Question

Last year, a manufacturer produced 1,850,000 DVD players. Of these, approximately 3% were defective. Assume that a simple random sample of n = 170 players is drawn. Use the Poisson approximation to the binomial distribution to compute the probability that fewer than four of the 170 DVD players were defective.

Answer 1

To find the probability of fewer than four defective DVD players in a sample of 170 with a defect rate of 3%, calculate the mean (μ=5.1) and use the Poisson CDF with that mean for values less than four.

Step-by-step explanation:

The question involves calculating the probability that fewer than four DVD players are defective in a random sample of 170 players, given a 3% defect rate from a manufacturer. To solve this, we use a Poisson approximation to the binomial distribution because the sample size is large (n=170) and the probability of finding a defective player is small (p=0.03).

First, we calculate the mean number of defective players (μ) by multiplying the sample size (n) by the defect rate (p):

µ = np = 170 * 0.03 = 5.1

Next, we use the Poisson distribution with a mean of 5.1 to find the probability of having fewer than four defective units. This can typically be done using a statistical calculator or software:

P(X < 4) ≈ Poisson CDF(5.1, 3)

It's important to note that a Poisson CDF function would be used here, but the actual numerical probability calculation isn't provided as it requires the use of a calculator or software.