A simple random sample of 85 analog circuits is obtained at random from an ongoing production process in which 21% of all circuits produced are defective. Let X be a binomial random variable corresponding to the number of defective circuits in the sample. Use the normal approximation to the binomial distribution to compute P ( 14 ≤ X ≤ 20 ) , the probability that between 14 and 20 circuits in the sample are defective. Report your answer to two decimal places of precision.

63.81% probability that between 14 and 20 circuits in the sample are defective.

Step-by-step 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:

The standard deviation of the binomial distribution is:

Normal probability distribution

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

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

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 , .

In this problem, we have that:

.

So

P ( 14 ≤ X ≤ 20 )

Using continuity correction, this is , which is the pvalue of Z when X = 20.5 subtracted by the pvalue of Z when X = 13.5. So

X  = 20.5

has a pvalue of 0.7611

X = 13.5

has a pvalue of 0.1230

0.7611 - 0.1230 = 0.6381

63.81% probability that between 14 and 20 circuits in the sample are defective.

32

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