Download Chapter 10. Sampling and Sampling Distributions Simple Random

Chapter 10. Sampling and Sampling
Distributions
Simple Random Sampling and Sampling
Distributions
„ An important step in statistical analysis is the
extraction of a “correct” sample. The
process of obtaining such a sample is called
sampling and sample design is the method
used to collect samples.
„ There are multiple methods of sampling:
„ If a sample of n is drawn from a population of
N, there are NCn possible samples and each
has the probability of 1/NCn being selected.
„ For an infinite population, the values of the
sample must be drawn independently from
the same population (distribution).
„ If we are interested in a particular parameter,
these different samples may lead to different
statistics. The distribution of these values is
referred to as the sampling distribution.
„ Simple random sampling
„ Systematic sampling
„ Stratified sampling
„ Cluster sampling
„ Simple random sampling is assumed here.
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Simple Random Sampling Distributions
2
Simple Random Sampling Distributions
n=2
„ Suppose an office has 5 people whose working
experiences are {3, 5, 7, 9, 11}. If a random
sample is taken to calculate the average
working experience, what is the distribution of
the sample means? Note that µ = 7 and σ2 =
8.
If a sample of size 2 is taken, there are 5C2 = 10
possible samples: {3,5}, {3,7}, {3,9}, {3,11},
{5,7}, {5,9}, {5,11}, {7,9}, {7,11}, and {9,11}; the
averages are: 4, 5, 6, 6, 7, 7, 8, 8, 9, and 10.
„ If a sample of size 4 is taken, there are 5C4 = 5
possible samples. Samples of sizes 3, 5 and 1?
„
n=4
n=1
x
Prob.
x
Prob.
x
Prob.
4
1/10
6
1/5
3
1/5
5
1/10
6.5
1/5
5
1/5
6
2/10
7
1/5
7
1/5
7
2/10
7.5
1/5
9
1/5
8
2/10
8
1/5
11
1/5
9
1/10
10
1/10
2/10
1/10
n=5
1/10
x
Prob.
7
1
4 5 6 7 8 9 10
3
4
The Mean and Variance of Sampling
Distributions
The Mean and Variance of Sampling
Distributions
„ Just like any probability distribution, we can
„ For random samples of size n taken from a
compute the means and variances of the
above sampling distributions as follows:
Sample Size
n=1
n=2
n=4
n=5
µx
2
Variance σ x
7
7
7
7
8
3
0.5
0
Mean
population with mean µ and standard
deviation σ, the mean and standard deviation
of the sampling distribution are
N −n
σ
σ
or σ x =
⋅
µ x = µ; σ x =
N −1
n
n
if n is not a small proportion of N. The factor
is called the finite population correction factor.
„ Also from now on, σ x is called the standard
error of the mean. Why is named so?
„ The mean of the sampling distribution equals
N −n
N −1
the population mean.
„ The variance of the sampling distribution is
smaller than the population variance.
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The Central Limit Theorem (CLT)
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CLT in Graph: the Average Number of Rolling Dice
„ When both n and N are small, there is not
much can be described about the sampling
distribution.
„ If a random sample of size n is drawn from an
infinite population with µ and σ, then when n
is large the sample mean x can be viewed as
a normal random variable. That is, as n → ∞,
x −µ
σ2
x → N ( µ, ) and z =
→ N (0,1)
n
σ/ n
„ How large is large enough? n ≥ 30 for x .
Monte Carlo simulations.
0.2
0.200
0.15
0.150
0.1
0.100
0.05
0.050
0.000
0
0
2
4
6
0
8
Roll a die once (6)
0.150
4
6
8
0.12
0.1
0.08
0.06
0.04
0.02
0
0.100
0.050
0.000
0
2
4
6
Roll a die 3 times (16)
7
2
Roll a die twice (11)
8
0
2
4
6
8
Roll a die 4 times (21)
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Sampling Distributions of Other Statistics
„ The sample median: the distribution tends to
be normal with mean being the population
median and standard deviation of 1.25σ / n .
This implies that the sample median has a
larger variation than the sample mean; it also
requires larger samples.
„ The sample variance: the distribution also
tends to normal with large samples; it follows
a different distribution for small samples.
„ HW: 10.33, 34, 36, 42, 43 and 45.
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