Question

Orchard Relief is a product that is designed to improve sleep at night. The company, Eli Orchard, is guessing that sales of the product is somewhat related to sleeping patterns of customers over the days of the week. Before mass production of the product, Eli Orchid has market-tested Orchid Relief in only Orange County over the past 8 weeks. The weekly demand is recorded. Eli Orchid is now trying to use the sales pattern over the past 8 weeks to predict sales in US for the upcoming few weeks, especially for days 57 and 60. An accurate forecast would be helpful in arrangements for the company’s production processes and design of price promotions over each week.

Taking seasonality into account and using the regression method, what is a forecast for day 60?

Number of Days	Daily Demand
1	297
2	293
3	327
4	315
5	348
6	447
7	431
8	283
9	326
10	317
11	345
12	355
13	428
14	454
15	305
16	310
17	350
18	328
19	366
20	460
21	427
22	291
23	325
24	354
25	322
26	405
27	442
28	450
29	318
30	298
31	355
32	355
33	374
34	447
35	463
36	291
37	319
38	333
39	339
40	416
41	475
42	459
43	319
44	326
45	356
46	340
47	395
48	465
49	453
50	307
51	324
52	350
53	348
54	384
55	474
56	485

Answer 1

We will fit a simple linear regression model to the daily demand data. The linear regression model is :30+ β where?,-daily demand. Tl-time period + ε .2,., To solve the least squares normal equations, we will need to setup the . X matrix and -y vector Model matrix, for time periods, X Observation vector, Daily Demand, y [297| 293 l 327 315 348 447 431 283 326 317| 345 | 355 X: y 1 10 1 12

「 56 15961 X .X=11596 60116] The X.X matrix 「 206691 X.y604009] and X" y vector We can find the least squares estimates of the parameters in the regression model as T[339.981] Therefore the regression model is 339.981-1.021-x, where y,-daily demand, 1,time periods, we have reported the regression coefficients in three decimal places The demand forecast for the time period 60 is x,-60 у:-339.981-1.021 .x,-278.721