Name                                       Shakeel Nouman Religion                                  Christian Domicile                            Punjab (Lahore) Contact #             0332-4462527. 0321-9898767 E.Mail                                sn_gcu@yahoo.com sn_gcu@hotmail.com

1. Sampling Distribution
Slide 1
Shakeel Nouman
M.Phil Statistics
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

2. Sampling Distributions
6.1
6.2
Slide 2
The Sampling Distribution of the Sample Mean
The Sampling Distribution of the Sample
Proportion
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

3. Sampling Distribution of the
Sample Mean
Slide 3
The sampling distribution of the sample mean is the
probability distribution of the population of the
sample means obtainable from all possible samples
of size n from a population of size N
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

4. Example: Sampling Annual % Return
on 6 Stocks #1
Slide 4
• Population of the percent returns from six
stocks
– In order, the values of % return are:
10%, 20%, 30%, 40%, 50%, and 60%
» Label each stock A, B, C, …, F in order of increasing %
return
» The mean rate of return is 35% with a standard deviation
of 17.078%
– Any one stock of these stocks is as likely
to be picked as any other of the six
» Uniform distribution with N = 6
» Each stock has a probability of being picked of 1/6 =
0.1667
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

5. Example: Sampling Annual %
Return
Slide 5
on 6 Stocks #2
Stock
Stock A
Stock B
Stock C
Stock D
Stock E
Stock F
Total
% Return
10
20
30
40
50
60
Frequency
1
1
1
1
1
1
6
Relative
Frequency
1/6
1/6
1/6
1/6
1/6
1/6
1
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

6. Example: Sampling Annual % Return
on 6 Stocks #3
Slide 6
• Now, select all possible samples of
size n = 2 from this population of
stocks of size N = 6
– Now select all possible pairs of stocks
• How to select?
– Sample randomly
– Sample without replacement
– Sample without regard to order
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

7. Example: Sampling Annual % Return
on 6 Stocks #4
Slide 7
• Result: There are 15 possible
samples of size n = 2
• Calculate the sample mean of each
and every sample
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

8. Example: Sampling Annual % Return
on 6 Stocks #5
Sample
Mean
15
20
25
30
35
40
45
50
55
Slide 8
Relative
Frequency Frequency
1
1/15
1
1/15
2
1/15
2
1/15
3
1/15
2
1/15
2
1/15
1
1/15
1
1/15
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

9. Observations
•
•
Slide 9
Although the population of N = 6 stock
returns has a uniform distribution, …
… the histogram of n = 15 sample mean
returns:
1. Seem to be centered over the
sample mean return of 35%, and
2. Appears to be bell-shaped and
less spread out than the
histogram of individual returns
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

10. General Conclusions
Slide 10
• If the population of individual items is
normal, then the population of all
sample means is also normal
• Even if the population of individual
items is not normal, there are
circumstances when the population of
all sample means is normal (Central
Limit Theorem)

11. General Conclusions Continued 11
Slide
• The mean of all possible sample means
equals the population mean
– That is, m = mx
• The standard deviation sx of all sample
means is less than the standard
deviation of the population
– That is, sx < s
» Each sample mean averages out the high and the
low measurements, and so are closer to m than
many of the individual population measurements

12. And the Empirical Rule
Slide 12
• The empirical rule holds for the sampling
distribution of the sample mean
– 68.26% of all possible sample means are within
(plus or minus) one standard deviation sx of m
– 95.44% of all possible observed values of x are
within (plus or minus) two sx of m
– 99.73% of all possible observed values of x are
within (plus or minus) three sx of m

13. Properties of the Sampling
Distribution of the Sample Mean #1
Slide 13
• If the population being sampled is normal, then so is
the sampling distribution of the sample mean, x
• The mean sx of the sampling distribution of x is
mx = m
•
That is, the mean of all possible sample means is the same
as the population mean
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

14. Properties of the Sampling
Distribution of the Sample Mean #2
Slide 14
• The variance s 2 of the sampling distribution of x is
x
s 
2
x
s2
n
 That is, the variance of the sampling distribution x
of
is
 directly proportional to the variance of the
population, and
 inversely proportional to the sample size
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

15. Properties of the Sampling
Distribution of the Sample Mean #3
Slide 15
• The standard deviation sx of the sampling distribution
of x is
sx 
s
n
 That is, the standard deviation of the sampling
distribution of x is
 directly proportional to the standard deviation of
the population, and
 inversely proportional to the square root of the
sample size
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

16. Notes
Slide 16
• The formulas for s2x and sx hold if the sampled
population is infinite
• The formulas hold approximately if the sampled
population is finite but if N is much larger (at
least 20 times larger) than the n (N/n ≥ 20)
– x is the point estimate of m, and the larger the
sample size n, the more accurate the estimate
– Because as n increases, sx decreases as 1/√n
» Additionally, as n increases, the more
representative is the sample of the population
• So, to reduce sx, take bigger samples!

