Question

Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 136 millimeters, and a variance of 36. If a random sample of 50 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 1.9 millimeters

Answer 1

0.025 = 2.5% probability that the sample mean would differ from the population mean by greater than 1.9 millimeters

Explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, 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.

Mean diameter of 136 millimeters, and a variance of 36.

This means that
`\mu = 136, \sigma = √(36) = 6`

Sample of 50:

This means that
`n = 50, s = (6)/(√(50)) = 0.8485`

What is the probability that the sample mean would differ from the population mean by greater than 1.9 millimeters?

This is the probability that it is less than 136 - 1.9 = 134.1 millimeters or more than 136 + 1.9 = 137.9 millimeters. Since the normal distribution is symmetric, there probabilities are equal, which means that we can find one of them and multiply by two.

Probability that it is less than 134.1 millimeters:

This is the pvalue of Z when X = 134.1. So

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

By the Central Limit Theorem

`Z = (134.1 - 136)/(0.8485)`

`Z = -2.24`

`Z = -2.24` has a pvalue of 0.0125

2*0.0125 = 0.025

0.025 = 2.5% probability that the sample mean would differ from the population mean by greater than 1.9 millimeters