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Cal Poly STAT 217 - t-procedures (Topics 19 and 20)

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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 MeanPenny agesPopulationSample (n = 30) Sampling distributionChangePopulationSample (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 15-5What’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 problemWe don’t usually know the population standard deviationInformallyA 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”xDemoSuppose 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 valuesBut 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” tailsModels 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 – 1As df increases, the heaviness of the tails decreases and the t distribution looks more and more like the normal distributionLess penalty for estimating  with sActivity 20-1 (p. 394)Two Central Limit Theorems (p. 295)Categorical (p-hat)Mean = SD = (1- )/nShape = approx normal if n > 10 and n(1- ) > 10Random sampleQuantitative (x-bar)Mean = SD = /nShape = normal if population normal or approximately normal if n > 30Random sampleTo turn in, with partnerActivity 20-1 (m) (n) (o)Handout (f) and (g)For WednesdayHW 5Activities 19-6, 20-3, 20-4Be working on Lab


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