Given below is a six-monthly timber

Question

demand for houses based on application
for building permits
Required
Using least square methods to :
Find the predicted model (𝑦ො)
Period
No of
Permits
Timber
Demand
1 30 10
2 45 17
3 52 21
4 47 18
5 55 24
6 50 21
7 46 17
8 40 12
9 38 10
10 29 8
11 37 11
12 40 13
13 44 16
14 51 20
Cont:
Step 1. Expand the Table
Step 2. Find b0 & b1
b1 = ௌௌ௑௒
ௌௌ௑
SSXY =∑ 𝑥𝑖𝑦𝑖 ௡
௜ୀଵ - ∑ ௫௜ (∑ ௬௜) ௡
= 9898 -(଺଴ସ)(ଶଵ଼) ଵସ
= 9898 -9405.142857 = 492.857143
Ssx = ∑ 𝑥𝑖 ௡ 2
௜ୀଵ - ∑ ௫௜ ଶ ௡
=26870 - ଺଴ସ ଶ
ଵସ
= 26870 – 26058.2857 =811.714329
So that b1 = ௌௌ௑௒
ௌௌ௑
= 𝟒𝟗𝟐.𝟖𝟓𝟕𝟏𝟒𝟑
𝟖𝟏𝟏.𝟕𝟏𝟒𝟐𝟗
= 0.60718
b0 =𝑦ത - b1𝑥̅
= 15.57142857 – 0.60718(43.14285714) = -10.624105
Step 3. Predicted Model
𝑦ො =b0 + b1Xi or
Period
No of Permits
(x)
Timber
Demand (y) X2 Y2
xy
1 30 10 900 100 300
2 45 17 2025 289 765
3 52 21 2704 441 1092
4 47 18 2209 324 846
5 55 24 3025 576 1320
6 50 21 2500 441 1050
7 46 17 2116 289 782
8 40 12 1600 144 480
9 38 10 1444 100 380
10 29 8 841 64 232
11 37 11 1369 121 407
12 40 13 1600 169 520
13 44 16 1936 256 704
14 51 20 2601 400 1020
SUM: 604 218 26870 3714 9898
Mean of y 15.57142857

Answer 1

Step 4. Predicted Model:

𝑦̂ = -10.624105 + 0.60718(X)

Step 5. Predict timber demand for the given periods:
- For Period 10:
𝑦̂ = -10.624105 + 0.60718(29) = 7.040274

- For Period 15:
𝑦̂ = -10.624105 + 0.60718(56) = 22.148548

- For Period 20:
𝑦̂ = -10.624105 + 0.60718(81) = 37.256822

These are the predicted timber demands for the given periods using the least square method.