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 75 with a standard deviation of 6.3. Assume that the distribution is approximately normal.
(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg?
(b) What is the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower?
play around with Z stuff at
http://davidmlane.com/hyperstat/z_table.html
(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg?
Well, if they're an elite athlete, they probably have a pretty high oxygen uptake. But let's do some math just to be sure.
To find the probability, we can use the standard normal distribution and calculate the z-score.
The z-score is calculated by subtracting the mean from the given value and then dividing by the standard deviation.
For an oxygen uptake of 65 ml/kg:
z = (65 - 75) / 6.3
z = -1.59
Now, we need to find the percentage of the distribution that is above this z-score. Lucky for us, there are tables that tell us these percentages! (Who needs math when you have tables?)
According to the standard normal distribution table, the area to the left of z = -1.59 is 0.0564. So the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg is approximately 1 - 0.0564 = 0.9436, or 94.36%.
So, there's a pretty good chance that an elite athlete will have a maximum oxygen uptake of at least 65 ml/kg. They need all that oxygen to fuel their athletic endeavors!
(b) What is the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower?
Okay, so now we're looking at a different oxygen uptake value. Let's see what the probability is for this one.
For an oxygen uptake of 70 ml/kg:
z = (70 - 75) / 6.3
z = -0.79
Using the standard normal distribution table, the area to the left of z = -0.79 is 0.2148. So the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower is approximately 0.2148, or 21.48%.
So, there's about a 21.48% chance that an elite athlete will have a maximum oxygen uptake of 70 ml/kg or lower. They're still pretty fit, but maybe they're not at the very top of their game. Keep training, elite athlete!
To solve these questions, we will use the z-score formula and the standard normal distribution table to find the probabilities.
The z-score formula is given by:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
(a) To find the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg, we need to find the area under the normal curve to the right of 65 ml/kg.
First, calculate the z-score:
z = (65 - 75) / 6.3
z = -10 / 6.3
z ≈ -1.59
Now, we can use the standard normal distribution table to find the probability corresponding to the z-score. The probability of an elite athlete having a maximum oxygen uptake of at least 65 ml/kg is the same as the probability of having a z-score of -1.59 or greater.
Using the standard normal distribution table, we find that the cumulative probability for z = -1.59 is approximately 0.0559.
Therefore, the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg is approximately 0.0559 or 5.59%.
(b) To find the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower, we need to find the area under the normal curve up to 70 ml/kg.
First, calculate the z-score:
z = (70 - 75) / 6.3
z = -5 / 6.3
z ≈ -0.79
Now, we can use the standard normal distribution table to find the probability corresponding to the z-score. The probability of an elite athlete having a maximum oxygen uptake of 70 ml/kg or lower is the same as the probability of having a z-score of -0.79 or less.
Using the standard normal distribution table, we find that the cumulative probability for z = -0.79 is approximately 0.2148.
Therefore, the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower is approximately 0.2148 or 21.48%.
To solve this problem, we can use the standard normal distribution since the problem states that the distribution is approximately normal. Let's break down the steps:
Step 1: State the given parameters:
Mean (μ) = 75
Standard deviation (σ) = 6.3
(a) Probability of a maximum oxygen uptake of at least 65 ml/kg:
To find this probability, we need to calculate the area under the curve to the right of 65 ml/kg.
Step 2: Convert the given value to a z-score:
To do so, we use the formula z = (x - μ) / σ, where x is the given value. In this case, x = 65.
z = (65 - 75) / 6.3
z = -1.59 (rounded to two decimal places)
Step 3: Look up the z-score in the standard normal distribution table:
If you have a z-table, you can look up the probability associated with the z-score -1.59. The z-table gives us the probability of a standard normal distribution being less than or equal to a certain z-score. However, we want the probability to the right of -1.59.
To find the probability to the right of -1.59, we subtract the probability to the left from 1:
P(Z > -1.59) = 1 - P(Z ≤ -1.59)
Looking up the z-score -1.59 in the table, we find that the value is approximately 0.0564. Remember that this value is the cumulative probability to the left, so we subtract it from 1 to find the probability to the right.
P(Z > -1.59) ≈ 1 - 0.0564
P(Z > -1.59) ≈ 0.9436
Therefore, the probability that an elite athlete has a maximum oxygen uptake of at least 65 ml/kg is approximately 0.9436 or 94.36%.
(b) Probability of a maximum oxygen uptake of 70 ml/kg or lower:
To find this probability, we need to calculate the area under the curve to the left of 70 ml/kg.
Step 2: Convert the given value to a z-score:
Using the formula z = (x - μ) / σ, where x is the given value, we calculate the z-score for x = 70.
z = (70 - 75) / 6.3
z = -0.79 (rounded to two decimal places)
Step 3: Look up the z-score in the standard normal distribution table:
To find the probability to the left of -0.79, we look up the z-score in the table. The cumulative probability to the left is approximately 0.2148.
P(Z ≤ -0.79) ≈ 0.2148
Therefore, the probability that an elite athlete has a maximum oxygen uptake of 70 ml/kg or lower is approximately 0.2148 or 21.48%.