Question

Monthly income, in thousands of dollars ($K), at the Acme Law Firm for the six-month period from October to March is listed in the table below.

month	income, $K
Oct	69.3
Nov	68.6
Dec	69.4
Jan	71.7
Feb	72.3
Mar	74.8

-what is the simple exponential smoothing forecast with  α = 0.1 for monthly income for April?

-what is the regression forecast for monthly income for April?

-what is the numerical value of the strength of the linear association between monthly income and month number?

-how much of the variation in monthly income is accounted for by the regression?

-what is the approximate average rate of change of monthly income?

-What forecast method is the most accurate?

what is the best forecast value for monthly income for April?

Answer 1

1)

Simple exponential smoothing forecast with  α = 0.1 for monthly income for April is computed below

A E 1 Month 2 Oct 3 Nov 4 Dec 5 Jan 6 Feb 7 Mar 8 Apr B Income $K 69.3 68.6 69.4 71.7 72.3 74.8 С Exponential smoothing forecast 69.30 69.30 69.23 69.25 69.49 69.77 70.28 D Absolute Deviation 0.00 0.70 0.17 2.45 2.81 5.03 9 10 11 MAD = 1.86 12 13 14

EXCEL FORMULA:

А B с D 1 Month Income $K Exponential smoothing forecast Absolute Deviation 2 Oct 69.3 =B2 =ABS(B2-C2) 3 Nov 68.6 =+B2*0.1+C2*(1-0.1) =ABS(B3-C3) 4 Dec 69.4 =+B3*0.1+C3*(1-0.1) =ABS(B4-C4) 5 Jan 71.7 =+B4*0.1+C4*(1-0.1) =ABS(B5-C5) 6 Feb 72.3 =+B5*0.1+C5*(1-0.1) =ABS(B6-C6) 7 Mar 74.8 =+B6*0.1+C6*(1-0.1) =ABS(B7-C7) 8 Apr =+B7*0.1+C7*(1-0.1) 9 10 11 MAD = =AVERAGE(D2:07) 12 12

Simple exponential smoothing forecast with  α = 0.1 for monthly income for April = 70.28

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2)

Regression forecast

X 1 2 3 4 5 6 21 69.3 68.6 69.4 71.7 72.3 74.8 426.1 x*y 69.3 137.2 208.2 286.8 361.5 448.8 1511.8 x? 1 4 9 16 25 36 91 y 4802.49 4705.96 4816.36 5140.89 5227.29 5595.04 30288.03 Σ = x = Month number (Independent variable) y = Income $ (dependent variable) 91 N = Σx = Σy = Σy = 6 21 426.1 30288.03 Σx = (Σx) = Σxy = (Σy)2 = 441 1511.8 181561.2 Slope (b) = 1.1686 (Ν Σxy - ΣΧΣy) = Ν Σx - (Σκ)? Intercept (a) = (Σy - bΣx) = 66.9266 N Regression Equation: y = 66.9266 + 1.1686x

Regression forecast for monthly income for April = 66.9266+1.1686*7 = 75.11

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3)

Numerical value of the strength of the linear association between monthly income and month number is indicated by correlation coefficient (r)

Coefficient of correlation, P = *Σ- (Σκ) Σν) (Σ)-(Σκ) (Σ)- (Σ») f= 0.9267

Value of r is 0.9267, which is positive and close to 1

Which means linear association between monthly income and month number is direct and  strong

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4)

How much of the variation in monthly income is accounted for by the regression

This is indicated by coefficient of determination, r2

Coefficient of determination, r2 is the measure of percentage of variability of relationship between the two variables. r2 = 0.8587

85.87 % variation in monthly income is accounted for by the regression

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