Sampling distributionschapter 6 ST 315

Nutan S. Mishra

Department of Mathematics and Statistics

University of South Alabama

Useful links

• http://oak.cats.ohiou.edu/~wallacd1/ssample.html

• http://garnet.acns.fsu.edu/~jnosari/05.PDF

• http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/

Sampling distributionIn chapter 2 we defined a population parameter as a function of all the population

values.Let population consists of N observations then population mean and population

standard deviation are parameters

For a given population, the parameters are fixed values.

NN

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N

xxxxf

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22

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Sampling distributionOn the other hand if we draw a sample of size n from a population of size N,

then a function of the sample values is called a statistics

For example sample mean and sample standard deviation are sample statistics.

Since we can draw a large number of samples from the population the value of sample statistic varies from sample to sample

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Sampling distributionSince value of a sample statistic varies from sample to sample, the

statistic itself is a random variable and has a probability distribution.

For Example sample mean is random variable and it has a probability distribution.

Example: Start with a toy example

Let the population consists of 5 students who took a math quiz of 5 points.

Name of the students and corresponding scores are as follows:

Name of the student A B C D E

Score 2 3 4 4 5

For this population mean µ = 3.6 and standard deviation σ = 1.02

x

Sampling distribution

Now we repeatedly draw samples of size three from the population of size 5. then the possible samples are 10 as listed below

The population parameters are µ = 3.6 and s.d. σ = 1.02

Sample sample Sample values s

1 A,B,C 2,3,4 3 1

2 A,B,D 2,3,4 3 1

3 A,B,E 2,3,5 3.33 1.53

4 A,C,D 2,4,4 3.33 1.16

5 A,C,E 2,4,5 3.67 1.53

6 A,D,E 2,4,5 3.67 1.53

7 B,C,D 3,4,4 3.67 .58

8 B,C,E 3,4,5 4 1

9 B,D,E 3,4,5 4 1

10 C,D,E 4,4,5 4.33 .58

x

Sampling distributionX= score of a student in the math quiz

Thus we see that the sample mean is a new random variable and has a probability distribution.

Question: What is the mean of this random variable and what is its variance?

x f P(x)

2 1 .2

3 1 .2

4 2 .4

5 1 .2

f P( )

3 2 .2

3.33 2 .2

3.67 3 .3

4 2 .2

4.33 1 .1

x xPopulation distribution

Sampling distribution of sample mean

x

Sampling distributionLet N be the size of the population and n be the size of the

sample

If n/N > .05

And if n/N ≤.05

1

mean sample ofdevation standard and

mean sample ofmean

x

x

N

nN

n

n

x

x

mean sample ofdevation standard and

mean sample ofmean

Sampling distribution of sample mean

Theorem

Let X be a random variable with population mean µ and population standard deviation σ . If we collect the samples of size n then the new random variable sample mean has the mean same as µ and standard deviation σ/√n

We can denote them as follows:

x

n

mean

x

x

x ofdeviation standard

x of

Sampling distribution of sample mean

n

mean

x

x

x ofdeviation standard

x of

It is easy to see that the standard deviation of sample mean decreases as the sample size increases.

The mean of the sample remains unaffected with the change in sample size.

Sample mean is called an estimator of the population mean.

Because whenever population mean is unknown we will use sample mean in place.

Sampling distribution of sample meanP( )

3 .2

3.33 .2

3.67 .3

4 .2

4.33 .1

x x

From the above table when we compute the mean and variance

They are

Sampling distribution of sample mean

We have seen that distribution of the sample mean is derived from the distribution of x

Thus distribution of x is called parent distribution.

The next question is to investigate what is the relationship between the parent distribution and the sampling distribution of .

x

x

Sampling distribution of sample mean

Let the distribution of x is normal with mean µ and standard deviation σ then it is equivalent to saying that

Let the parent population is normal with mean µ and standard deviation σ

If we draw a sample of size n from such a population then • Mean of that is is equal to the mean of the

population µ.• Standard deviation of that is is equal to σ/√n

• The shape of the distribution of is normal whatever be the value of n

xx

x x

x

Sampling distribution of sample mean

If X~ N(µ, σ) then

~ N ((µ, σ/√n)

Where n is size of the sample drawn from the population

x

Central Limit Theorem

For a large sample size, the sampling distribution of is approximately normal, irrespective of the shape of the population distribution.

What size of the sample is considered to be large?

A sample of size ≥ 30 is considered to be large.

x

Sampling distribution of sample meanAssume that population standard deviation σ is

known

If the random sample comes from a normal population, the sampling distribution of sample mean is normal regardless the size of the sample.

If the shape of the parent population is not known or not normal then distribution of sample mean is approximately normal when ever n is large (≥30).(this is central limit theorem)

If the shape of the parent population is not known or not normal and sample size is small then we can not say readily about the shape of sample distribution

Sampling distribution of sample meanWhen population standard deviation is

unknown• If the sample size is large the sampling

distribution of sample mean is still approximately normal

• If the sample size is small then

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nSX

t

1

2 2

1

)(S where

1.-n parameter with

ondistributi- thaving variablerandom a is

About t-distribution• t is a special continuous distribution• Its symmetric about zero• Has bell shaped curve like normal• Its variance depends on the parameter • is called degrees of freedom and is the only

parameter of t-distribution.• Variance of t approaches 1 as n ∞• In other words t approaches Z as n ∞• The t-values are tabulated for different values of

the right tail areas and degrees of freedom

Sampling distribution of sample mean

Sampling distribution of sample mean

σ knownσ unknown

n>30 n<30 n>30 n<30

Normal normal Approx. normal t

For t-distribution :assume that parent population is approximately normal