Sampling Sampling DistributionsDistributions

Sampling DistributionSampling Distribution

Is the Theoretical probability distribution of a Is the Theoretical probability distribution of a

sample statisticsample statistic

A sample statistic is a random variable:A sample statistic is a random variable: E.g. Sample mean, sample proportionE.g. Sample mean, sample proportion

For the mean:For the mean:

It shows how It shows how samplesample means are distributed in means are distributed in

relation to the relation to the true population true population meanmean

Why Study Sampling Distributions?Why Study Sampling Distributions?

Sample statistics are used to estimate population Sample statistics are used to estimate population parametersparameters e.g.: estimates the population meane.g.: estimates the population mean

Problems: Different samples provide different Problems: Different samples provide different estimatesestimates Large samples give better estimates; large sample Large samples give better estimates; large sample

costs morecosts more How good is the estimate?How good is the estimate?

Approach: Approach: Sampling distributionSampling distribution tells us how tells us how close our estimate close our estimate should beshould be to the true value to the true value

50X

Developing Sampling DistributionsDeveloping Sampling Distributions

Simple Example Case:Simple Example Case:

Assume there is a population …Assume there is a population …

Population size N=4Population size N=4

Random variable, Random variable, XX,,

is age of individualsis age of individuals

Values of Values of XX: 18, 20,: 18, 20,

22, 24 measured in22, 24 measured in

yearsyearsA

B C

D

1

2

1

18 20 22 2421

4

2.236

N

ii

N

ii

X

N

X

N

.3

.2

.1

0 A B C D (18) (20) (22) (24)

Uniform Distribution

P(X)

X

Developing Sampling DistributionsDeveloping Sampling Distributions(continued)

Summary Measures for the Population DistributionSummary Measures for the Population Distribution

Instead of considering the population of ages, Instead of considering the population of ages, let’s look at the let’s look at the SAMPLE MEANSSAMPLE MEANS

1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

All Possible Samples of Size n=2All Possible Samples of Size n=2

16 Different 16 Different Samples Can Be Samples Can Be

Taken Taken

16 Sample Means

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3

P(X)

X

Sample Means

Distribution

16 Sample Means

_

Developing Sampling DistributionsDeveloping Sampling Distributions (continued)

1

2

1

2 2 2

18 19 19 2421

16

18 21 19 21 24 211.58

16

N

ii

X

N

i Xi

X

X

N

X

N

Summary Measures of Summary Measures of Sampling DistributionSampling Distribution

(continued)

Mean of Sample Means

Standard Deviation of Sample Means

Developing Sampling DistributionsDeveloping Sampling Distributions

Comparing the Population with its Comparing the Population with its

Sampling DistributionSampling Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distribution

n = 2

A B C D (18) (20) (22) (24)

0

.1

.2

.3

PopulationN = 4

P(X)

X_

21 2.236 21 1.58X X

When the Population is NormalWhen the Population is Normal

Central TendencyCentral Tendency

VariationVariation

Sampling with Replacement

Population Distribution

Sampling Distributions

X

Xn

X50X

4

5X

n

16

2.5X

n

50

10

When the Population is Not NormalWhen the Population is Not Normal

Population Distribution

Sampling Distributions

X

Xn

X50X

4

5X

n

30

1.8X

n

50

10 Central TendencyCentral Tendency

VariationVariation

Sampling with Replacement

Central Limit TheoremCentral Limit Theorem

As Sample Size Gets “Large Enough”

Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X

Population ProportionsPopulation Proportions

Categorical variableCategorical variable e.g.: Gender, voted for bush, college degreee.g.: Gender, voted for bush, college degree

Proportion of population that has a characteristicProportion of population that has a characteristic

Sample proportion provides an estimateSample proportion provides an estimate

If two outcomes, X has a binomial distributionIf two outcomes, X has a binomial distribution Possess or do not possess characteristicPossess or do not possess characteristic

number of successes

sample sizeS

Xp

n

p

p

Sampling Distribution of a Sample ProportionSampling Distribution of a Sample Proportion

Approximated by Approximated by

normal distribution normal distribution if:if:

andand

Mean of samples:Mean of samples:

Standard error of proportionStandard error of proportion: :

p p = = population population proportionproportion

Sampling DistributionP(ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1ps

5np 1 5n p

Spp

1Sp

p p

n

Standardizing Sampling Distribution of Standardizing Sampling Distribution of

ProportionProportion

1S

S

S p S

p

p p pZ

p p

n

Sampling Sampling DistributionDistribution

Standardized Standardized Normal Normal

DistributionDistribution

Sp 1Z

Sp Sp Z0Z

ExampleExample 200 .4 .43 ?Sn p P p

.43 .4.43 .87 .8078

.4 1 .4

200

S

S

S pS

p

pP p P P Z

Sampling Sampling DistributionDistribution

Standardized Standardized Normal Normal

DistributionDistribution

Sp 1Z

Sp

Sp Z0.43 .87

Election Polling Example:Election Polling Example:

In the Gore/Bush Election, the In the Gore/Bush Election, the

proportion voting for Bush was proportion voting for Bush was

practically practically pp=0.5 . If =0.5 . If ABC NewsABC News took a took a

poll of 1067 adults, what is the poll of 1067 adults, what is the

probability that the probability that the sample proportion sample proportion

(p(p is within .03 of the population is within .03 of the population

proportion?proportion?

We are interested in designing a study to estimate We are interested in designing a study to estimate a given population parameter (MEAN) with certain a given population parameter (MEAN) with certain precision.precision.

d zn

nz

d

2 2

2

Estimate mean weight of 2576 babies born in the hospital Estimate mean weight of 2576 babies born in the hospital with a 99% CI. Estimate of with a 99% CI. Estimate of = 1. = 1.

You need to weight at least 27 You need to weight at least 27 babies to obtain an estimate to babies to obtain an estimate to be 99% confident that the error be 99% confident that the error will be will be 0.5 pounds0.5 pounds

n ( . ) ( )

( . ).

2 5 7 6 1

0 52 6 5 4 2 7

2 2

2

n ( . ) ( )

( . ).

1 9 6 1

0 51 5 3 7 1 6

2 2

2

You need to weight at least 16 You need to weight at least 16 babies to obtain an estimate to babies to obtain an estimate to be 95% confident that the error be 95% confident that the error will be will be 0.5 pounds0.5 pounds

We are interested in designing a study to estimate a We are interested in designing a study to estimate a given population parameter (PROPORTION) with given population parameter (PROPORTION) with certain precisioncertain precision

Determine proportion of adults living with hepatitis B virus. N Determine proportion of adults living with hepatitis B virus. N required to estimate it within 0.03 with 95% confidence. In a required to estimate it within 0.03 with 95% confidence. In a similar area this proportion is 0.20. What should be n if such similar area this proportion is 0.20. What should be n if such estimate is not available?estimate is not available?

You need to recruit at least 683 You need to recruit at least 683 people to obtain an estimate people to obtain an estimate within 0.03 being 95% confidentwithin 0.03 being 95% confident

n ( . ) ( . )( . )

( . ).

1 9 6 0 2 0 0 8 0

0 0 36 8 2 9 5 6 8 3

2

2

n ( . ) ( . )

( . ).

1 9 6 0 5

0 0 31 0 6 7 11 1 0 6 8

2 2

2

You need to recruit at most 1068 You need to recruit at most 1068 people to obtain an estimate people to obtain an estimate within 0.03 being 95% confident within 0.03 being 95% confident -> p=0.5 = Worst case scenario!-> p=0.5 = Worst case scenario!

d z nz p p

d

( )12

2n

pp )1(

THE THE

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