Download Demo of Sampling Distribution of Statistics

Demo of Sampling Distributions of Statistics
We simulate random samples from a probability distribution using Minitab and examine
the sampling distributions of some statistics via the histograms of the statistics under
repeated random sampling. We will mainly compare each histogram of realized values of
a statistic in the simulation with its theoretical counterpart.
Random sampling
Suppose that the probability distribution for a population is a uniform distribution on
[0,1]. It has a population mean .5 and standard deviation .28867. We generate, say, 1,000
random samples of size 25 from the distribution. Use the menu sequence Calc > Random
Data > Uniform in Minitab and specify the number of rows (1,000 for this example) and
columns (c1-c25) to store 25 values in each sample.
Histograms of statistics
Consider the following three statistics and their histograms when random sampling is
repeated 1,000 times. (a) X1, (b) sample mean = (X1 +...+X5)/5, and (c) Sample mean =
(X1 +...+X25)/25. Use the menu sequence, Calc > Row Statistics to compute the sample
mean of the first 5 observations for each sample and store 1,000 sample means in a
variable xbar5. Similarly compute the sample mean of 25 observations in each row and
store the values in a variable xbar25.
What are the theoretical values of the mean and standard deviation of these statistics?
Compare the mean and standard deviation of each generated distribution with those of the
respective theoretical distribution.
Histogram of C1, xbar5, xbar25
Normal
C1
60
xbar5
80
45
60
30
40
Frequency
15
0
20
0.0
0.2
0.4
0.6
0.8
1.0
xbar25
80
60
40
20
0
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
0
0.125 0.250 0.375 0.500 0.625 0.750 0.875
Mean
StDev
N
C1
0.4974
0.2878
1000
xbar5
Mean 0.5025
StDev 0.1320
N
1000
xbar25
Mean
0.5053
StDev 0.05626
N
1000
Repeat the above simulation for an Exponential(1) random variable with the probability
density function f ( x ) = e− x , x > 0 (0 elsewhere). Note that the population mean is 1 and
the standard deviation is 1 for Exponential(1).
Histogram of C1, xbar5, xbar25
Normal
C1
160
xbar5
100
Frequency
120
75
80
50
40
25
0
Mean
StDev
N
-1.2 0.0
1.2
2.4
3.6
4.8
6.0
0
7.2
C1
0.9765
0.9556
1000
xbar5
Mean
1.015
StDev 0.4512
N
1000
-0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8
xbar25
xbar25
Mean
1.006
StDev 0.2000
N
1000
100
75
50
25
0
0.6
0.8
1.0
1.2
1.4
1.6
The following is simulation results for the standard normal distribution as a population.
Histogram of C1, xbar5, xbar25
Normal
C1
100
80
20
25
0
0
-2.4 -1.6 -0.8 0.0 0.8 1.6 2.4
xbar25
100
75
50
25
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
C1
0.01180
0.9494
1000
xbar5
Mean
0.01954
StDev
0.4522
N
1000
50
40
Frequency
Mean
StDev
N
75
60
0
xbar5
-1.6 -1.2 -0.8 -0.4 0.0
0.4 0.8
1.2
xbar25
Mean
0.008064
StDev
0.1989
N
1000