The physical fitness of an athlete is often measured by how much oxygen the athlete takes in (which is recorded in milliliters per kilogram, ml/kg). The mean maximum oxygen uptake for elite athletes has been found to be 68.5 with a standard deviation of 7.5. Assume that the distribution is approximately normal. a) Find the probability that an elite athlete has a maximum oxygen uptake of at least 90.3 ml/kg.
b) Find the probability that an elite athlete has a maximum oxygen uptake of at most 59.5 ml/kg.
To answer both questions, we need to find the corresponding probabilities using the standard normal distribution. To do this, we'll use the z-score formula:
z = (x - μ) / σ
where:
- x is the value we're interested in,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.
a) Find the probability that an elite athlete has a maximum oxygen uptake of at least 90.3 ml/kg:
First, let's find the z-score for 90.3 ml/kg:
z = (90.3 - 68.5) / 7.5
z = 21.8 / 7.5
z ≈ 2.91
Using a standard normal distribution table or calculator, we can find the probability associated with a z-score of 2.91. Looking up the closest value, we find that the area to the left of 2.91 is approximately 0.9988.
However, we want to find the probability that an athlete has a maximum oxygen uptake of at least 90.3 ml/kg, which means we want the area to the right of 2.91. To find this, we subtract the area to the left from 1:
P(x ≥ 90.3) ≈ 1 - 0.9988
P(x ≥ 90.3) ≈ 0.0012
Therefore, the probability that an elite athlete has a maximum oxygen uptake of at least 90.3 ml/kg is approximately 0.0012 or 0.12%.
b) Find the probability that an elite athlete has a maximum oxygen uptake of at most 59.5 ml/kg:
Similarly, let's find the z-score for 59.5 ml/kg:
z = (59.5 - 68.5) / 7.5
z = -9 / 7.5
z = -1.2
Using the standard normal distribution table or calculator, we can find the probability associated with a z-score of -1.2. Looking up the closest value, we find that the area to the left of -1.2 is approximately 0.1151.
To find the probability that an athlete has a maximum oxygen uptake of at most 59.5 ml/kg, we can use this area directly:
P(x ≤ 59.5) ≈ 0.1151
Therefore, the probability that an elite athlete has a maximum oxygen uptake of at most 59.5 ml/kg is approximately 0.1151 or 11.51%.