Day 11

MATH 313: Survey Design and Sampling

Bastola

Practices for Simple Random Sampling

Example 1: Find the sample sizes required for the following estimations.

A. Estimating the population mean if the population size is 8000 , the data has a range of 100 , and the \(98 \%\) bound on the error of estimation should be 5.

B. Estimating the population total if the population size is 3000 , previous study provides \(s^2=1000\), and the \(92 \%\) bound on the error of estimation should be 8000.

C. Estimating the population proportion if the population size is 10000 and the \(90 \%\) bound on the error of the estimation should be 0.05

Example 2: The populations of states within western and eastern U.S.A. are presented in the following tables.

Population (in thousand) - Western U.S.A
Index Name Pop
1 Alaska 733
2 Arizona 7151
3 California 39538
4 Colorado 5733
5 Hawaii 1455
6 Idaho 1839
7 Kansas 2938
8 Montana 1084
9 Nebraska 1962
10 Nevada 3105
11 New Mexico 2117
12 North Dakota 779
13 Oklahoma 3959
14 Oregon 4237
15 South Dakota 887
16 Texas 29146
17 Utah 3272
18 Washington 7705
19 Wyoming 577
Population (in thousand) - Eastern U.S.A
Index Name Index Name Pop
1 Alabama 5024 17 Mississippi 2961
2 Arkansas 3011 18 Missouri 6155
3 Connecticut 3605 19 New Hampshire 1377
4 Delaware 990 20 New Jersey 9289
5 Florida 21538 21 New York 20201
6 Georgia 10712 22 North Carolina 10439
7 Illinois 12813 23 Ohio 11799
8 Indiana 6786 24 Pennsylvania 13003
9 Iowa 3190 25 Rhode Island 1097
10 Kentucky 4506 26 South Carolina 5118
11 Louisiana 4658 27 Tennessee 6911
12 Maine 1362 28 Vermont 643
13 Maryland 6177 29 Virginia 8631
14 Massachusetts 7030 30 West Virginia 1794
15 Michigan 10077 31 Wisconsin 5894
16 Minnesota 5706

A. Use the random number table, select a simple random sample with size 8 from the list of states in Western U.S.A., using the first two digit on the 5 th line and 11 th column as the starting point.

B. Use the random number table, select a simple random sample with size 10 from the list of states in Eastern U.S.A., using the first two digit on the 4th line and 5th column as the starting point.

C. Use the data provided, estimate the average population per state for Western U.S.A.

Solution 
# western
ny <- 8
Ny <- 19
y <- c(733, 1839, 3105, 2117, 779, 4237, 887, 7705) 
y.bar <- mean(y); y.bar
[1] 2675.25
sy <- sd(y); sy
[1] 2375.441
Vy.bar <- (1 - ny/Ny)* sy^2/ny
CI <- y.bar + c(-1,1)*qt(0.975, df = ny-1)*sqrt(Vy.bar); CI
[1] 1164.193 4186.307

D. Use the data provided, estimate the average population per state for Eastern U.S.A.

Solution 
# eastern
nx <- 10
Nx <- 31
x <- c(5024, 10712, 12813, 4658, 7030, 10077, 5706, 6155, 20201, 6911)
x.bar <- mean(x); x.bar
[1] 8928.7
sx <- sd(x); sx
[1] 4782.982
Vx.bar <- (1 - nx/Nx)* sx^2/nx
CI <- x.bar + c(-1,1)*qt(0.975, df = nx-1)*sqrt(Vx.bar); CI
[1]  6112.586 11744.814

E. Use the data provided, estimate the total population for Western and Eastern U.S.A, respectively.

Solution 
tau.y <- Ny*y.bar; tau.y # western
[1] 50829.75
tau.x <- Nx*x.bar; tau.x # eastern
[1] 276789.7
V.tau.y <- Ny^2 * Vy.bar
V.tau.x <- Nx^2 * Vx.bar
CI.tau.y <- tau.y + c(-1,1)*qt(0.975, df = ny-1)*sqrt(V.tau.y); CI.tau.y
[1] 22119.67 79539.83
CI.tau.x <- tau.x + c(-1,1)*qt(0.975, df = nx-1)*sqrt(V.tau.x); CI.tau.x
[1] 189490.2 364089.2

F. Compare the average populations per state between Western and Eastern U.S.A.

Solution 
delta <- y.bar - x.bar; delta # western - eastern
[1] -6253.45
V.delta <- sy^2/ny + sx^2/nx
df <- (sy^2/ny + sx^2/nx)^2/((sy^2/ny)^2/(ny -1) + (sx^2/nx)^2/(nx -1)); df
[1] 13.72751
CI <- delta + c(-1,1)*qt(0.975, df = df-1)*sqrt(V.delta); CI
[1] -9999.12 -2507.78