Stat 301 – Day 20 Sampling Distributions for a sample proportion (cont.) (4.3)

Announcements Lab 3 due Tuesday, lab 2 returned Thank you for mid-quarter feedback  Practice problems Not on Fridays Not an expectation for every PP (see recent scoring) Time length Regularity More worked out examples, wording…  What’s missing in text/index… examples at end of chapters  Using R

Top 3 lessons from Friday? Sampling variability  Value of sample statistic varies from sample to sample, just due to "random sampling error” Sampling distribution has a predictable pattern (with random samples)  Statistics are random variables, have a probability distribution (unknown parameters), expected value  Unbiased => centered at population parameter  Variability decreases as increase sample size

Simulation With  =.45, n = 25, and 500 samples With  =.45, n = 75, and 500 samples

Top 3 Lessons from Friday? Often the count follows a binomial distribution (sampling from process) or is well approximated by a binomial distribution (sampling from large population) Often the count and proportion are well modeled by a normal distribution…

Sampling Distribution of When  =.45: Center: mean.45 Spread: std dev.10 68% of samples have within +.10 of.45 95% of samples have within +.20 of.45 Would be pretty surprised to get less than 25% or more than 65% orange candies in a random sample of 25 candies from this population/process.

Theoretical results (4.3.2)

Why does this work? X = number of successes in n trials  Probability of success, , is constant  Trials are independent X follows a Binomial distribution (Ch. 3)  P(X=x) = C(n,x)  x (1-  n-x  E(X) =  x C(n,x)  x (1-  n-x = n   V(X) =  (x-n  ) 2 C(n,x)  x (1-  n-x = n  (1-  )

Why does this work? = X/n is also a random variable E( ) = E(X/n) = 1/n E(X) = n  /n =  V( ) = V(X/n) = 1/n 2 V(X) = n  (1-  )/n 2 =  (1-  )/n

The Central Limit Theorem (CLT) p. 309: The sampling distribution of sample proportions will approximately normal with mean equal to  and standard deviation equal to as long as  Random sample from large population or process (with constant probability of success)  Sample size is “large” n  > 10 and n(1-  ) > 10

“Technical conditions” n = 25,  =.45 n = 5,  =.75

Three approaches… (p. 314) To calculate probabilities for statistics arising from a binomial process (n,  ): Exact binomial calculations Simulation Normal probability approximation to binomial  Mean =  SD =  /n  Valid for “large” n

PP (p. 320) Swain vs. Alabama

For Tuesday Lab 3 PP in BB Inv (a)-(c)