商务统计学英文版教学课件第7章

1、Sampling DistributionsChapter 7Sampling Chapter 7ObjectivesIn 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 TheoremObjectivesIn this chapter, youSampling DistributionsA

2、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 compute a

3、 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.DCOVASampling DistributionsA sampliDeveloping a Sampling DistributionAssume there is a population Population size N=4Random variable, X,is age of individ

4、ualsValues of X: 18, 20,22, 24 (years)ABCDDCOVADeveloping a Sampling Distribu.3.2.1 0 18 20 22 24 A B C DUniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing a Sampling DistributionDCOVA.3.2.1 0 18 20 16 possible samples (sampling with replacement)Now consi

5、der all possible samples of size n=2(continued)Developing a Sampling Distribution16 Sample Means1stObs2nd Observation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2224,24DCOVA16 possible samples (sampling Sampling Distribution of All Sample Means18 19 20 21

6、 22 23 240 .1 .2 .3 P(X) XSample Means Distribution16 Sample Means_Developing a Sampling Distribution(continued)(no longer uniform)_DCOVASampling Distribution of All SSummary Measures of this Sampling Distribution:Developing A Sampling Distribution(continued)DCOVANote:Here we divide by 16 because th

7、ere are 16different samples of size 2.Summary Measures of this SamplComparing the Population Distributionto the Sample Means Distribution18 19 20 21 22 23 240 .1 .2 .3 P(X) X 18 20 22 24 A B C D0 .1 .2 .3 PopulationN = 4P(X) X_Sample Means Distributionn = 2_DCOVAComparing the Population DistrSample

8、Mean Sampling Distribution:Standard Error of the MeanDifferent 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

9、or sampling is without replacement from an infinite population)Note that the standard error of the mean decreases as the sample size increasesDCOVASample Mean Sampling DistributSample Mean Sampling Distribution:If the Population is NormalIf a population is normal with mean and standard deviation , t

10、he sampling distribution of is also normally distributed with andDCOVASample Mean Sampling DistributZ-value for Sampling Distributionof the MeanZ-value for the sampling distribution of :where:= sample mean= population mean= population standard deviation n = sample sizeDCOVAZ-value for Sampling Distr

11、ibutNormal Population DistributionNormal Sampling Distribution (has the same mean)Sampling Distribution Properties (i.e. is unbiased )DCOVANormal Population DistributionSampling Distribution Properties As n increases, decreasesLarger sample sizeSmaller sample size(continued)DCOVASampling Distributio

12、n PropertiDetermining An Interval Including A Fixed Proportion of the Sample MeansFind 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 int

13、ervalSince the interval is symmetric 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.DCOVADetermining An Interval IncludDetermining An In

14、terval Including A Fixed Proportion of the Sample MeansCalculating 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)DCOVADetermining An Interval IncludSample Mean Sampling Distribution:If the P

15、opulation is not NormalWe 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: andDCOVASample Mean Sampling DistributnCentral Limit Th

16、eoremAs the sample size gets large enough the sampling distribution of the sample mean becomes almost normal regardless of shape of populationDCOVAnCentral Limit TheoremAs the Population DistributionSampling Distribution (becomes normal as n increases)Central TendencyVariationLarger sample sizeSmall

17、er sample sizeSample Mean Sampling Distribution:If the Population is not Normal(continued)Sampling distribution properties:DCOVAPopulation DistributionSamplinHow Large is Large Enough?For most distributions, n 30 will give a sampling distribution that is nearly normalFor fairly symmetric distributio

18、ns, n 15For a normal population distribution, the sampling distribution of the mean is always normally distributedDCOVAHow Large is Large Enough?For ExampleSuppose a population has mean = 8 and standard deviation = 3. Suppose a random sample of size n = 36 is selected. What is the probability that t

19、he sample mean is between 7.8 and 8.2?DCOVAExampleSuppose a population haExampleSolution: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)DCOVAExamp

20、leSolution:(continued)DCOExample Solution (continued):(continued)Z7.8 8.2-0.4 0.4Sampling DistributionStandard Normal DistributionPopulation Distribution?SampleStandardizeXDCOVAExample Solution (continued)Population Proportions = the proportion of the population having some characteristicSample prop

21、ortion (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)DCOVAPopulation Proportions = tSampling Distribution of pApproximated by ano

22、rmal distribution if: where and(where = population proportion)Sampling DistributionP( ps).3.2.1 0 0 . 2 .4 .6 8 1pDCOVASampling Distribution of pApprZ-Value for ProportionsStandardize p to a Z value with the formula:DCOVAZ-Value for ProportionsStandarExampleIf the true proportion of voters who support Proposition A is = 0.4, what is the probability that a sample of size 200 yields a s