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?
(1 point)
16%
68%
34%

Answer 1

To find the percentage of batteries with lifetimes longer than 561 days, we can use the properties of the normal distribution.

1. First, we calculate the z-score for 561 days using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) = 561 days,
- \(\mu\) = mean = 500 days,
- \(\sigma\) = standard deviation = 61 days.

2. Plugging in the values:
\[
z = \frac{(561 - 500)}{61} = \frac{61}{61} = 1
\]

3. Now, we look up the z-score of 1 in the standard normal distribution table (or use a calculator). The area to the left of \(z = 1\) is approximately 0.8413, which means that about 84.13% of batteries have a lifetime of 561 days or less.

4. To find the percentage of batteries that have lifetimes longer than 561 days, we calculate:
\[
1 - 0.8413 = 0.1587
\]

5. Converting to a percentage:
\[
0.1587 \times 100 \approx 15.87\%
\]

Thus, approximately 16% of batteries have lifetimes longer than 561 days.

The answer is **16%**.