Question

The lifetime of a certain lightbulb is a random variable with the expectation of 8000 hours and a standard deviation of 200 hours. What is the probability that the average lifetime of 500 of these light bulbs is less than 8250 hours?

Answer 1

Answer:

100% probability that the average lifetime of 500 of these light bulbs is less than 8250 hours

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 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:

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

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.

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

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:

$\mu = 8000, \sigma = 200, n = 500, s = \frac{200}{\sqrt{500}} = 8.94$

What is the probability that the average lifetime of 500 of these light bulbs is less than 8250 hours?

This is the pvalue of Z when X = 8250. So

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

By the Central Limit Theorem

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

$Z = \frac{8250 - 8000}{8.94}$

$Z = 27.95$

$Z = 27.95$ has a pvalue of 1.

100% probability that the average lifetime of 500 of these light bulbs is less than 8250 hours