Stat 217 – Day 20Recap: Quantitative variableRecall: Body TemperatureDistribution of Sample MeansLast Time – Distribution of x-barBut what if I don’t know s?t distributiont distributionWhat about one individual?ExampleSlide 11To DoStat 217 – Day 20t proceduresRecap: Quantitative variableDescribe shape (e.g., symmetric or skewed to the right or skewed to the left), center, spreadGraphical summary = dotplots, histogramsNumerical summary = mean “xbar” (center), standard deviation s (spread) For reasonably symmetric, mound-shaped distributionsParameter of interest: = population mean = population standard deviationRecall: Body TemperatureSuppose the population mean is 98.6.Would 98.248 be surprising for one person?Would 98.248 be surprising for average temperature 130 people?Distribution of Sample MeansPenny agesPopulationSample (n = 30) Sampling distributionChangePopulationSample (n = 30)Sampling distributionObs unit = sampleVariable = sample meanLast Time – Distribution of x-barCentral Limit Theorem for Sample Mean 1. Sampling distribution is (approximately) normal2. Sampling distribution mean equals population mean3. Sampling distribution standard deviation equals /nTechnical conditions1. Random sample2. Either large sample (n>30) or normal population (be told or look at sample)But what if I don’t know ?What really matters is the distribution of the standardized valuesBut what happens if we use s instead of ?nxdevstdmeannobservatio/nsxdevstdmeannobservatio/t distribution The “t distribution” is symmetric and mound-shaped like the normal distribution but has “heavier” tailsModels the extra variation we have with the additional estimation of by st distributiont distributionA family of distributions, characterized by “degrees of freedom” (df)df = n – 1As df increases, the heaviness of the tails decreases and the t distribution looks more and more like the normal distributionLess penalty for estimating with sWhat about one individual?ExampleEthan Allen October 5, 2005Are several explanations, could excess passenger weight be one?ExampleThe boat can hold a total of 7500 lbs (or an average of 159.57 lbs over 47 passengers)CDC: weights of adult Americans have a mean of 167 lbs and SD 35 lbs. What’s the probability the average weight of 47 passengers will exceed 159.57 lbs?To DoFinish
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