17. Reasoning from the Sampling
Distribution
Slide 17
• Recall from Chapter 2 mileage example,
x = 31.5531 mpg for a sample of size n=49
– With s = 0.7992
• Does this give statistical evidence that the
population mean m is greater than 31 mpg
– That is, does the sample mean give evidence that m
is at least 31 mpg
• Calculate the probability of observing a
sample mean that is greater than or equal to
31.5531 mpg if m = 31 mpg
– Want P(x > 31.5531 if m = 31)

18. Reasoning from the Sampling
Distribution Continued
Slide 18
• Use s as the point estimate for s so that
sx 
s
n

0.7992
 0.1143
49
• Then

31.5531 m x 

Px  31.5531if m  31  P z 


sx


31.5531 31 

 P z 

0.1143 

 Pz  4.84
• But z = 4.84 is off the standard normal table
• The largest z value in the table is 3.09, which has a right hand
tail area of 0.001
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

19. Reasoning from the Sampling
Distribution #3
Slide 19
• z = 4.84 > 3.09, so P(z ≥ 4.84) < 0.001
• That is, if m = 31 mpg, then fewer than 1 in 1,000
of all possible samples have a mean at least as
large as observed
• Have either of the following explanations:
– If m is actually 31 mpg, then very unlucky in
picking this sample
OR
– Not unlucky, but m is not 31 mpg, but is really
larger
• Difficult to believe such a small chance would
occur, so conclude that there is strong evidence
that m does not equal 31 mpg
– Also, m is, in fact, actually larger than 31 mpg
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

20. Central Limit Theorem
Slide 20
• Now consider sampling a non-normal population
• Still have: m x  m
and
sx  s n
– Exactly correct if infinite population
– Approximately correct if population size N finite but much
larger than sample size n
» Especially if N ≥ 20  n
• But if population is non-normal, what is the shape of the
sampling distribution of the sample mean?
– Is it normal, like it is if the population is normal?
– Yes, the sampling distribution is approximately normal if the
sample is large enough, even if the population is non-normal
» By the “Central Limit Theorem”
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

21. The Central Limit Theorem #2 21
Slide
• No matter what is the probability distribution
that describes the population, if the sample
size n is large enough, then the population of
all possible sample means is approximately
normal with mean m x  m and standard
deviation s x  s n
• Further, the larger the sample size n, the closer
the sampling distribution of the sample mean
is to being normal
– In other words, the larger n, the better the
approximation
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

22. How Large?
Slide 22
• How large is “large enough?”
• If the sample size is at least 30, then for most
sampled populations, the sampling distribution of
sample means is approximately normal
– Here, if n is at least 30, it will be assumed that the
sampling distribution of x is approximately normal
» If the population is normal, then the sampling
distribution of x is normal no regardless of the sample
size

23. Unbiased Estimates
Slide 23
• A sample statistic is an unbiased point estimate
of a population parameter if the mean of all
possible values of the sample statistic equals the
population parameter
• x is an unbiased estimate of m because mx=m
– In general, the sample mean is always an
unbiased estimate of m
– The sample median is often an unbiased estimate
of m
» But not always
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

24. Unbiased Estimates Continued 24
Slide
• The sample variance s2 is an unbiased estimate of s2
– That is why s2 has a divisor of n – 1 and not n
• However, s is not an unbiased estimate of s
– Even so, the usual practice is to use s as an estimate of s
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

25. Minimum Variance Estimates 25
Slide
• Want the sample statistic to have a small
standard deviation
– All values of the sample statistic should be
clustered around the population parameter
» Then, the statistic from any sample should be close to the
population parameter
» Given a choice between unbiased estimates, choose one with
smallest standard deviation
» The sample mean and the sample median are both unbiased
estimates of m
» The sampling distribution of sample means generally has a
smaller standard deviation than that of sample medians
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

26. Finite Populations
Slide 26
• If a finite population of size N is sampled
randomly and without replacement, must use
the “finite population correction” to calculate
the correct standard deviation of the sampling
distribution of the sample mean
– If N is less than 20 times the sample size, that is,
if N < 20  n
– Otherwise
sx 
s
n
but instead s x 
s
n
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

27. Finite Populations ContinuedSlide 27
• The finite population correction is
N n
N 1
• and the standard error is
sx 
s
n
N n
N 1
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

28. Sampling Distribution of the
Sample Proportion
Slide 28
The probability distribution of all possible sample
proportions is the sampling distribution of the sample
proportion
ˆ
p
If a random sample of size n is taken from a population
then the sampling distribution of is
ˆ
p
 approximately normal, if n is large
 has mean m ˆ  p
p
p1  p 
 has standard deviation s ˆp 
n
p
where p is the population proportion andˆ is a sampled
proportion

29. Slide 29
Name
Religion
Domicile
Contact #
E.Mail
M.Phil (Statistics)
Shakeel Nouman
Christian
Punjab (Lahore)
0332-4462527. 0321-9898767
sn_gcu@yahoo.com
sn_gcu@hotmail.com
GC University, .
(Degree awarded by GC University)
M.Sc (Statistics)
Statitical Officer
(BS-17)
(Economics & Marketing
Division)
GC University, .
(Degree awarded by GC University)
Livestock Production Research Institute
Bahadurnagar (Okara), Livestock & Dairy Development
Department, Govt. of Punjab
Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer