Question

A new extended-life light bulb has an average service life of 750 hours, with a standard deviation of 50 hours. the shape of this distribution is unknown. from a sample of 100 light bulbs, about what percent of the bulbs will last more than 700 hours?

Answer 1

Using the central limit theorem for a sample of 100 light bulbs with unknown distribution, we calculate a z-score for 700 hours, leading to the conclusion that approximately 100% of the light bulbs will last more than 700 hours.

Step-by-step explanation:

To find what percent of the light bulbs will last more than 700 hours, we can use the concept of standard deviation and the empirical rule, which applies to normally distributed data. However, since the shape of the distribution is unknown, without assuming normality we cannot use the empirical rule directly. Instead, we can still apply the central limit theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed, especially with a large enough sample size (in this case, n = 100).

First, we calculate the z-score for 700 hours to see how many standard deviations away it is from the mean. The z-score is calculated as:

Z = (X - μ) / (σ / √n), where X is the value of interest (700 hours), μ is the population mean (750 hours), σ is the population standard deviation (50 hours), and n is the sample size (100).

Z = (700 - 750) / (50 / √100) = -50 / 5 = -10

This is an extreme z-score, and we can safely say that nearly all of the light bulbs will last more than 700 hours, as a z-score of -10 would position the value far in the tail of the normal distribution.

Therefore, approximately 100% of the sample of 100 light bulbs will last more than 700 hours.