Stat 217 – Day 18 t-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 distribution Obs unit = sample Variable = sample mean

Last Time – Distribution of x-bar Central Limit Theorem for Sample Mean (p. 282) 1. Sampling distribution is (approximately) normal 2. Sampling distribution mean equals population mean 3. Sampling distribution standard deviation equals  /  n Technical conditions 1. Random sample 2. Either large sample (n>30) or normal population (be told or look at sample)

Activity 15-5 (p. 300) Ethan Allen October 5, 2005 Are 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 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 lbs?

Activity 15-5 What’s the probability the average weight of 47 passengers will exceed lbs? n > 30 so we can apply the CLT 1. Shape is approximately normal 2. mean will equal 167 lbs 3. standard deviation = 35/  47 = lbs Z = ( )/5.105 = Above: % chance of an overweight boat!

One small problem We don’t usually know the population standard deviation

Informally 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? “standard error”

Demo 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  ?

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 s t distribution

t 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 s

Activity 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 6