Four hundred retired athletes participate in a 10-km run. The runner's ages are normally distributed with a mean of 54 years and a standard deviation of 5 years.
Answer the following questions:
What percent of the runners were between 49 and 69 years old?
What is the probability that a runner was at most 64 years old?
There is a 47.5% probability that the age of a randomly selected runner is greater than 54 and less than x years. What would be the value of x?
To solve these questions, we will use the Z-score formula: Z = (X - μ) / σ, where X is the value we are looking for, μ is the mean, and σ is the standard deviation.
1. What percent of the runners were between 49 and 69 years old?
Z1 = (49 - 54) / 5 = -1
Z2 = (69 - 54) / 5 = 3
Using a Z-table, we can find the percentage corresponding to each Z-score:
P(Z < -1) = 0.1587
P(Z < 3) = 0.9987
The percentage of runners between 49 and 69 years old is:
P(-1 < Z < 3) = P(Z < 3) - P(Z < -1) = 0.9987 - 0.1587 = 0.84 or 84%
2. What is the probability that a runner was at most 64 years old?
Z = (64 - 54) / 5 = 2
Using the Z-table, we find:
P(Z < 2) = 0.9772
The probability that a runner was at most 64 years old is 97.72%.
3. What would be the value of x if the probability that the age of a randomly selected runner is greater than 54 and less than x years is 47.5%?
First, find the Z-score corresponding to the 47.5% probability:
P(Z < Zx) - P(Z < 0) = 0.475
From the Z-table, we find that P(Z < 0) = 0.5000, so:
P(Z < Zx) - 0.5000 = 0.475
P(Z < Zx) = 0.475 + 0.5000 = 0.975
Now find the Z-score corresponding to 0.975:
Zx = 1.96
Now solve for X:
1.96 = (X - 54) / 5
X - 54 = 1.96 * 5
X - 54 = 9.8
X = 63.8
Therefore, the value of x for the given probability is 63.8 years old.