Download Chapter 8: Sampling distributions

CHAPTER 8:
SAMPLING
DISTRIBUTIONS
By: Wandi Ding
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8.1: DISTRIBUTION OF THE SAMPLE MEAN
Sampling distribution: the sampling distribution of a statistic is a
probability distribution for all possible values of the statistic
computed from a sample of size n.
Sampling distribution of the sample mean:
The sampling distribution of the sample mean x is the probability
distribution of all possible values of the random variable
computed from a sample of size n from a population with mean
and standard deviation .
The idea is as follows obtaining the sampling distribution:
Step 1: obtain a simple random sample of size n
Step 2: compute the sample mean .
Step 3: assuming that we are sampling from a finite population,
repeat steps 1 and 2 until all distinct simple random samples of n
have been obtained.
Note: once a particular sample is obtained, it cannot be obtained a
second time.
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8.1: DISTRIBUTION OF THE SAMPLE MEAN
Ex: the below is to only illustrate the sampling
distribution :
a small population including 4 individuals is shown
below and also list all sample with size=3.
Population: 10 12 18 20 and
population mean   15 and   4.12
all samples with size=3 listed below:
Sample
Sample
mean x
Sample 1
10
12
18
13.3
Sample 2
10
12
20
14
Sample 3
12
18
20
16.7
Sample 4
10
18
20
16
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8.1: DISTRIBUTION OF THE SAMPLE MEAN
The mean and standard deviation of the sampling
distribution of x:
Suppose that a simple random sample of size n is drawn
from a large population with mean  and standard
deviation  . The sampling distribution of x will have






mean x
and standard deviation
called
x
n
the standard error of the mean. (there is a note on the
bottom of page 381, keep in mind! And then using this
correction to redo our example again.)
Ex: for the previous example, we calculate the standard
deviation of the sampling mean is:

4.12
x 

 2.38
n
3
also called standard
error of the mean.
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8.1: DISTRIBUTION OF THE SAMPLE MEAN
Comment:
If a random variable X is normally distributed, the
distribution of the sample mean, x , is normally
distributed. Its mean is the same as the population mean
and the standard deviation of sample mean is  x  
n
Central limit theorem:
Regardless of the shape of the underlying population, the
sampling distribution of sample mean, x , becomes
approximately normal as the sample size, n, increases.
Comments:
1. when parent distribution is normally distributed, the
distribution of the sample means will be normally
distributed for any sample size, n.
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8.1: DISTRIBUTION OF THE SAMPLE MEAN
2. if the parent distribution is not normal, the sample size
n≥30, the distribution of the sample means is kind of
normal, as n increases, the distribution of sample n will
approach normal.
Let’s take a look an question and some figures on page
386.
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