Question

Suppose a geyser has a mean time between eruptions of 99 minutes. If the interval of time between the eruptions is normally distributed with standard deviation 24 minutes​, answer the following questions.

​(a) What is the probability that a randomly selected time interval between eruptions is longer than 110 ​minutes?

​(b) What is the probability that a random sample of 10 time intervals between eruptions has a mean longer than 110 minutes?

(c) What is the probability that a random sample of 29 time intervals between eruptions has a mean longer than 110 ​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 29 time intervals between eruptions has a mean longer than 110 minutes?

Answer 1

Mean, = 99 minutes

Standard deviation, = 24 minutes

a) When an individual value X is taken,

P(X < A) = P(Z < (A - )/)

P(a randomly selected time interval between eruptions is longer than 110 ​minutes) = P(X > 110)

= 1 - P(X < 110)

= 1 - P(Z < (110 - 99)/24)

= 1 - P(Z < 0.46)

= 1 - 0.6772

= 0.3228

b) When a sample mean, is considered from a sample of size n,

P( < A) = P(Z < (A - )/)

n = 10

= = 99 minutes

=

=

= 7.5895

P(a random sample of 10 time intervals between eruptions has a mean longer than 110 minutes) = P( > 110)

= 1 - P( < 110)

= 1 - P(Z < (110 - 99)/7.5895)

= 1 - P(Z < 1.45)

= 1 - 0.9265

= 0.0735

c) n = 29

= = 99 minutes

=

=

= 4.4567

P(a random sample of 10 time intervals between eruptions has a mean longer than 110 minutes) = P( > 110)

= 1 - P( < 110)

= 1 - P(Z < (110 - 99)/4.4567)

= 1 - P(Z < 2.47)

= 1 - 0.9932

= 0.0068

d) As sample size increase, the standard error decreases. This means, the sample mean is more likely to be closer to the population mean. Therefore, as sample size increases, the probability that mean exceeds 110 minutes reduces.

e) The probability that mean linger length is greater than 110 minutes is unusual since it is less than 0.05. Therefore, from this observation, it can be said that that the population mean of 99 minutes is more likely to be an incorrect value.