Question

The batteries from a certain manufacturer have a mean lifetime of 810 hours, with a standard deviation of 90 hours. Assuming that the lifetimes are normally distributed, complete the following statements.

(a) Approximately ___% of the batteries have lifetimes between 540 hours and 1080 hours.

Answer 1

Approximately 99.7% of the batteries have lifetimes between 540 hours and 1080 hours, as this range falls within three standard deviations from the mean in a normal distribution.

Step-by-step explanation:

The batteries from a certain manufacturer have a mean lifetime of 810 hours, with a standard deviation of 90 hours. Assuming that lifetimes are normally distributed, to find the percentage of batteries that have lifetimes between 540 hours and 1080 hours, we use properties of the normal distribution.

In a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The range of 540 to 1080 hours is three standard deviations below and above the mean (810 - 270 to 810 + 270). Therefore, approximately 99.7% of the batteries have lifetimes between 540 hours and 1080 hours.