Business Statistics: A Decision-Making Approach 6th EditionChapter 6 Introduction to Sampling DistributionsChap 6-1Chapter GoalsAfter completing this chapter, you should be able to: nDefine the concept of sampling errornDetermine the mean and standard deviation for the sampling distribution of the sample mean, xnDetermine the mean and standard deviation for the sampling distribution of the sample proportion, pnDescribe the Central Limit Theorem and its importancenApply sampling distributions for both x and p_2Sampling Errorn Sample Statistics are used to estimate Population Parametersex: X is an estimate of the population mean, n Problems: nDifferent samples provide different estimates of the population parameternSample results have potential variability, thus sampling error exits3Calculating Sampling ErrornSampling Error:The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a populationExample: (for the mean)where:4ReviewnPopulation mean:Sample Mean:where: = Population mean x = sample mean xi = Values in the population or sample N = Population size n = sample size5ExampleIf the population mean is = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is6Sampling ErrorsnDifferent samples will yield different sampling errorsnThe sampling error may be positive or negative( may be greater than or less than )nThe expected sampling error decreases as the sample size increases7Sampling DistributionnA sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population8Developing a Sampling DistributionnAssume there is a population nPopulation size N=4nRandom variable, x, is age of individualsnValues of x: 18, 20, 22, 24 (years)ABCD9.3.2 .1018 20 22 24A B C D Uniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing a Sampling Distribution1016 possible samples (sampling with replacement)Now consider all possible samples of size n=2(continued)Developing a Sampling Distribution16 Sample Means11Sampling Distribution of All Sample Means18 19 20 21 22 23 240 .1 .2 .3 P(x) xSample Means Distribution16 Sample Means_Developing a Sampling Distribution (continued)(no longer uniform)12Summary Measures of this Sampling Distribution:Developing a Sampling Distribution (continued)13Comparing the Population with its Sampling Distribution18 19 20 21 22 23 240 .1 .2 .3 P(x) x 18 20 22 24A B C D0 .1 .2 .3 Population N = 4P(x) x_Sample Means Distribution n = 214If the Population is Normal(THEOREM 6-1)If a population is normal with mean and standard deviation , the sampling distributionof is also normally distributed withand15z-value for Sampling Distribution of xnZ-value for the sampling distribution of :where:= sample mean = population mean = population standard deviationn = sample size16Finite Population CorrectionnApply the Finite Population Correction if:nthe sample is large relative to the population(n is greater than 5% of N) andnSampling is without replacementThen17Normal Population DistributionNormal Sampling Distribution (has the same mean)Sampling Distribution Propertiesn (i.e. is unbiased )18Sampling Distribution PropertiesnFor sampling with replacement:As n increases, decreasesLarger sample sizeSmaller sample size(continued)19If the Population is not NormalnWe can apply the Central Limit Theorem:nEven if the population is not normal,nsample means from the population will be approximately normal as long as the sample size is large enoughnand the sampling distribution will haveand20nCentral Limit TheoremAs the sample size gets large enough the sampling distribution becomes almost normal regardless of shape of population21Population DistributionSampling Distribution (becomes normal as n increases)Central TendencyVariation(Sampling with replacement)Larger sample sizeSmaller sample sizeIf the Population is not Normal(continued)Sampling distribution properties:22How Large is Large Enough?nFor most distributions, n 30 will give a sampling distribution that is nearly normalnFor fairly symmetric distributions, n 15nFor normal population distributions, the sampling distribution of the mean is always normally distributed23ExamplenSuppose a population has mean = 8 and standard deviation = 3. Suppose a random sample of size n = 36 is selected. nWhat is the probability that the sample mean is between 7.8 and 8.2?24ExampleSolution:nEven if the population is not normally distributed, the central limit theorem can be used (n 30)n so the sampling distribution of is approximately normaln with mean = 8 nand standard deviation (continued)25ExampleSolution