Question

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes?

Answer 1

The probability that a randomly selected time interval between eruptions of Old Faithful is longer than 95 minutes is approximately 31.92%, calculated using the z-score method in a normal distribution.

Step-by-step explanation:

To answer the question regarding the probability that the time interval between eruptions of Old Faithful is longer than 95 minutes, considering a normally distributed mean time of 85 minutes between eruptions with a standard deviation of 21.25 minutes, we must use standard normal distribution methods.

First, calculate the z-score for 95 minutes using the formula:
z = (X - μ) / σ where X is the value of interest (95 minutes), μ is the mean (85 minutes), and σ is the standard deviation (21.25 minutes).

Z = (95 - 85) / 21.25 = 0.47.

Using a z-table, locate the area to the left of z-score 0.47 and subtract it from 1 to find the probability of a time interval being longer than 95 minutes.

The area to the left of z = 0.47 is approximately 0.6808. Therefore, the probability of a time interval being longer than 95 minutes is:

P(X > 95) = 1 - P(Z ≤ 0.47) = 1 - 0.6808 = 0.3192.

Thus, there is approximately a 31.92% chance that a randomly selected time interval between eruptions is longer than 95 minutes.