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 b Without replacement Trials are not independent Find the probability that exactly 2 nut bars were selected 15 7 nut 2 7 C 2 8 C 4 21 70 0 2937 5 005 15 C 6 8 plain 4 OR 7 6 8 7 6 5 6 C 2 15 14 13 12 11 10 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 S N S x n x S C x N S Cn x P X x N N Cn n OR n S S 1 S x 1 N S N S 1 N S n x 1 P X x N x 1 N x N x 1 N n 1 x N N 1 max 0 n S N x min n S c Find the probability that at most 2 nut bars were selected C C C C C C P X 2 7 0 8 6 7 1 8 5 7 2 8 4 15 C 6 15 C 6 15 C 6 d 1 28 7 56 21 70 0 3776 5 005 5 005 5 005 Find the probability that at least 4 nut bars were selected C C C C C C P X 4 7 4 8 2 7 5 8 1 7 6 8 0 15 C 6 15 C 6 15 C 6 6 35 28 21 8 7 1 0 23077 5 005 5 005 5 005 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 a N 10 S 4 2 1 3 C 2 0 40 0 60 0 288 b 4 C 2 6 C 1 0 30 10 C 3 N 100 S 40 2 1 3 C 2 0 40 0 60 0 288 c without replacement 40 C 2 60 C 1 0 289425 100 C 3 N 1 000 S 400 2 1 3 C 2 0 40 0 60 0 288 400 C 2 600 C 1 0 288144 1000 C 3 Binomial Hypergeometric with replacement without replacement Probability n P X x p x 1 p n x x Expected Value E X n p Variance Var X n p 1 p S N S x n x P X x N n S E X n N S S N n Var X n 1 N N N 1 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 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 b A fair coin is tossed 3 times X of H s Yes c n 3 p 0 50 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 d n 7 p 1 6 The probability of success is not the same for all trials 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 f n 10 p 0 20 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 g n 7 p 10 40 0 25 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 h Trials are not independent 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 i Trials are not independent However Binomial distribution can be used as an approximation 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 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 l n 400 p 0 10 More than two possible outcomes for each trial 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 n x p 1x 1 p 2x 2 p k k x 1 x 2 x k 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 30 0 30 10 0 20 8 0 15 7 0 15 3 0 10 2 0 10 0 10 8 7 3 2 0 b What is the probability that 10 are brown …
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