What is the input value for the following function if the output value is 5?

A simple random sample of 90 analog circuits is obtained at random from an ongoing production process in which 25% of all circui

kozerog [31]

Answer:

76.10% probability that between 19 and 28 circuits in the sample are defective

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

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:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

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:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

In this problem, we have that:

$n = 90, p = 0.25$

So

$\mu = E(X) = np = 90*0.25 = 22.5$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{90*0.25*0.75} = 4.11$

P(19 ≤ X ≤ 28)

Using continuity correction, this is $P(19-0.5 \leq X \leq 28+0.5) = P(18.5 \leq X \leq 28.5)$ , which is the pvalue of Z when X = 28.5 subtracted by the pvalue of Z when X = 18.5. So