SamplingDistributionsSamplingDistributions
Sampling DistributionSampling Distribution
Is the Theoretical probability distribution of asample statisticIs the Theoretical probability distribution of asample 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 sample means are distributed inrelation to the true population meanIt shows how sample means are distributed inrelation to the true population mean
Why Study Sampling Distributions?Why Study Sampling Distributions?
Sample statistics are used to estimate populationparametersSample statistics are used to estimate populationparameters
e.g.: estimates the population meane.g.: estimates the population mean
Problems: Different samples provide differentestimatesProblems: Different samples provide differentestimates
Large samples give better estimates; large samplecosts moreLarge samples give better estimates; large samplecosts more
How good is the estimate?How good is the estimate?
Approach: Sampling distribution tells us howclose our estimate should be to the true valueApproach: Sampling distribution tells us howclose our estimate should be to the true value
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, X,is age of individualsRandom variable, X,is age of individuals
Values of X: 18, 20,22, 24 measured inyearsValues of X: 18, 20,22, 24 measured inyears
A
B
C
D
.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,let’s look at the SAMPLE MEANSInstead of considering the population of ages,let’s look at the SAMPLE MEANS
All Possible Samples of Size n=2All Possible Samples of Size n=2
16 Different SamplesCan Be Taken16 Different SamplesCan Be Taken
16 Sample Means
Sampling Distribution of All Sample Means
18 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
Sample MeansDistribution
16 Sample Means
_
Developing Sampling DistributionsDeveloping Sampling Distributions
(continued)
Summary Measures of Sampling DistributionSummary Measures of Sampling Distribution
(continued)
Mean of
Sample Means
StandardDeviation of
Sample Means
Developing Sampling DistributionsDeveloping Sampling Distributions
Comparing the Population with itsSampling DistributionComparing the Population with itsSampling Distribution
18 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
Sample Means Distribution
n = 2
A B C D
(18) (20) (22) (24)
0
.1
.2
.3
Population
N = 4
P(X)
X
_
When the Population is NormalWhen the Population is Normal
Central TendencyCentral Tendency
VariationVariation
Sampling withReplacement
Population Distribution
Sampling Distributions
When the Population is Not NormalWhen the Population is Not Normal
Population Distribution
Sampling Distributions
Central TendencyCentral Tendency
VariationVariation
Sampling withReplacement
Central Limit TheoremCentral Limit Theorem
As SampleSize Gets“LargeEnough”
SamplingDistributionBecomesAlmostNormalRegardlessof Shape ofPopulation
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
Sampling Distribution of a Sample ProportionSampling Distribution of a Sample Proportion
Approximated bynormal distribution if:Approximated bynormal distribution if:
and and
Mean of samples:Mean of samples:
Standard error of proportion:Standard error of proportion:
p = population proportion p = population proportion
Sampling Distribution
P(ps)
.3
.2
.1
0
0 . 2 .4 .6 8 1
ps
Standardizing Sampling Distribution ofProportionStandardizing Sampling Distribution ofProportion
Sampling DistributionSampling Distribution
StandardizedNormal DistributionStandardizedNormal Distribution
ExampleExample
Sampling DistributionSampling Distribution
StandardizedNormal DistributionStandardizedNormal Distribution
Election Polling Example:Election Polling Example:
In the Gore/Bush Election, theproportion voting for Bush waspractically p=0.5 . If ABC News took apoll of 1067 adults, what is theprobability that the sample proportion(p is within .03 of the populationproportion?In the Gore/Bush Election, theproportion voting for Bush waspractically p=0.5 . If ABC News took apoll of 1067 adults, what is theprobability that the sample proportion(p is within .03 of the populationproportion?
We are interested in designing a study to estimate agiven population parameter (MEAN) with certainprecision.We are interested in designing a study to estimate agiven population parameter (MEAN) with certainprecision.
Estimate mean weight of 2576 babies born in the hospitalwith a 99% CI. Estimate of = 1.Estimate mean weight of 2576 babies born in the hospitalwith a 99% CI. Estimate of = 1.
You need to weight at least 27babies to obtain an estimate tobe 99% confident that the errorwill be 0.5 poundsYou need to weight at least 27babies to obtain an estimate tobe 99% confident that the errorwill be 0.5 pounds
You need to weight at least 16babies to obtain an estimate tobe 95% confident that the errorwill be 0.5 poundsYou need to weight at least 16babies to obtain an estimate tobe 95% confident that the errorwill be 0.5 pounds
We are interested in designing a study to estimate agiven population parameter (PROPORTION) withcertain precisionWe are interested in designing a study to estimate agiven population parameter (PROPORTION) withcertain precision
Determine proportion of adults living with hepatitis B virus. Nrequired to estimate it within 0.03 with 95% confidence. In asimilar area this proportion is 0.20. What should be n if suchestimate is not available?Determine proportion of adults living with hepatitis B virus. Nrequired to estimate it within 0.03 with 95% confidence. In asimilar area this proportion is 0.20. What should be n if suchestimate is not available?
You need to recruit at least 683people to obtain an estimatewithin 0.03 being 95% confidentYou need to recruit at least 683people to obtain an estimatewithin 0.03 being 95% confident
You need to recruit at most 1068people to obtain an estimatewithin 0.03 being 95% confident-> p=0.5 = Worst case scenario!You need to recruit at most 1068people to obtain an estimatewithin 0.03 being 95% confident-> p=0.5 = Worst case scenario!
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