Hypergeometric Distribution

1. Hypergeometric Distribution • Example*: Automobiles arrive in a dealership in lots of 10. Five out of each 10 are inspected. For one lot, it is know that 2 out of 10 do not meet prescribed safety standards. What is probability that at least 1 out of the 5 tested from that lot will be found not meeting safety standards? *from Complete Business Statistics, 4th ed (McGraw-Hill)

2. This example follows a hypergeometric distribution: • A random sample of size n is selected without replacement from N items. • k of the N items may be classified as “successes” and N-k are “failures.” • The probability associated with getting x successes in the sample (given k successes in the lot.) Where, k = number of “successes” = 2 n = number in sample = 5 N = the lot size = 10 x = number found = 1 or 2

3. Hypergeometric Distribution • In our example, = _____________________________

4. Expectations of the Hypergeometric Distribution • The mean and variance of the hypergeometric distribution are given by • What are the expected number of cars that fail inspection in our example? What is the standard deviation? μ =___________ σ2 =__________ , σ =__________

5. Your turn … A worn machine tool produced defective parts for a period of time before the problem was discovered. Normal sampling of each lot of 20 parts involves testing 6 parts and rejecting the lot if 2 or more are defective. If a lot from the worn tool contains 3 defective parts: • What is the expected number of defective parts in a sample of six from the lot? • What is the expected variance? • What is the probability that the lot will be rejected?

6. Binomial Approximation • Note, if N >> n, then we can approximate this with the binomial distribution. For example: Automobiles arrive in a dealership in lots of 100. 5 out of each 100 are inspected. 2 /10 (p=0.2) are indeed below safety standards. What is probability that at least 1 out of 5 will be found not meeting safety standards? • Recall: P(X≥ 1) = 1 – P(X< 1) = 1 – P(X = 0)