Suppose that an automobile parts wholesaler claims that .5 percent of the car batteries in a shipment are defective. A random sample of 200 batteries is taken, and four are found to be defective.

(a) Use the Poisson approximation to find the probability that four or more car batteries in a random sample of 200 such batteries would be found to be defective, if we assume that the wholesaler's claim is true. (Round your answer to 4 decimal places.)

Any help would be greatly apriciated. Thanks

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To solve this problem, we will use the Poisson approximation. The Poisson distribution can be used to model rare events when the number of trials is large and the probability of success is small.

First, let's calculate the average number of defective batteries in a sample of 200, assuming that the wholesaler's claim is true. We can do this by multiplying the proportion of defective batteries claimed by the sample size:

Average number of defects = 0.005 * 200 = 1

Now, we will use the Poisson distribution to find the probability of observing four or more defective batteries in a sample of 200, given an average of one defective battery.

The probability mass function (PMF) of the Poisson distribution is given by the formula:

P(X = k) = (e^(-λ) * λ^k) / k!

where X is the random variable representing the number of defects, λ is the average number of defects, e is the base of the natural logarithm, and k is the number of defects.

To find the probability of four or more defects, we need to sum the probabilities of having four, five, six, and so on, up to infinity.

P(X ≥ 4) = 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3)

To calculate each of these probabilities, we will substitute the values of λ and k into the formula above and sum them.

P(X = 0) = (e^(-1) * 1^0) / 0! ≈ 0.3679
P(X = 1) = (e^(-1) * 1^1) / 1! ≈ 0.3679
P(X = 2) = (e^(-1) * 1^2) / 2! ≈ 0.1839
P(X = 3) = (e^(-1) * 1^3) / 3! ≈ 0.0613

Now we can compute the probability of four or more defects:

P(X ≥ 4) = 1 - 0.3679 - 0.3679 - 0.1839 - 0.0613

P(X ≥ 4) ≈ 0.0190

So, the probability that four or more car batteries in a random sample of 200 would be found to be defective, if we assume that the wholesaler's claim is true, is approximately 0.0190 (or 0.019 when rounded to 4 decimal places).

I hope this helps! Let me know if you have any further questions.