77 21200 1. Modeling the data linearly - a. Generate a linear model for this dat

Question

77

21200

1. Modeling the data linearly -

a. Generate a linear model for this data by choosing two points.

b. Generate a least squares linear regression model for this data.

c. How good is this regression model?

d. What is the marginal revenue for this department using the linear model with two data points and the regression model? Note that marginal revenue is the same as the first derivative of the revenue (sales) function.

e. Compare the two models. Which do you feel is better?

2. Modeling the data quadratically -

a. Generate a quadratic model for this data.

b. What is the marginal revenue for this department using this model?

c. Calculate the model generated relative max/min value. Show backup analytical work.

d. Compare actual and model generated relative max/min value.

3. Comparing models

a. Which model do you feel best predicts future trends? Explain your rationale.

b. Based on the model selected, what type of seasonal adjustments, if any, would be required to meet customer needs?

4. Identify holiday periods or special events that cause spikes in the original data. WalMart weeks start the beginning of February. So, for example, Walmart week 30 in the 2002 is actually week 34 (30 + 4) in the calendar year 2002 which equates to the end of August 2002. To make the weeks continuous, week 53 is actually WalMart week 1 in 2003 and this equates to week 5 (53 – 52 +4) or the first week in February 2003. Week 72 is week 24 (72 – 52 + 4) in the year 2003 or mid June 2003.

Dry Goods Sales The data is for weekly sales in the dry goods department at a Wal*Mart store in the Northeast.  Peak values, I.e. spikes, usually occur at holiday periods.  Week 1 is the first week of February 2002.  To show continuity, week 1 of 2003 is represented as week 54 since week 53 represents the end of fiscal 2002 and start of the 2003 fiscal year. Dollar values are adjusted in order to disguise true sales figures, but trends in the data are retained for analysis puposes. Week Sales in $ 26 15200 27 15600 28 16400 29 15600 30 14200 31 14400 32 16400 33 15200 34 14400 35 13800 36 15000 37 14100 38 14400 39 14000 40 15600 41 15000 42 14400 43 17800 44 15000 45 15200 46 15800 47 18600 48 15400 49 15500 50 16800 51 18700 52 21400 53 20900 54 18800 55 22400 56 19400 57 20000 58 18100 59 18000 60 19600 61 19000 62 19200 63 18000 64 17600 65 17200 66 19800 67 19600 68 19600 69 20000 70 20800 71 22800 72 23000 73 20800 74 25000 75 30600 76 24000

77

21200

Explanation / Answer

1. a. Let us consider a linear model such that it satisfied the normality condition. Then the model is

Y = weeks

X= sales

Y =a + bX

when a and b are estiamted by OLS.

or

Y = Sales, X=Weeks

Y= c + d X

b. least squares linear regression model for this data is

We find by Excel

THe model is

Y =-13.425+0.00359X

Y = 8741.97+180.999X

C. On the basis of r-squared we conclude that 65% data are linear.

D. The marginal revenue function is the first derivative of the total revenue function. so

Y = 8741.97+180.999X

Total revenue=(8741.97+180.999X)X

marginal revenue= 8741.97+361.99X

e. After comparing the two models, I find that the least square regression model is better. Although this is a simple linear least squares regression model because there is only one variable, it is still more effective and complete as compared to the linear model. The estimates of the unknown parameters obtained from linear least squares regressions are the optimal estimates from a broad class of possible parameter estimates under the usual assumptions used for process modeling.

2. For quadartic model.

Y = a + bX +cX2 where a,b,c are estimated by OLS.

SUMMARY OUTPUT Regression Statistics Multiple R 0.806575 R Square 0.650563 Adjusted R Square 0.643574 Standard Error 9.047601 Observations 52 ANOVA df SS MS F Significance F Regression 1 7620.045 7620.045 93.08734 5.29E-13 Residual 50 4092.955 81.85909 Total 51 11713 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -13.4251 6.845226 -1.96123 0.055433 -27.1741 0.323968 X Variable 1 0.003594 0.000373 9.648178 5.29E-13 0.002846 0.004343

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