Answer to Question #214563 in Statistics and Probability for Getahun

Question #214563

1. 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


1
Expert's answer
2021-07-07T16:41:08-0400

The 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]"

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

"\\sigma=\\sqrt193.8058=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"


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