 + (0.5)2 + (1.5)2 + (2.5)2Σ(Xi - X-mean)2 = 17.5Now, we will substitute the values of Numerator and Denominator in the formula for slope of the least squares trend line:b = Σ((Xi - X-mean)(Yi - Y-mean)) / Σ(Xi - X-mean)2b = 70.83 / 17.5b = 4.05So, the monthly rate of change (slope) of the least squares trend line for these data is 4.05. Therefore, the correct option is (c) 8.

User FinalDark
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