To determine the percentage of batteries that have lifetimes longer than 561 days, we first need to calculate the z-score. The z-score is calculated using the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \(X\) is the value we are interested in (561 days),
- \(\mu\) is the mean (500 days),
- \(\sigma\) is the standard deviation (61 days).
Substituting the values:
\[ z = \frac{(561 - 500)}{61} = \frac{61}{61} = 1 \]
Next, we look up the z-score of 1 in the z-score table, which gives us the area to the left of the z-score. The area corresponding to \(z = 1\) is approximately 0.8413. This means that about 84.13% of the batteries have lifetimes less than 561 days.
To find the percentage of batteries with lifetimes longer than 561 days, we subtract this value from 1:
\[ 1 - 0.8413 = 0.1587 \]
To express this as a percentage:
\[ 0.1587 \times 100 \approx 15.87% \]
Thus, approximately 16% of the batteries have lifetimes longer than 561 days.
The final answer is: 16%