Question

The age, L, in years, of a wolf can be modelled by the normal distribution L~ N(8, 4)

A. Find the probability that a wolf selected at random is at least 5 years old.

Eight wolves are independently selected at random and their ages recorded

Answer 1

The probability that a randomly selected wolf is at least 5 years old, given that the age distribution follows a normal distribution with mean 8 years and standard deviation 2 years, is approximately 93.32%.

Step-by-step explanation:

To calculate the probability that a randomly selected wolf is at least 5 years old, when the age, L, in years, of a wolf can be modelled by the normal distribution L~ N(8, 4), we need to use the properties of the normal distribution. Here, the mean (μ) is 8 years and the standard deviation (σ) is 2 years (since the variance given is 4, and standard deviation is the square root of variance).

First, we need to find the z-score for L = 5:

Z = (L - μ) / σ = (5 - 8) / 2 = -1.5

Now we look up the z-score in a standard normal distribution table or use a calculator with normal distribution functions. We want the probability that L is greater than 5, which is the same as finding 1 minus the probability that L is less than 5.

P(L ≥ 5) = 1 - P(Z < -1.5)

After looking up the z-score, we find the cumulative probability up to -1.5 is approximately 0.0668. So, the probability that a wolf is at least 5 years old is:

P(L ≥ 5) = 1 - 0.0668 = 0.9332

Therefore, there is a 93.32% chance that a randomly selected wolf will be at least 5 years old.