Question

Suppose a geyser has a mean time between eruptions of 71 minutes.Let the interval of time between the eruptions be normally distributed with standard deviation 24 minutes.

Complete parts ​(a) through ​(e) below.

​(a) What is the probability that a randomly selected time interval between eruptions is longer than 83 ​minutes?

The probability that a randomly selected time interval is longer than 83 minutes is approximately ____

​ (Round to four decimal places as​ needed.)

​(b) What is the probability that a random sample of 8 time intervals between eruptions has a mean longer than 83 ​minutes?

The probability that the mean of a random sample of 8 time intervals is more than 83 minutes is approximately ______

​(Round to four decimal places as​ needed.)

​(c) What is the probability that a random sample of 24 time intervals between eruptions has a mean longer than 83​ minutes?

The probability that the mean of a random sample of 24 time intervals is more than 83 minutes is approximately ____

​(Round to four decimal places as​ needed.)

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

If the population mean is less than 83 ​minutes, then the probability that the sample mean of the time between eruptions is greater than

83 minutes (________________)because the variability in the sample mean(___________)as the sample size (_____________)

​(e) What might you conclude if a random sample of 24 time intervals between eruptions has a mean longer than 83 ​minutes? Select all that apply

A.The population mean is71​,and this is an example of a typical sampling result.

B.The population mean is 71​,and this is just a rare sampling.

C.The population mean may be greater than 71

D.The population mean must be more than 71​,since the probability is so low.

E.The population mean cannot be 71​,since the probability is so low.

F.The population mean must be less than 71 since the probability is so low.

G.The population mean may be less than 71

Answer 1

Answer a)

The probability that a randomly selected time interval is longer than 83 minutes is approximately 0.3085

Answer b)

e the standard normal table to conclude that:

P (Z>1.41) = 0.0793

The probability that the mean of a random sample of 8 time intervals is more than 83 minutes is approximately 0.0793

Answer c)

Use the standard normal table to conclude that:

P (Z>2.45) = 0.0071

The probability that the mean of a random sample of 24 time intervals is more than 83 minutes is approximately  0.0071

Answer d)

If the population mean is less than 83 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 83 minutes (decreases) because the variability in the sample mean (decreases) as the sample size (increases)

Answer e)

B.The population mean is 71​, and this is just a rare sampling.

C.The population mean may be greater than 71

E.The population mean cannot be 71​,since the probability is so low.