Question

Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days, and the standard deviation is 61 days. Approximately what percentage of batteries have lifetimes longer than 561 days? Use the z-score table to determine the percentage.

(1 point)
Responses

68%
68%

34%
34%

84%
84%

16%
16%

Answer 1

To determine the percentage of batteries that have lifetimes longer than 561 days, we first need to calculate the z-score for 561 days using the formula:

\[
z = \frac{(X - \mu)}{\sigma}
\]

where:
- \(X\) is the value of interest (561 days),
- \(\mu\) is the mean (500 days),
- \(\sigma\) is the standard deviation (61 days).

Now plug in the values:

\[
z = \frac{(561 - 500)}{61} = \frac{61}{61} = 1
\]

Next, we look up the z-score of 1.00 in the z-table, which provides the area to the left of the z-score. The area for \(z = 1.00\) is approximately 0.8413, meaning that about 84.13% of the batteries have a lifetime of 561 days or less.

To find the percentage of batteries that have lifetimes longer than 561 days, we can subtract this value from 1 (or 100%).

\[
1 - 0.8413 = 0.1587
\]

This means that approximately 15.87% of batteries have a lifetime longer than 561 days. Rounding that gives us about 16%.

Therefore, the answer is:

**16%**