Stat 217 – Day 18Last Time – Sampling Distn for MeanLast Time – Distribution of x-barActivity 15-5 (p. 300)Slide 5Activity 15-5One small problemInformallyDemoDemo, cont.t distribution (p. 376)Slide 12Activity 20-1 (p. 394)Two Central Limit Theorems (p. 295)Slide 15Stat 217 – Day 18t-procedures (Topics 19 and 20)Last Time – Sampling Distn for MeanPenny agesPopulationSample (n = 30) Sampling distributionChangePopulationSample (n = 30)Sampling distributionObs unit = sampleVariable = sample meanLast Time – Distribution of x-barCentral Limit Theorem for Sample Mean (p. 282)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)Activity 15-5 (p. 300)Ethan Allen October 5, 2005Are several explanations, could excess passenger weight be one?Activity 15-5 (p. 300)The 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?Activity 15-5What’s the probability the average weight of 47 passengers will exceed 159.57 lbs? n > 30 so we can apply the CLT1. Shape is approximately normal2. mean will equal 167 lbs3. standard deviation = 35/47 = 5.105 lbsZ = (159.57-167)/5.105 = -1.46Above: .927293% chance of an overweight boat!One small problemWe don’t usually know the population standard deviationInformallyA conjecture for the value of is not plausible if it falls more than 2 SD = 2 / n from the observed sample mean ( )Standardize:Small problem: don’t know either! Easy solution? nxdevstdmeannobservatio/nsxdevstdmeannobservatio/“standard error”xDemoSuppose we have a population with mean = 10 and standard deviation = 5. What does the sampling distribution of samples of size n=5 look like?Demo, cont.What really matters is the distribution of the standardized valuesBut what happens if we use s instead of ?nxdevstdmeannobservatio/nsxdevstdmeannobservatio/t distribution (p. 376)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 distribution (p. 376)A 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 sActivity 20-1 (p. 394)Two Central Limit Theorems (p. 295)Categorical (p-hat)Mean = SD = (1- )/nShape = approx normal if n > 10 and n(1- ) > 10Random sampleQuantitative (x-bar)Mean = SD = /nShape = normal if population normal or approximately normal if n > 30Random sampleTo turn in, with partnerActivity 20-1 (m) (n) (o)Handout (f) and (g)For WednesdayHW 5Activities 19-6, 20-3, 20-4Be working on Lab
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