sampling distributions chapter 7. the concept of a sampling distribution repeated samples of the...
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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.
The Sampling ProcessThe Sampling Process
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
The Sampling ProcessThe Sampling Process
POPULATIONMean = 50
Calculate Mean = 47.9
30, 42, 48, 49, 61, 54, 41, 38, 59, 57
Plot the Sample Mean
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Population Histogram
SAMPLE
40 45 50 55 60
mean
Means of the Separate Samples Dot Plot
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Population Histogram
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
Means of the Separate Samples Dot Plot
What about this sampling What about this sampling distribution?distribution?
40 45 50 55 60
mean
Means of the Separate Samples Dot Plot
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.
40 45 50 55 60
mean
Means of the Separate Samples Dot Plot
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
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 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
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 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