Let's break down the problem step-by-step:
1. Understand the Problem:
- We need to find the time that corresponds to the top 20% of runners in a field where the mean time is 10.06 seconds and the standard deviation is 0.07 seconds.
2. Identify the Distribution:
- The problem suggests that the distribution of times follows a normal distribution.
3. Determine the Percentile:
- We are looking for the time under which the top 20% of runners fall, which corresponds to the 80th percentile in the distribution of times. This is because the top 20% of runners have the fastest times, and thus we need times below the 80th percentile.
4. Find the Z-Score for the 80th Percentile:
- The Z-score for a given percentile can be found using the standard normal distribution table or statistical software. The 80th percentile corresponds to a Z-score.
- Looking up the 80th percentile (0.80) in a Z-table, we find that the Z-score is approximately 0.84.
5. Transform the Z-Score to the Time:
- We will use the Z-score formula for a normal distribution:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\( X \)[/tex] is the value we are trying to find, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
- Rearrange the formula to solve for [tex]\( X \)[/tex]:
[tex]\[
X = Z \cdot \sigma + \mu
\][/tex]
6. Substitute the Values:
- Substitute [tex]\( Z = 0.84 \)[/tex], [tex]\( \mu = 10.06 \)[/tex], and [tex]\( \sigma = 0.07 \)[/tex] into the formula:
[tex]\[
X = 0.84 \cdot 0.07 + 10.06
\][/tex]
7. Calculate the Time:
- Perform the multiplication:
[tex]\[
0.84 \cdot 0.07 = 0.0588
\][/tex]
- Add this result to the mean time:
[tex]\[
10.06 + 0.0588 = 10.1188
\][/tex]
So, to be in the top 20%, a runner must achieve a time below 10.1188 seconds.
