Question

Suppose that the lifetime of a certain type of light bulb has a mean of 1000 hours and a standard deviation of 1000 hours. a. Is it reasonable to assume that the lifetime has a normal distribution, and why? b. Can we calculate the probability that the average lifetime exceeds 1200 hours for a random sample of three bulbs? c. What's the probability that the average lifetime exceeds 1200 hours for a random sample of fifty bulbs? d. What's the probability that the average lifetime exceeds 1200 hours for a random sample of five hundred bulbs?

Answer 1

The question addresses using normal distribution and Central Limit Theorem in statistics to calculate probabilities based on sample size, considering the means and standard deviations provided.

Step-by-step explanation:

The question relates to probability and statistics, particularly the use of the normal distribution and the Central Limit Theorem (CLT) to calculate the probabilities of various outcomes based on the sample size.

While the lifetime of the light bulbs given cannot be assumed to be normally distributed due to the large standard deviation equal to the mean, the CLT allows us to assume that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough.

Calculation Steps

For a small sample size from a non-normal distribution, the distribution of the sample means might not be normal.

For larger samples (usually n > 30), thanks to the CLT, we can assume the sample mean will have a normal distribution.

We use Z-scores and the standard error of the mean to calculate the probabilities.