Question

When Khloe runs the 400 meter dash, her finishing times are normally distributed with a mean of 81 seconds and a standard deviation of 1 second. What percentage of races will her finishing time be slower than 83 seconds, to the nearest tenth?

Answer 1

To determine the percentage of races in which Khloe's finishing time is slower than 83 seconds, we will use the properties of the normal distribution.

1. **Identify the parameters**:
- Mean (\(\mu\)) = 81 seconds
- Standard Deviation (\(\sigma\)) = 1 second

2. **Calculate the z-score for 83 seconds**:
The z-score is calculated using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) = 83 seconds
- \(\mu\) = 81 seconds
- \(\sigma\) = 1 second

Substituting the values:
\[
z = \frac{83 - 81}{1} = \frac{2}{1} = 2
\]

3. **Find the probability corresponding to the z-score**:
We will use the z-score to find the probability that Khloe's finishing time is less than 83 seconds. We can look up the z-score of 2.0 in the standard normal distribution table, or use a calculator or software.

Looking up the z-score of 2.0, we find that the cumulative probability is approximately 0.9772. This means that about 97.72% of the time, her finishing time will be less than 83 seconds.

4. **Convert to percentage**:
To express this probability as a percentage:
\[
0.9772 \times 100 \approx 97.72\%
\]

5. **Round to the nearest tenth**:
Finally, rounding to the nearest tenth gives us:
\[
\text{Percentage} \approx 97.7\%
\]

Thus, approximately **97.7%** of races will have Khloe finishing in less than 83 seconds.