Sampling Distributions Chapter 7. The Concept of a Sampling Distribution Repeated samples of the same size are selected from the same population. Repeated

Sampling DistributionsSampling DistributionsChapter 7 Chapter 7

The Concept of a The Concept of a Sampling DistributionSampling Distribution

Repeated samples of the same size Repeated samples of the same size are selected from the same are selected from the same population.population.

The same sample statistic is The same sample statistic is calculated from the data in EACH calculated from the data in EACH sample.sample.

The distribution of the sample The distribution of the sample statistics is the SAMPLING statistics is the SAMPLING DISTRIBUTION of that sample DISTRIBUTION of that sample statistic.statistic.

The Sampling ProcessThe Sampling Process

1 2, , ... , nx x x

SAMPLE

POPULATION

μ

x a nd s

The Sampling DistributionThe Sampling Distribution

Repeated Sampling

1 2 3, , , ... , Nx x x x

POPULATION μ Sampling

Distribution

What is Standard Error?What is Standard Error?

Standard Error Standard Error has been identified has been identified as a quantity that is not as a quantity that is not understood.understood.

Is it a Standard Deviation?Is it a Standard Deviation?

Standard Error of what?Standard Error of what?

What does it tell us?What does it tell us?

The purpose of this presentation is The purpose of this presentation is to make the concept of to make the concept of Standard Standard Error Error clearer and more clearer and more understandable.understandable.


POPULATIONMean = 50

Calculate Mean = 47.9

30, 42, 48, 49, 61, 54, 41, 38, 59, 57

This sample mean is an ESTIMATE of the population mean.

We should not be surprised that the estimate does not equal the true mean for

the population!

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Population Histogram

SAMPLE


POPULATIONMean = 50

Calculate Mean = 47.9

30, 42, 48, 49, 61, 54, 41, 38, 59, 57

Plot the Sample Mean

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SAMPLE

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mean

Means of the Separate Samples Dot Plot

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The Sampling DistributionThe Sampling Distribution

Repeated Sampling

Sampling Distribution of the Sample Means

Calculate means for each sample

m1, m2, …

Plot All Sample Means

40 45 50 55 60

mean


What about this sampling What about this sampling distribution?distribution?

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mean


Each dot represents a mean from one of the samples.

Each sample mean is an ESTIMATE of the population mean.

Notice that center of this graph is around 50 and the spread ranges from 45 to 55.

What about this sampling What about this sampling distribution?distribution?

The mean of sampling distribution (that is, the mean of the sample means) is the MEAN of the population!

AND…

We call the standard deviation of the distribution of sample means the STANDARD ERROR OF THE ESTIMATE OF THE POPULATION MEAN.

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In SummaryIn Summary

STANDARD DEVIATION is a measure STANDARD DEVIATION is a measure of the spread of of the spread of datadata in a in a population or in a sample.population or in a sample.

STANDARD ERROR is a measure of STANDARD ERROR is a measure of the spread of the the spread of the ESTIMATESESTIMATES of of a measure of a population a measure of a population calculated from repeated calculated from repeated sampling.sampling.

In short…In short…

STANDARDSTANDARD

DEVIATIONDEVIATION VariationVariation

in DATAin DATA

STANDARDSTANDARD

ERRORERROR Variation in Variation in ESTIMATES FROM ESTIMATES FROM

SAMPLESSAMPLES

Point EstimatorsPoint Estimators

When inferences are made from the When inferences are made from the sample to the population, the sample to the population, the sample mean is viewed as an sample mean is viewed as an estimator of the mean of the estimator of the mean of the population from which the sample population from which the sample was selected.was selected.

Similarly, the proportion of Similarly, the proportion of successes in a sample is an successes in a sample is an estimator of the proportion of estimator of the proportion of successes in the population.successes in the population.

