Sampling distribution

Sampling Distribution

Slide 1

Shakeel Nouman
M.Phil Statistics

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

Sampling Distributions
6.1
6.2

Slide 2

The Sampling Distribution of the Sample Mean
The Sampling Distribution of the Sample
Proportion


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


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


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


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


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


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


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


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)

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

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

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



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



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


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!

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)

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

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

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”


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

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

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


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


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


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


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 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

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

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