Download the sampling distribution of the mean

THE SAMPLING DISTRIBUTION OF THE SAMPLE
MEAN
Provides linkage between descriptive statistics, probability
concepts, random variables, & normal or binomial distributions.
First step towards inferential statistics.
Recall: Simple Random Sampling
A random-sampling procedure for which each possible sample
of a given size is equally likely to be the one selected.
Example: Construct a simple random sample of size 2 from the
population of 4 letters: A, B, C, & D.
Solution:
Population Size (N)
Sample Size (n)
= 4
= 2
Possible Sample Space:
S = {AB, AC, AD, BC, BD, CD}
The probability of each item is 1/6
 4
Alternatively, use the combination formula:    6
2
 
Sampling Error
The error resulting from using a sample instead of a census to
estimate a population quantity.
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -1-
Sampling Distribution of the Mean
Recall:
Population mean:
Sample mean:
x
N
x
x
n

Example: From a population of 5 heights: 76, 78, 79, 81, & 86
inches we wish to construct a sample of size 4.
Recall:
1. The number of samples that can be constructed is
5 
 
 4
=5
2. Each sample will be equally likely and each sample will
have its own mean which is also equally.
The samples, their mean, and the probability of the mean is
shown below:
Sample
Mean P(mean)
76,78,79,81 78.50
0.2
76,78,79,86 79.75
0.2
76,78,81,86 80.25
0.2
76,79,81,86 80.50
0.2
78,79,81,86 81.00
0.2
If we now wish to determine:
P(79  x  81) = P( x = 79.75, 80.25, 80.50, 81.00)
= P(79.75) + P(80.25) + P(80.50) + P(81.00)
= 4 * 0.2
= 0.8
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -2-
There is an 80% chance that the mean of the sample selected
will be within one inch of the population mean.
NOTE: The bigger the sample the higher the probability of the
sample mean being within one inch of the population mean.
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -3-
The Mean Of x
The mean of x (that is the mean of the mean) is the estimate for
the population mean. That is,
 x     xP ( x )
The Standard Deviation of x
The standard deviation of the random variable x is
approximately equal to the standard deviation of the population
divided by the square root of the sample size. That is,
2
2
 x   x P( x)   x 

n
Thus the population standard deviation is:   n x
Example
Mean
78.50
79.75
80.25
80.50
81.00
Totals
P(mean)
0.2
0.2
0.2
0.2
0.2
1.0
Mean*P(mean)
15.70
15.95
16.05
16.10
16.20
80.00
Mean2*P(mean)
1232.450
1272.013
1288.013
1296.050
1312.200
6400.726
Mean of x = 80.00
Std. Deviation =
6400 .726  80 2
= 0.862
Which implies that the population standard deviation is approximately
(0.862 * √4) = 1.724
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -4-
Sampling Distribution For Normally Distributed Variables
If variable x is normally distributed then the sampling
distribution of the mean and standard deviation for a random
sample of size n is:
x  
and  x 

n
Example:
Given  = 50 and  = 8 what is the sampling distribution of
the mean for a random sample of size 3?
Solution:
 x   = 50, and
x

n
= 8 / Sqrt(3) = 4.62
Standardized Form
z
x

n
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -5-
The Central Limit Theorem
For a relatively large sample size, the random variable x is
approximately normally distributed, regardless of how the
population is distributed. The approximation becomes better
with increasing sample size.
This is especially true for n  30.
Some Visual Displays
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -6-
Practical Uses
1. For samples of size n larger than 30, the distribution of the
sample means can be approximated reasonably well by a
normal distribution. The approximation gets better as the
sample size n becomes larger.
2. If the original population is itself normally distributed,
then the sample means will be normally distributed for any
sample size n.
Notation for the Central Limit Theorem
Mean:
x = 
Standard Deviation of the Sample Means
(Standard Error of the Mean):  x = /n
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -7-
The Theorem
Given:
1. The random variable x has a distribution with a mean  and
standard deviation .
2. Samples of size n are randomly selected from this population.
Then:
1. The distribution of sample means will, as the sample size
increases, approach a normal distribution.
2. The mean of the sample means will be the population mean
.
3. The standard deviation of the sample means will be /n
Most Important Consequence
As the sample size increases, the sampling distribution of
sample means approaches a normal distribution
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -8-
Chapter 7: Sampling Distribution of the Sample Mean
Class Notes to accompany: Introductory Statistics, 9th Ed, By Neil A. Weiss
Prepared by: Nina Kajiji
Page -9-