Question

Homework: Section 4.3 Homework Save 13 of 15 (4 complete) | > w score: 26.67%, 4 of Score: 0 of 1 pt 4.3.25 Question Help An automobile manufacturer finds that 1 in every 2000 automobiles produced has a particular manufacturing defect (a) Use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 6500 cars. (b) The Poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. Repeat (a) using a Poisson distribution and compare the results. (a) The probability using the binomial distribution is (Round to five decimal places as needed.)

Answer 1

Given the probability that the automobiles produced has a particular manufacturing defect is p=1/2000=0.0005

Let X be the number of defects in a random sample of 6500 cars.

Therefore,

X ~ Bin (n 6500. P 0.0005)

a) The probability of finding 4 cars with the defect in a random sample of 6500 cars given by:

P(X 4) = (6500 (0.0005)4* (10.0005)500-4)

P(X = 4) = 4-64961 (0.0005)4 (0.9995)6496

6500 6499 k 6498 6497 24 (0.0005)4 (0.9995)6496 P(X = 4) =

By using calculator:

P(X = 4) = 0.18029

b) Since sample size is large i.e n=6500 and probability of success p=0.0005 (small)

The distribution of X follows approximately Poisson with mean=np

Therefore the probability of finding 4 cars with the defect in a random sample of 6500 cars given by:

e-nP (np) (np)' P(X = 4) = P(X = 4) =一一41

-6500-0.005 (6500 0.005) 24

P(X = 4) = 25(3.25)4 4!

0.03877 111.5664 24

4.325429 24 P(X = 4)-

P(X = 4) = 0.1802262

From part a) and b) the probability is approximately same.