Math A160 - CooleyIntroduction to StatisticsOCCSECTION 8.1 – Distribution of the Sample MeanSampling DistributionThe sampling distribution of a statistic is a probability distribution for all possible values of the statisticcomputed from a sample of sizen.Sampling Distribution of the Sample MeanThesampling distribution of the sample meanxis the probability distribution of all possible values of therandom variablexcomputed from a sample of sizenfrom a population with meanμand standard deviation.The Mean and Standard Deviation of the Sampling Distribution ofxSuppose that a simple random sample of sizenis drawn from a population with meanμand standard deviation.The sampling distribution ofxhas meanxand standard deviationnx. The standarddeviation of the sampling distribution ofx,x, is called thestandard error of the mean.The Shape of the Sampling Distribution ofxIfXIs NormalIf a random variableXis normally distributed, the distribution of the sample mean,x, is normally distributed.The Central Limit TheoremRegardless of the shape of the underlying population, the sampling distribution ofxbecomes approximatelynormal as the sample size,n, increases.Note:The more skewed the distribution of the population is, the larger the sample size needed to invoke the Central Limit Theorem. We will erron the side of caution and say that, if the distribution of the population is unknown or not normal, then the distribution of the samplemean is approximately normal provided that the sample size is greater than or equal to 30.