A quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard de

Question

3 years ago

7

A quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard de

viation 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

Mathematics

1 answer:

Evgesh-ka [11] · Answer 1 · 2022-04-15T09:06:07+03:00

Answer:

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

Step-by-step 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:

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

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 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 $s = \frac{\sigma}{\sqrt{n}}$ .