Question

1. For a national comparison, a real estate magazine is interested in estimating the average price of a house in an area of Raleigh. The area is composed of three neighborhoods consisting of 200, 250 and 400 homes, respectively. In each neighborhood, a SRS of homes is taken. The data are summarized in the following table.

Neighborhood	Sample size	Sample Mean	Sample Variance
1	12	150,000	400,000
2	15	223,000	210,000
3	18	318,000	450,000

1. Using an unbiased estimator, give an estimate of the mean price of a house in this area of Raleigh and determine the bound on the error of estimation.

2. A Raleigh real estate company would like to obtain its own sample to estimate the average selling price of a house in this area of Raleigh with a bound of $150 on the error of estimation. To access the county database to get the data, the real estate company will have to pay the county c₁ = $5, c₂ =$3, and c₃ = $8 for each observation from neighborhoods 1, 2, and 3, respectively. Use Neyman optimal allocation to determine the total sample size n and the sample sizes n₁, n₂, and n₃ from each stratum. Use the data in the table above to approximate the variances.

Answer 1

- 2010. Given that • pop? size for each stratum. (Neighbourhood). ie. - Mi = 200, M2 2 250, M3-400 So unbiased estimate on pop? moon y is u Yst -11 200x150000+ ost 50 250 x 228000+ 400x318 Vol Sample i variance Neighbour Suwple, Sample Thood sizzin meer 150000 புல 223000 210000 Istana (300X1057557.5X105+ 1272x1057 Jel = $( 2129.5x105] Tel = 2.505290x105 18 319000 450000 2: ElJet) - T - @ we and want to know por 1992 = 250529.4 Ane:: 2 st = ? Estimate of popn , mean | Bound at on the error of Estimation is given by (BOE) cysto know as straitified 18:0€. = t. s vamiya weighted mcorno strata sauple mean. Assuming that ist is normally E = Ni Ini = ² witni from normal dist? distributed so it can be read 121 - Ginen that K23, N= Net Myt 4₂ = 850 - Mi = 2.00; 122250, 0477 400 - Un, = 1 Sovod, You 223000 Ing = 318000 And if only few degree of freedom are provided by each stratum, the t value are obtained from student .t- distribution. Here sample size 7730 80 we we 2-score. Take x=0.05

N . variance 200X12 md vañtyst) = , P. ) Civen toch Clehene eliza Mi, gan sauple 80.{. = 150 Then in cost of each sdradim Marc You) = (20)? (2.00-12.) 400,0270 9 = 5, C = 3, 28 si - 400,000, +- (250)? / 250-157 , +(850) Gavia). 210.000 + 1%,2 = 210,000 1832 = USovevo 40? 1.00 -18 450000 ( 850 ) ( uod X15 ) Degree of freedom opa ): 0-3 o vartust) 81734.68+1138.34 Then found on +634.646 Error is (220.0,5) varlyst) = 3507.48 16.05, ng vona ) = 150 Vüañiyet) = 59.22 17. 54.22 = 160 : Then Bound on Ermor to 150 = 2.53 S9.22 Estimate at 95% is it = 2.53 = Za Svarligst we see t-table cet al 954 at which t=2.53 = 1.96 X 59122 for a degree of freedom = 116.07 » toros, 5) = 2157 13.0.. = 116.07. So 3 - 5

and: Newman allocation. whin cost of each stralum & variaues are given then Ini . n. Misina / inval li: BT Misip n n. Thon 200x s Fouono 2002 /4000W + 150 210000 + 400 1450000 mi m . X 56568.5424 - 2007 " 217580.6549 In, x2] my, na Similarly, we can calculate 8 , 250 / 210000 8 x 66143,7827 mga 212540.6sug 217580.6549 [12__2.43] ex 400 /usado 6x 94868.32 98 - = 3,488 217580.6549 217580.6549 Sexo m3 = x 100 luşoudo IM3 cu m=2, M2- 2, no cuan.