Question

Model II: Table of Parameter Estimates Parameter Estimates Standard Error P-value 12.324 0.0001 3.858 0.0119 0.5550 65.095 31 β2 0.026 0.027 0.017 Model II: ANOVA Table SourceDf SS MS F Value P-value Model (4) 7) (10) (1 0.0162 Error (5) (8) 13.65 Total (6) (9) iv. In a city that has a population of 100,000 people, what is the expected change amount in expenditure (Y) with $1,000 increase in the average annual family income (X2)? Provide an estimate based on Model 11, together with a 95% confidence interval (Hint: Be careful with the measurement unit for each variable.) (c) Now consider the interaction model IID which gives the following fit Model III Table of Parameter Estimates Parameter Estimates Standard Error P-value 7.3295 0.0018 4.8344 0.0084 3.6314 0.0221 3.4874 0.0252 Bo 45.8404 0.3359 0.1230 一0.0012 6.2542 0.0695 0.0339 0.0003 β3 i. Complete the ANOVA table Model III ANOVA Table SourceDfSSMS F Value P-value Model (1) (4) (6)(8.0042 Error (2) 16.8957 (7) Total (3 ii. In a city that has a population of 100,000 people, what is the expected change amount in expenditure (Y) with $1,000 increase in the average annual family income (X2)? Provide an point estimate based on Model III iii. Test if all the terms involving x2 can be dropped from the above model, namely, Ho 33-0 at the significance level α-0.05 iv. Compute the coefficient of determination R2 and interpret

Answer 1

Model II: Standard error of estimate of Bo 65.095/12.324 5.2820 Estimate of β1 = 0.026 * 3.858 0.1003 t-value corresponding β2 = 0.017/0.027 = 0.6296 SSR = sum of square due to regression (65.095)2 (0.1003)2(0.017)2 4237.369 SSE-sm of square due to error (10000 313.65136459.1 SST = sum of square due to total = SSE + SSR = 136459.1 4237.369 140696.5 Model II: ANOVA Table Source Df MS F Value P-value Model 24237.369 4237.369/2- 2118.684 2118.684/13.65 155.2149 0.0000 Error 9997 136459.1 13.65 Total 9999 140696.5 Expected change amount in erpenditure (Y) with $1000 increase in the average annual family income (X2) = 0.017 * 1000 = 17 t0·025.9997 1.96 95% C. 1, for Bo is : (Bo_ to 025.9997 se (k), Я to 0259997 se(3)) = (54.7423. 75.4477)

95% C.1. for (0.0493, 0.1513) 95% C.1. for β2 ïs : (-0.0359, 0.0699) is : (A-t 0.025.9997 se (β1), βι + to.025.99975e(31)) 228. (β2-t0.025.9997 se(β2), (3) β2+ t0.0259997 se(β2))

SSR- (45.8404)(0.3359)(0.1230)2(-0.0012)2 2101.47 Model III: ANOV A Table: Source Df MS F Value P-value Model 3 2101.47 16.8957 2101.47/3 700.49 700.49/0.0017-412052.9 0.0000 Error 9996 16.8957/9996 0.0017 Total 9999 2101.4716.8957-2118.366 Expected change amount in expenditure (Y) with $1000 increase in the average annual family income (2)0.1230 1000 123 0025,9997 = 1.96 95% C.1, for A -(45.8404-1.96 * 6.2542, 45.8404 + 1.96 * 6.2542) (33.5822, 58.0986)

95% C. 1. for βι = (0.3359-1.96 * 0.0695, 0.3359 + 1.96 * 0.0695) = (0.1997, 0.4721) 95% C.1. for β2 = (0.1230-1.96 * 0.0339, 0.1230 + 1.96 * 0.0339) (0.0566, 0.1894) 95% C.1, for = (一0.00 12-1.96 * 0.0003. -0.00 12 + 1.96 * 0.0003) (一0.0018. 一0.0006) от

iii. Since p - values corresponding to B2 and p3 < 0.05 so we reject Ho 820, B3 0. it. R-SSR/SST-0.9920 i.e. 99.20% of total variation in Y is explained by reqression equation.