Question

A quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard deviation of 86 minutes with a mean life of 505 minutes. Of the claim is true, in a sample of 120 batteries, what is the probability that the mean battery life would be greater than 523.8 minutes

Answer 1

0.0082 = 0.82% probability that the mean battery life would be greater than 523.8 minutes

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.

Standard deviation of 86 minutes with a mean life of 505 minutes.

This means that
`\sigma = 86, \mu = 505`

Sample of 120:

This means that
`n = 120, s = (86)/(√(120))`

What is the probability that the mean battery life would be greater than 523.8 minutes?

This is 1 subtracted by the p-value of Z when X = 523.8. So

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

By the Central Limit Theorem

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

`Z = (523.8 - 505)/((86)/(√(120)))`

`Z = 2.39`

`Z = 2.39` has a p-value of 0.9918.

1 - 0.9918 = 0.0082

0.0082 = 0.82% probability that the mean battery life would be greater than 523.8 minutes