STAT 400 Lecture AL1 Examples for 2.4 (Extended) Spring 2015 Dalpiaz 5. Suppose that on Halloween 6 children come to a house to get treats. A bag contains 8 plain chocolate bars and 7 nut bars. Each child reaches into the bag and randomly selects 1 candy bar. Let X denote the number of nut bars selected. a) Is the Binomial model appropriate for this problem? No. Without replacement ⇒ Trials are not independent. b) Find the probability that exactly 2 nut bars were selected. 15 005570 2161548 27 ,CCC⋅⋅= ≈ 0.2937. 7 8 nut plain 2 4 OR ⋅⋅⋅⋅⋅⋅10511612713814615726 C ≈ 0.2937. Hypergeometric Distribution: N = population size, S = number of “successes” in the population, N – S = number of “failures” in the population, n = sample size. X = number of "successes" in the sample when sampling is done without replacement. Then nxnxNSNSXCCCNSNSnxnxx−−⋅=−−⋅== )(P OR ( )( )+−+−−−−−−−−−+−+−−−==⋅⋅⋅⋅⋅⋅⋅⋅11111111PnxnxxxxxnxNSN...NSNNSNNS...NSNSX max(0, n + S – N ) ≤ x ≤ min(n, S).c) Find the probability that at most 2 nut bars were selected. P( X ≤ 2 ) = 61548 2761558 1761568 07 CCCCCCCCC⋅⋅⋅++ = 005570 21005556 7005528 1,,,⋅⋅⋅++ ≈ 0.3776. d) Find the probability that at least 4 nut bars were selected. P( X ≥ 4 ) = 61508 6761518 5761528 47 CCCCCCCCC ⋅⋅⋅++ = 005,51 7005,58 21005,528 35 ⋅⋅⋅++ ≈ 0.23077. 6. A jar has N marbles, S of them are orange and N – S are blue. Suppose 3 marbles are selected. Find the probability that there are 2 orange marbles in the sample, if the selection is done … with replacement without replacement a) N = 10, S = 4; ( ) ( )12 60.040.023 ⋅⋅C = 0.288. 31016 24 CCC ⋅ = 0.30. b) N = 100, S = 40; ( ) ( )12 60.040.023 ⋅⋅C = 0.288. 3100160 240 CCC ⋅ ≈ 0.289425. c) N = 1,000, S = 400; ( ) ( )12 60.040.023 ⋅⋅C = 0.288. 310001600 2400 CCC ⋅ ≈ 0.288144.Binomial Hypergeometric with replacement without replacement Probability ( )xnx)Xppxnx−−⋅⋅== 1P( −−⋅==nxnxxNSNSX )(P Expected Value E(X) = n ⋅ p E(X) = NSn⋅ Variance Var(X) = n ⋅ p ⋅ (1 – p) Var(X) = 11−−⋅−⋅⋅NNNSNSnn If the population size is large (compared to the sample size) Binomial Distribution can be used regardless of whether sampling is with or without replacement. 6 ½ . In each of the following cases, is it appropriate to use Binomial model? If yes, what are the values of its parameters n and p (if known)? If no, explain why Binomial model is not appropriate. a) A fair 6-sided die is rolled 7 times. X = # of 6’s. Yes. n = 7, p = 1/6 . b) A fair coin is tossed 3 times. X = # of H’s. Yes. n = 3, p = 0.50. c) An exam consists of 10 questions, the first 4 are True-False, the last 6 are multiple choice questions with 4 possible answers each, only one of which is correct. A student guesses independently on each question. X = # of questions he answers correctly. No. The probability of success is not the same for all trials. d) Suppose 20% of the customers at a particular gas station select Premium gas X = # of customers at this gas station on a particular day who selected Premium gas. No. The number of trials is not fixed.e) Suppose 20% of the customers at a particular gas station select Premium gas X = # of customers in the first 10 at a gas station on a particular day who selected Premium gas. Yes. n = 10, p = 0.20. f) A box contains 40 parts, 10 of which are defective. A person takes 7 parts out of the box with replacement. X = # of defective parts selected. Yes. n = 7, p = 10/40 = 0.25. g) A box contains 40 parts, 10 of which are defective. A person takes 7 parts out of the box without replacement. X = # of defective parts selected. No. Trials are not independent. h) A box contains 400,000 parts, 100,000 of which are defective. A person takes 7 parts out of the box without replacement. X = # of defective parts selected. No. Trials are not independent. However, Binomial distribution can be used as an approximation. i) Seven members of the same family are tested for a particular food allergy. X = # of family members who are allergic to this particular food. Yes if we can assume independence, No if we cannot. j) In Neverland, 10% of the labor force is unemployed. A random sample of 400 individuals is selected. X = # of individuals in the sample who are unemployed. Yes. n = 400, p = 0.10. k) Suppose that 5% of tax returns have arithmetic errors. 25 tax returns are selected at random. X = # of arithmetic errors in those 25 tax returns. No. More than two possible outcomes for each trial. l) Suppose that 5% of tax returns have arithmetic errors. 25 tax returns are selected at random. X = # of tax returns among those 25 with arithmetic errors. Yes. n = 25, p = 0.05.Multinomial Distribution: • The number of trials, n, is fixed. • Each trial has k possible outcomes, with probabilities p 1 , p 2 , … , p k , respectively. ( p 1 + p 2 + … + p k = 1 ) • The trials are independent. • X 1 , X 2 , … , X k represent the number of times outcome 1, outcome 2, … , outcome k occur, respectively. ( X 1 + X 2 + … + X k = n ) Then P ( X 1 = x 1 , X 2 = x 2 , … , X k = x k ) = kxkxxkpppxxxn ... ! ... ! !!2121 2 1 , x 1 + x 2 + … + x k = n. 7. A particular brand of candy-coated chocolate comes in six different colors. Suppose 30% of all pieces are brown, 20% are blue, 15% are red, 15% are yellow, 10% are green, and 10% are orange. Thirty pieces are selected at random. a) What is the probability that 10 are brown, 8 are blue, 7 are red, 3 are yellow, 2 are green, and none are orange? ( ) ( ) ( ) ( ) ( ) ( )0237810
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