A runner has a average race time of 4.5 minutes with a standard deviation of 0.14.

What time do 75% of his race times fall and what time would he have to achieve to be in the top 5% of his race times?

To answer these questions, we will use the concept of z-scores and the normal distribution.

1. Finding the time when 75% of his race times fall:
To find the time when 75% of his race times fall, we need to determine the z-score corresponding to the 75th percentile.

The z-score formula is given by:
z = (x - μ) / σ

Where:
- x is the observed value (time in this case),
- μ is the mean (average race time),
- σ is the standard deviation.

First, we need to find the z-score using the z-table or a statistical calculator. Since we want the time when 75% of his race times fall, we are looking for the z-score such that 75% of the area under the normal distribution curve is to the left of it.

By looking up the z-table, we find that the z-score corresponding to the 75th percentile is approximately 0.674.

Now, we can rearrange the formula to solve for the observed value (time):
x = z * σ + μ
x = 0.674 * 0.14 + 4.5
x ≈ 4.596 minutes

Therefore, approximately 75% of his race times fall below 4.596 minutes.

2. Finding the time to be in the top 5% of his race times:
For this question, we need to determine the z-score corresponding to the 95th percentile since we want to find the time when only 5% of race times are faster.

Using the z-table, we find that the z-score corresponding to the 95th percentile is approximately 1.645.

Again, rearranging the formula to solve for the observed value (time):
x = z * σ + μ
x = 1.645 * 0.14 + 4.5
x ≈ 4.732 minutes

Therefore, he would need to achieve a race time of approximately 4.732 minutes or faster to be in the top 5% of his race times.