Final answer:
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.