Question

A manufacturing process is designed to produce bolts with a 0.25-in. diameter. Once each day, a random sample of 36 bolts is selected and the diameters recorded. If the resulting sample mean is less than 0.230 in. or greater than 0.270 in., the process is shut down for adjustment. The standard deviation for diameter is 0.04 in. What is the probability that the manufacturing line will be shut down unnecessarily

Answer 1

0.26% probability that the manufacturing line will be shut down unnecessarily

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are 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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 0.25, \sigma = 0.04, n = 36, s = (0.04)/(√(36)) = 0.0067`

What is the probability that the manufacturing line will be shut down unnecessarily

Sample mean more than 2 inches from 0.25-in, that is, lesser than 0.23 or greater than 0.27. Since 0.23 and 0.27 have the same distance from the mean of 0.25, and the normal distribution is symmetric, these probabilities are equal, so we find one of them and multiply by 2.

Less than 0.23

pvalue of Z when X = 0.23. So

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

By the Central Limit Theorem

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

`Z = (0.23 - 0.25)/(0.0067)`

`Z = -3`

`Z = -3` has a pvalue of 0.0013

2*0.0013 = 0.0026

0.26% probability that the manufacturing line will be shut down unnecessarily