Answer to Question #322932 in Statistics and Probability for thea

Question

Given a sample of 100 projector bulbs from a company has a mean

length of life of 20.5 hours with a standard deviation of 1.6 hours, how do I find a 95% confidence interval for the average length of life of those bulbs and then interpret the results?

Answer 1

The formula to calculate a confidence interval for a population mean is as follows:

$CI=\bar{x}\pm z\cdot\cfrac{\sigma}{\sqrt{n}},$

where:

• $\bar{x}=20.5$ : sample mean
• z:  the chosen z-value, for a 95% confidence interval z = 1.96
• $\sigma=1.6$ :  sample standard deviation
• n = 100:  sample size.

So,

$CI=20.5 \pm 1.96\cdot\cfrac{1.6}{\sqrt{100}}=20.5\pm0.314=(20.186, 20.814).$

There is a 95% chance that the confidence interval of (20.186, 20.814) contains the true population mean length of life of bulbs.

Another way of saying the same thing is that there is only a 5% chance that the true population mean lies outside of the 95% confidence interval. That is, there’s only a 5% chance that the true population mean length of life of bulbs is greater than 20.814 hours or less than 20.186 hours.