Introduction to Sampling DistributionsChapter GoalsSampling ErrorSlide 4ExampleSlide 6Slide 7Sampling Distribution of MeanSlide 9ReviewSampling ErrorsSampling DistributionProperties of a Sampling DistributionTheorem 2If the Population is Normalz-value for Sampling Distribution of xSampling Distribution PropertiesConsistent EstimatorCentral Limit TheoremSlide 20Slide 21Example 2Population ProportionsPopulation Proportions, πSampling Distribution of pz-Value for ProportionsSlide 27Slide 28Slide 29ExampleMean and Standard ErrorSlide 32Slide 33Estimating Single Population ParametersWhy?Confidence IntervalsPoint and Interval EstimatesPoint EstimatesSlide 39Confidence Interval EstimateGeneral FormulaConfidence LevelConfidence Level, (1-)Slide 44Confidence Interval for μ (σ Known)Finding the Critical ValueCommon Levels of ConfidenceMargin of ErrorSlide 49Slide 50InterpretationConfidence Interval for μ (σ Unknown)Slide 53Student’s t DistributionDegrees of Freedom (df)Student’s t Tablet distribution valuesSlide 58If σ is unknownSlide 60Confidence Intervals for the Population Proportion, πSlide 62Confidence interval endpointsSlide 64Slide 65Slide 66Changing the sample sizeQuestions?Introduction to Sampling DistributionsChapter GoalsAfter completing this chapter, you should be able to: Define the concept of sampling errorDetermine the mean and standard deviation for the sampling distribution of the sample mean, xDetermine the mean and standard deviation for the sampling distribution of the sample proportion, pApply sampling distributions for both x and p____Sampling Error Sample Statistics are used to estimate Population Parametersex: X is an estimate of the population mean, μ Problems: Different samples provide different estimates of the population parameterSample results have potential variability, thus sampling error exitsSampling ErrorVenice Beach vs. KCDecisions are based on samplesExercise Design – to get optimal answer, not “what do you want it to be?”Goal – sample that accurately represents the populationError = difference between sample and corresponding population parameterError = - μChap 7-4x© Walden, 2010ExampleProject Square Feet1 114,5602 202,3003 78,6004 156,7005 134,6006 88,2007 117,3008 155,3009 214,20010 303,80011 125,20012 156,900Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-5Exampleμ = (Σx)/Nμ = 158,972 s.f. = parameter So what?Sample: © Walden, 2010Chap 7-6project s.f.5 134,6004 156,7001 114,5608 155,3009 214,200total 775,360mean 155,072Exampleμ = Σx/Nμ = 158,972 s.f. = parameter So what?Sample: © Walden, 2010Chap 7-7project s.f.3 78,6006 88,2001 114,56011 125,20012 156,900total 563,460mean 112,692Sampling Distribution of MeanMost likely: Random sample ≠ populationSize of sampling error depends on sample selectedSampling error can be + or –Different mean for every samplePossible number of combinations = n!/((x!)(n-x)!)Chap 7-8© Walden, 2010Sampling ErrorShopping center example – # of possible samples = (12x11x10x9x8x7x6x5x4x3x2x1) ((5X4X3X2X1)(7X6X5X4X3X2X1)= 792 possible samples of 5; 3960 possible samples of 4Therefore, sample size affects # of combinations and influences sample errorBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-9Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-10ReviewPopulation mean: Sample Mean:Nxμiwhere:μ = Population meanx = sample meanxi = Values in the population or sampleN = Population sizen = sample sizenxxiBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-11Sampling ErrorsDifferent samples will yield different sampling errorsThe sampling error may be positive or negative ( may be greater than or less than μ)The expected sampling error decreases as the sample size increasesx© Walden 2010Chap 7-12Sampling DistributionA sampling distribution is a distribution of all possible values of a statistic for a given sample size – given that the sample is randomly selectedBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-13For any population, the average value of all possible sample means computed from all possible random samples of a given size from the population is equal to the population mean:The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size:Properties of a Sampling DistributionμμxnσσxTheorem 1Theorem 2Theorem 2If you take every sample possible, you now have the total population so,μx = μσx = σBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-14Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-15If the Population is NormalIf a population is normal with mean μ and standard deviation σ, the sampling distributionof is also normally distributed with andxμμxnσσxTheorem 3Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-16z-value for Sampling Distributionof xZ-value for the sampling distribution of :where: = sample mean= population mean= population standard deviation n = sample sizexμσnσμ)x(zxBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-17Normal Population DistributionNormal Sampling Distribution (has the same mean)Sampling Distribution PropertiesThe sample mean is an unbiased estimatorxxμμxμxμConsistent EstimatorUnbiased estimator if:The difference between the estimator and the parameter becomes smaller as sample becomes bigger© Walden, 2010Chap 7-18Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-19n↑Central Limit TheoremAs the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of populationxBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-20ExampleSuppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?Exampleμ = 8σ = 3Z= (7.8-8)/(3/√36)