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, answer the following questions:

(a) What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes?

(b) What is the probability that a random sample of 20 time intervals between eruptions has a mean longer than 95 minutes?

(c) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?

(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.

(e) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?

(f ) On a certain day, suppose there are 22 time intervals for Old Faithful. Treating these 22 eruptions as a random sample, the likelihood the mean length of time between eruptions exceeds _________ minutes is 0.20.