Question

Consider eruptions of Old Faithful geyser in Yellowstone National Park. The distribution of eruptions is said to be roughly normal and based on a study done in the park in 2011 has a mean time between eruptions of 93 minutes with a standard deviation of 9.5 minutes.

A)You have watched the Geyser erupt, but your friend missed it. What is the probability that he will have to wait less than an hour to see it?

B)What is the probability your friend has to wait more than 100 minutes to see the eruption?

C)Suppose you decide to study this geyser over an extended period. You watch 20 eruptions and record the time between each. What is the probability that the mean from your sample is more than 100 minutes (assuming that the original data is true)?

D)What would our probability be if we studied 60 eruptions instead?

Answer 1

a)

The probability that he will have to wait less than an hour i.e.,60 minutes

b)  Probability your friend has to wait more than 100 minutes

P( X >100)

c)

probability that the mean from your sample is more than 100 minutes

Given sample n =20

P(X > 100) = P( X > x - μ/ σ / √n )

= 100 - 93 / 9.5 /√20

= 3.2952

P( Z > 3.2952 ) = 0.00049

d)

probability be if we studied 60 eruptions instead

.Using continutity correction

P(X = 60) = P( 60 - 0.5< x < 60 -0.5 )

            = P( 59.5 < X < 60.5)
= P( 59.5 -93 /9.5 < X <60.5 -93/9.5)
= P( z < -3.42 ) - P( z < -3.52)
= 0.000313 - 0.000215
= 0.000097