Question

An insurance company wishes to examine the relationship between income (in $,000) and the amount of life insurance in $,000) held by families. The company drew a simple random sample of families and obtained the following results: Family Income Amount of life insurance A 40 110 B 80200 110 220 C lewO ILI What is the least squares estimate of the slope? Answer should be to four decimal places e.g. 1.2345. Please do NOT include any units in your answer. Answer: 1.922 What is the least squares estimate of the Y intercept? Answer should be to four decimal places e.g. 1.2345 Please do NOT include any units in your answer. Answer: 24.396

Answer 1

of To Examine the relationship between income (in 8000) and the amount of life insurance (in 1.000) held by families we sit a regression Equation of amount of life insurance (4) on relationship between income (x) det xi .. and Yi respectively denote the income and amount of life insurance of the ith family, i= A,B,C... J. we fit the Equation, Y; = at brit E = A,B,C.. I to the data using the method of least squares. Deline, Number of observations, n=10 mean of x, n = + Eni. CA mean of ÿ ý Venance of x, var(w) = + $0*** Šu Voptance of Y, var(y) = + Ž (:-u)?= + Šys? Covesume ? Coy (w.) = + (-)(4-5) = { -k Vario U bek୧୧୦ । X andy J

of slope b and intercept a are The least square Estimate respectively given by o b = Covlny) I and â = g - bh Variw Family Income (x) Amount of life, Insudance (4) ya2 200 4H00 16000 24200 12000 220 150 170 1600 6400 12100 6400 6400 12100 40,000 ५४५०० 22500 28900 72900 13600 al Histit o nos 270 32400 14400 8400 19600 3600 10000 240 , + 2H000 57600 3600 22500 Yso 200 9000 18000 8100 40000 • 1850 162000 172600 364500 Total 820 Average 82 185 1:162007260 36450

Therefore, Cov(x, y) = 16200-82 * 185 - 1030 - 536 1922 Varcx) = 7260-85^2 = 536 Var(4) = 36450-18512= 2225 The least squase estimate of slope 5 = 1030 56=1922] & The least square estimate of y-intercept is â = 185-1922*82 â = 24.396) * The prediction for the amount of life insurance for a family whose income is $85,000 ie prediction ofy when x=8s is ŷ = âtôn 24.396+ 1.922 * 85 19 = 187.766 or $187766 ) & when x=90. ŷ = 24.396+ 1.922 +90=197.376 * Thus, the prediction of life insurance when family In come is $90,000 is $ 197,376 The actual amount of life insurance when Income is $90000 is $200,000.

Thus, the residual (erron) in prediction 197396 - 200,000 where o com- Please give an upvote.