Chapter 5 A.ppt
- 1. Binomial Distribution The binomial distribution is a discrete distribution.
- 2. Binomial Experiment A binomial experiment has the following properties: experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure P(success) = p P(failure) = q = 1 - p for all trials The random variable of interest, X, is the number of successes in the n trials. X has a binomial distribution with parameters n and p
- 3. EXAMPLES A coin is flipped 10 times. Success = head. X = n = p = Twelve pregnant women selected at random, take a home pregnancy test. Success = test says pregnant. X = n = p = ? Random guessing on a multiple choice exam. 25 questions. 4 answers per question. Success = right answer. X = n = p =
- 4. Examples when assumptions do not hold Basketball player shoots ten free throws Feedback affects independence and constant p Barrel contains 3 red apples and 4 green apples; select 4 apples without replacement; X = # of red apples. Without replacement implies dependence
- 5. What is P(x) for binomial? x n x q p x n x n x P )! ( ! ! ) (
- 6. Mean and Standard Deviation The mean (expected value) of a binomial random variable is The standard deviation of a binomial random variable is np npq
- 7. Example Random Guessing; n = 100 questions. Probability of correct guess; p = 1/4 Probability of wrong guess; q = 3/4 Expected Value = On average, you will get 25 right. Standard Deviation = 1 np 100 25 4 1 3 npq np 1 p 100 4.33 4 4
- 8. Example Cancer Treatment; n = 20 patients Probability of successful treatments; p = 0.7 Probability of no success; q = ? Calculate the mean and standard deviation.
- 9. Normal Approximation For large n, the binomial distribution can be approximated by the normal, is approximately standard normal for large n. npq np X Z