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Unformatted text preview: table below). For example if x = (0; 2)
0+2
=1
x=
2
1 A. Zhensykbaev Sampling distributions Sample 0; 0 0; 2 0; 10 2; 0 2; 2 2; 10 10; 0 10; 2 10; 10
Mean x 0
1
5
1
2
6
5
6
10
2. Determine probability each sample: sample space consists of 9 samples.
Hence the probability of each of them equals to 1/9.
3. Determine the number of simple events containing in the event: because the mean x = 0 occurs in one sample, P (x = 0) = 1/9; similarly x = 1
occurs in two samples and P (x = 1) = 2/9 etc. So, we have the sampling
distribution of the sample mean x
Mean x 0
1
2
5
6
10
P (x) 1/9 2/9 1/9 2/9 2/9 1/9
If a sample statistic has a sampling distribution with the mean equal to
the population parameter the statistic is intended to estimate, the statistic
is called an unbiased estimator of the parameter.
If the mean of the sampling distribution is not equal to the population
parameter, the statistic is called a biased estimator of the parameter.
In previous example mean of the population is
1
µ = (0 + 2 + 10) = 4.
3
The expected value of the random variable x equals
1
2
1
2
2
1
+ 1 · + 2 · + 5 · + 6 · + 10 · = 4.
9
9
9
9
9
9
In view of E (x) = µ we conclude: x is a unbiased estimator of the mean in
this situation.
E (x) = xP (x) = 0 · The statistic that is unbiased and has the smallest variance among all
unbiased estimators is called minimum variance unbiased estimator.
Properties of the sampling distribution of the sample mean.
1. The mean of the sampling distribution of x is equal to the population
mean:
E (x) = µ.
2 A. Zhensykbaev Sampling distributions 2. The variance of the sampling distribution of x is:
2
V ar(x) = σx = σ2
n where n is the sample size.
√
Note that standard deviation σx = σ/ n is often referred to as the
standard error of the mean. Thus, x is an unbiased estimator of µ.
In previous example variance of the population is
1
56
σ 2 =...
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This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.
 Spring '13
 ChristopherStocker
 Math

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