Properties of Point Estimators Properties of Point Estimators The summary statistic should be The summary statistic should be UNBIASED, that is the mean of the UNBIASED, that is the mean of the sampling distribution is equal to the sampling distribution is equal to the value you would get if you computed the value you would get if you computed the summary statistic using the entire summary statistic using the entire population. More formally, an estimator population. More formally, an estimator is unbiased if its expected value equals is unbiased if its expected value equals the parameter being estimated.the parameter being estimated.

The summary statistic should have as The summary statistic should have as little variability as possible (be more little variability as possible (be more precise than other estimates) and should precise than other estimates) and should have a standard error that decreases as have a standard error that decreases as the sample size increases. the sample size increases.

HomeworkHomework

Pg 418-421Pg 418-421 AP style P2,P3,E5AP style P2,P3,E5 E2,4,6,8,10E2,4,6,8,10

PopulatioPopulation n ParameterParameter

Sample Sample StatistStatisticic

Sampling Sampling DistributiDistributionon

Mean Mean

Standard Standard DeviationDeviation

Size Size

X

X

X

Properties of the Sampling Properties of the Sampling Distribution of the Distribution of the Sample MeanSample Mean

If a random sample of size n is selected If a random sample of size n is selected from a population with mean from a population with mean µ and standard µ and standard deviation deviation σσ, then , then

The mean of the sampling The mean of the sampling distribution of equals the mean of distribution of equals the mean of the population µthe population µ

X X


If a random sample of size n is selected from a If a random sample of size n is selected from a population with mean population with mean µ and standard deviation µ and standard deviation σσ, then, then

The standard deviation, , of the sampling The standard deviation, , of the sampling distribution of , sometimes called the distribution of , sometimes called the standard error of the mean, equals the standard standard error of the mean, equals the standard deviation of the population deviation of the population σσ, divided by the , divided by the square root of the sample size n:square root of the sample size n:

*Only used when N>10n*Only used when N>10n

XX


If a random sample of size n is selected If a random sample of size n is selected from a population with mean from a population with mean µ and standard µ and standard deviation deviation σσ, then, then

The shape of the sampling distribution will The shape of the sampling distribution will be approximately normal if the population is be approximately normal if the population is approximately normal; for other populations, approximately normal; for other populations, the sampling distribution becomes more the sampling distribution becomes more normal as n increasesnormal as n increases

This property is called the CENTRAL LIMIT This property is called the CENTRAL LIMIT THEOREM (CLT)THEOREM (CLT)

Reasonably Likely Averages Reasonably Likely Averages

Mean Mean ± 1.96(SE)± 1.96(SE)

1.96 is the z-score and comes 1.96 is the z-score and comes from the cut off point of the from the cut off point of the middle 95% of a normal middle 95% of a normal distribution distribution

IfIf the Sampling Distribution is the Sampling Distribution is known…known…

Probability questions about sample Probability questions about sample statistics can be answered. For statistics can be answered. For example,example,

A simple random sample of 50 is A simple random sample of 50 is selected from a normal population selected from a normal population with a mean of 50 and a standard with a mean of 50 and a standard deviation of 10. What is the deviation of 10. What is the probability that the sample mean probability that the sample mean will be greater than 53?will be greater than 53?

The Answer…The Answer…

A simple random sample of 50 is selected A simple random sample of 50 is selected from a normal population with a mean of 50 from a normal population with a mean of 50 and a standard deviation of 10. What is the and a standard deviation of 10. What is the probability that the sample mean will be probability that the sample mean will be greater than 53?greater than 53?

