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1、Sampling Distributions,Chapter 7,Sampling Chapter 7,Objectives,In this chapter, you learn:The concept of the sampling distributionTo compute probabilities related to the sample mean and the sample proportionThe importance of the Central Limit Theorem,ObjectivesIn this chapter, you,Sampling Distribut
2、ions,A sampling distribution is a distribution of all of the possible values of a sample statistic for a given sample size selected from a population.For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of size 50, you will co
3、mpute a different mean for each sample. We are interested in the distribution of all potential mean GPAs we might calculate for any sample of 50 students.,DCOVA,Sampling DistributionsA sampli,Developing a Sampling Distribution,Assume there is a population Population size N=4Random variable, X,is age
4、 of individualsValues of X: 18, 20,22, 24 (years),A,B,C,D,DCOVA,Developing a Sampling Distribu,.3,.2,.1,0,18 20 22 24 A B C D,Uniform Distribution,P(x),x,(continued),Summary Measures for the Population Distribution:,Developing a Sampling Distribution,DCOVA,.3.2.1 0 18 20,16 possible samples (samplin
5、g with replacement),Now consider all possible samples of size n=2,(continued),Developing a Sampling Distribution,16 Sample Means,DCOVA,16 possible samples (sampling,Sampling Distribution of All Sample Means,18 19 20 21 22 23 24,0,.1,.2,.3,P(X),X,Sample Means Distribution,16 Sample Means,_,Developing
6、 a Sampling Distribution,(continued),(no longer uniform),_,DCOVA,Sampling Distribution of All S,Summary Measures of this Sampling Distribution:,Developing A Sampling Distribution,(continued),DCOVA,Note:Here we divide by 16 because there are 16different samples of size 2.,Summary Measures of this Sam
7、pl,Comparing the Population Distributionto the Sample Means Distribution,18 19 20 21 22 23 24,0,.1,.2,.3,P(X),X,18 20 22 24 A B C D,0,.1,.2,.3,PopulationN = 4,P(X),X,_,Sample Means Distributionn = 2,_,DCOVA,Comparing the Population Distr,Sample Mean Sampling Distribution:Standard Error of the Mean,D
8、ifferent samples of the same size from the same population will yield different sample meansA measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:(This assumes that sampling is with replacement or sampling is without replacement from an infinite po
9、pulation)Note that the standard error of the mean decreases as the sample size increases,DCOVA,Sample Mean Sampling Distribut,Sample Mean Sampling Distribution:If the Population is Normal,If a population is normal with mean and standard deviation , the sampling distribution of is also normally distr
10、ibuted with and,DCOVA,Sample Mean Sampling Distribut,Z-value for Sampling Distributionof the Mean,Z-value for the sampling distribution of :,where:= sample mean= population mean= population standard deviation n = sample size,DCOVA,Z-value for Sampling Distribut,Normal Population Distribution,Normal
11、Sampling Distribution (has the same mean),Sampling Distribution Properties,(i.e. is unbiased ),DCOVA,Normal Population Distribution,Sampling Distribution Properties,As n increases, decreases,Larger sample size,Smaller sample size,(continued),DCOVA,Sampling Distribution Properti,Determining An Interv
12、al Including A Fixed Proportion of the Sample Means,Find a symmetrically distributed interval around that will include 95% of the sample means when = 368, = 15, and n = 25.Since the interval contains 95% of the sample means 5% of the sample means will be outside the intervalSince the interval is sym
13、metric 2.5% will be above the upper limit and 2.5% will be below the lower limit.From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96.,DCOVA,Determining An Interval Includ,Determining An Interval Including A Fixed Pr
14、oportion of the Sample Means,Calculating the lower limit of the intervalCalculating the upper limit of the interval95% of all sample means of sample size 25 are between 362.12 and 373.88,(continued),DCOVA,Determining An Interval Includ,Sample Mean Sampling Distribution:If the Population is not Norma
15、l,We can apply the Central Limit Theorem:Even if the population is not normal,sample means from the population will be approximately normal as long as the sample size is large enough.Properties of the sampling distribution: and,DCOVA,Sample Mean Sampling Distribut,n,Central Limit Theorem,As the samp
16、le size gets large enough,the sampling distribution of the sample mean becomes almost normal regardless of shape of population,DCOVA,nCentral Limit TheoremAs the,Population Distribution,Sampling Distribution (becomes normal as n increases),Central Tendency,Variation,Larger sample size,Smaller sample
17、 size,Sample Mean Sampling Distribution:If the Population is not Normal,(continued),Sampling distribution properties:,DCOVA,Population DistributionSamplin,How Large is Large Enough?,For most distributions, n 30 will give a sampling distribution that is nearly normalFor fairly symmetric distributions
18、, n 15For a normal population distribution, the sampling distribution of the mean is always normally distributed,DCOVA,How Large is Large Enough?For,Example,Suppose a population has mean = 8 and standard deviation = 3. Suppose a random sample of size n = 36 is selected. What is the probability that
19、the sample mean is between 7.8 and 8.2?,DCOVA,ExampleSuppose a population ha,Example,Solution:Even if the population is not normally distributed, the central limit theorem can be used (n 30) so the sampling distribution of is approximately normal with mean = 8 and standard deviation,(continued),DCOV
20、A,ExampleSolution:(continued)DCO,Example,Solution (continued):,(continued),Z,7.8 8.2,-0.4 0.4,Sampling Distribution,Standard Normal Distribution,Population Distribution,?,?,?,?,?,?,?,?,?,?,?,?,Sample,Standardize,X,DCOVA,Example Solution (continued),Population Proportions, = the proportion of the pop
21、ulation having some characteristicSample proportion (p) provides an estimate of :0 p 1p is approximately distributed as a normal distribution when n is large(assuming sampling with replacement from a finite population or without replacement from an infinite population),DCOVA,Population Proportions =
22、 t,Sampling Distribution of p,Approximated by anormal distribution if: where and,(where = population proportion),Sampling Distribution,P( ps),.3.2.1 0,0 . 2 .4 .6 8 1,p,DCOVA,Sampling Distribution of pAppr,Z-Value for Proportions,Standardize p to a Z value with the formula:,DCOVA,Z-Value for Proport
23、ionsStandar,Example,If the true proportion of voters who support Proposition A is = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?,i.e.: if = 0.4 and n = 200, what is P(0.40 p 0.45) ?,DCOVA,ExampleIf the true proportion,Example,if = 0.4 and n
24、 = 200, what is P(0.40 p 0.45) ?,(continued),Find :,Convert to standardized normal:,DCOVA,Example if = 0.4 an,Example,Z,0.45,1.44,0.4251,Standardize,Sampling Distribution,Standardized Normal Distribution,if = 0.4 and n = 200, what is P(0.40 p 0.45) ?,(continued),Utilize the cumulative normal table:P
25、(0 Z 1.44) = 0.9251 0.5000 = 0.4251,0.40,0,p,DCOVA,ExampleZ0.451.440.4251Standard,Chapter Summary,In this chapter we discussed:The concept of a sampling distributionComputing probabilities related to the sample mean and the sample proportionThe importance of the Central Limit Theorem,Chapter SummaryIn this chapter,
