Expectation and Hypothesis TestingDate: 08/24/99 at 11:27:24 From: Dave Wright Subject: Statistics: Multinomial, Neg Binom, Hyp Test I have a few questions I'm stuck on: 1. A production line produces good articles with probability .7, average ones with probability of .2, and defective ones with probability .1. Ten articles are selected. a) What is the probability of 8 good ones and 1 defective? b) What is the probability that there is an equal number of good and defective articles? Answers in back of text: a) 0.1036 b) 0.006165 I attempted to solve this using Multinomial Distribution. 2. A person decides to throw a pair of dice until he gets 2 sixes. What is the expected number of throws until he stops? Answer in back of text: 36 I attempted to solve this using Negative Binomial Distribution. 3. A transistor manufacturer claims its product has 10% defectives. A sample of 15 transistors is examined and 3 are found to be defective. Would you reject their claim with alpha <= 0.05? Answer in back of text: No 4. An experiment is set to test the hypothesis that a given coin is unbiased. The decision rule is the following: Accept the hypothesis if the number of heads in a sample of 200 tosses is between 90 and 110 inclusive, otherwise reject the hypothesis. a) Find the probability of accepting the hypothesis when it is correct. b) Find the probability of rejecting the hypothesis when it is actually correct. Answer in back of text: a) 0.8612 b) 0.1388 Thanks in advance. Dave Wright Date: 08/24/99 at 18:35:39 From: Doctor Anthony Subject: Re: Statistics: Multinomial, Neg Binom, Hyp Test 1. a) What is the probability of 8 good ones and 1 defective? 10! P(8 good, 1 av, 1 defective) = -------- x 0.7^8 x 0.2 x 0.1 8! 1! 1! = 90 x 0.7^8 x 0.2 x 0.1 = 0.103766 b) What is the probability that there is an equal number of good and defective articles? Work out the following probabilities (in order good, average, defective): 0, 10, 0 1, 8, 1 2, 6, 2 3, 4, 3 4, 2, 4 5, 0, 5 2. A person decides to throw a pair of dice until he gets 2 sixes. What is the expected number of throws until he stops? You can use a difference equation. Let E = expected number of throws to a double 6 You MUST throw at least once and there is 35/36 probability of returning to the start point: E = 1 + (35/36)E E(1 - 35/36) = 1 E(1/36) = 1 E = 36 So the expected number of throws is 36. 3. A transistor manufacturer claims its product has 10% defectives. A sample of 15 transistors is examined and 3 are found to be defective. Would you reject their claim with alpha <= 0.05? Proportion defective = 1/5 = 0.2 while claim is 0.1 p = 0.1 pq/n = 0.1 x 0.9/15 = 0.006 sqrt(pq/n) = 0.07746 We test 0.2 - 0.1 z = --------- = 1.291 0.07746 and we compare this with 1.645 (single tailed test), and see that it is not significant. So do not reject null hypothesis. 4. a) Find the probability of accepting the hypothesis when it is correct. p=q = 1/2 npq = (1/2)(1/2)(200) = 50 sqrt(npq) = 7.071 110 - 100 10 z = --------- = ----- = 1.4142 7.071 7.071 A(z) = 0.9214 so area from mean to this value is 0.4214. By symmetry area other side of the mean is also 0.4214. Total area corresponding to range 90 - 110 is 0.8428, so we can accept null hypothesis with probability 0.8428. b) Find the probability of rejecting the hypothesis when it is actually correct. This is simply 1 - the previous answer. 1 - 0.8428 = 0.1572 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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