( , ) ,

1050,10 50,50

53 502.12 .017

1050

Normal Sampling Distribution Normaln

Normal Sampling Distribution Normal

xz p

n

Properties of the Sampling Properties of the Sampling Distribution of the sum of a SampleDistribution of the sum of a Sample

If a random sample of size n is selected from If a random sample of size n is selected from a distribution with mean a distribution with mean µ and standard µ and standard deviation deviation σσ, then , then

The mean of the sampling distribution of the The mean of the sampling distribution of the sum issum is

The standard error of the sampling The standard error of the sampling

distribution of the sum isdistribution of the sum is

CLT appliesCLT applies

HOMEWORK HOMEWORK

Pg 440 Pg 440 E15,16,17,19,20,21,22,25,26,29,E15,16,17,19,20,21,22,25,26,29,30,31,32, 33,34 30,31,32, 33,34

Sampling Distribution of the Sampling Distribution of the Sample ProportionSample Proportion

We will now move from studying We will now move from studying the behavior of the sample mean the behavior of the sample mean to studying the behavior of the to studying the behavior of the sample proportions (the sample proportions (the proportion of proportion of ““successessuccesses”” in in the sample)the sample)

Properties of the Sampling Distribution of Properties of the Sampling Distribution of the Number of Successesthe Number of Successes

If a random sample of size n is selected If a random sample of size n is selected from a population with proportion of from a population with proportion of successes, p, then the sampling successes, p, then the sampling distribution of the number of successes Xdistribution of the number of successes X

Has mean Has mean Has standard error Has standard error Will be approximately normal as long as n Will be approximately normal as long as n is large enoughis large enough

As a guideline both np and n(1-p) As a guideline both np and n(1-p) are at least 10are at least 10

np≥10 and n(1-np≥10 and n(1-p) ≥10 p) ≥10

Example Example

The use of seat belts continues to The use of seat belts continues to rise in the United States, with rise in the United States, with overall seat belt usage of 82%. overall seat belt usage of 82%. Mississippi lags behind the rest of Mississippi lags behind the rest of the nation—only about 60% wear seat the nation—only about 60% wear seat belts. Suppose you take a random belts. Suppose you take a random sample of 40 Mississippians. How sample of 40 Mississippians. How many do you expect will wear seat many do you expect will wear seat belts? What is the probability that belts? What is the probability that 30 or more of the people in your 30 or more of the people in your sample wear seat belts?sample wear seat belts?

Solution Solution

Sampling Distributions of the Sampling Distributions of the Sample ProportionSample Proportion

True proportion of successes is True proportion of successes is represented by represented by ““pp ””

Sample proportion of successes Sample proportion of successes is represented by is represented by ““ ””

= (# of = (# of successes)/(sample size)successes)/(sample size)

Sampling Distribution of p-hatSampling Distribution of p-hat

How does behave? To study How does behave? To study the behavior, imagine taking the behavior, imagine taking many random samples of size n, many random samples of size n, and computing a p-hat for each and computing a p-hat for each of the samples.of the samples.

Then we plot this set of Then we plot this set of with a histogram.with a histogram.

Sampling Distribution of p-hatSampling Distribution of p-hat

Properties of p-hatProperties of p-hat When sample sizes are fairly large, the When sample sizes are fairly large, the shape of the distribution will be shape of the distribution will be normal.normal.

The mean of the distribution is the value The mean of the distribution is the value of the population parameter p.of the population parameter p.

The standard deviation of this distribution The standard deviation of this distribution is the square root of p(1-p)/n.is the square root of p(1-p)/n.

n

pppsd

)1()ˆ(

As a guideline use np ≥ 10 and n(1-p) ≥ 10

Example Example

About 60% of Mississippians use seat About 60% of Mississippians use seat belts. Suppose your class conducts belts. Suppose your class conducts a survey of 40 randomly selected a survey of 40 randomly selected Mississippians.Mississippians.A.A.What is the chance that 75% or What is the chance that 75% or more of those selected wear seat more of those selected wear seat belts?belts?B.B.Would it be quite unusual to find Would it be quite unusual to find that fewer than 25% of that fewer than 25% of Mississippians selected wear seat Mississippians selected wear seat belts?belts?

Solution Solution

Homework Homework

Pg 435 Pg 435 E35-44E35-44

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Sampling Distributions Chapter 7. The Concept of a Sampling Distribution Repeated samples of the same size are selected from the same population. Repeated