Question

A shipment of 50,000 transistors arrives at a manufacturing plant. The quality control engineer at the plant obtains a random sample of 500 resistors and will reject the entire shipment if 10 or more of the resistors are defective. Suppose that 4% of the resistors in the whole shipment are defective. What is the probability the engineer accepts the shipment

Answer 1

0.82% probability the engineer accepts the shipment

Explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

`n = 500, p = 0.04`

So

`E(X) = np = 500*0.04 = 20`

`√(V(X)) = √(np(1-p)) = √(500*0.04*0.96) = 4.3818`

What is the probability the engineer accepts the shipment?

Less than 10 defective. Using continuity correction, this is
`P(X < 10 - 0.5) = P(X < 9.5)`, which is the pvalue of Z when X = 9.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (9.5 - 20)/(4.3818)`

`Z = -2.4`

`Z = -2.4` has a pvalue of 0.0082

0.82% probability the engineer accepts the shipment