Question

3. The following data refers to the price of a good ‘P’ and the quantity of the good supplied, ‘S’.

P 2 7 5 1 4 8 2 8

S 15 41 32 9 28 43 17 40

a. Estimate the linear regression line

b. Estimate the standard errors of

c. Test the hypothesis that price influences supply

d. Obtain a 95% confidence interval for

Answer 1

The linear regression line is $S=\alpha+\beta P+E$

E is random error

and $E(s)=\alpha+\beta P$

$S_{SS}=\sum S^2=\displaystyle\sum_{i=1}^8(S_i-S)^2=1205$

$S_{PP}=\sum P^2= \displaystyle\sum_{i=1}^8(P_i-P)^2=55.9$

$S_{SP}=\sum(SP)=\displaystyle\sum_{i=1}^8(S_i-S)(P_i-P)=22.4$

$\alpha=S-P\beta$ and $\beta=\frac{(\sum SP)}{(\sum S^2)^{1/2}\times (\sum P^2)^{1/2}}$ $=\frac{(225.4)}{(1205)^{1/2}\times (55.9)^{1/2}}=0.8685$

from the table $S=\sum S_i/n=225/8=28.125$

$P=\sum P_i/n=37/8=4.625$

$\alpha=28.125-4.625\times0.8685=24.1082$

a) the estimated regression line is, $S=24.1082+0.8685P_i$

b) the standard error (SE) of $\alpha$ and $\beta$ are

$SE(\alpha)=\sigma \sqrt{1/n+P^2/S_{pp}}$

and $SE(\alpha)=\sigma/\sqrt{S_{pp}}$

$\sigma^2=1/(n-2)SSE=1/(n-2)[S_{ss}-\beta^2S_{pp}]=1/(8-2)[1205-0.8685^2\times 55.9]$

$=1/6\times1162.8351=193.8058$

$\sigma=\sqrt{193.8058}=13.9214$

$SE(\alpha)=\sigma \sqrt{1/n+P^2/S_{pp}}=13.9214/\sqrt{55.9}=13.9214/7.4766=1.86199$

c) testing for hypothesis 

$H_0:\beta=0 \space versus \space H_1:\beta \not= 0$

at $\alpha=0.05$

$t=(\beta-0)/(SE(\beta))$

$t=0.8685/1.86199=0.4664$

$t_{critical}=2.4469$

$|t|\lt2.4469$

we fail to reject $H_0$ , that means at $\alpha=0.05, P\space doesn't\space affect \space S$