Question

Suppose a geyser has a mean time between eruptions of 79 minutes. Let the interval of time between the eruptions be normally distributed with standard deviation 23 minutes. Complete parts ​(a) through ​(e) below.

The probability that a randomly selected time interval is longer than 89 minutes is approximately ____. ​(Round to four decimal places as​ needed.)

​(b) What is the probability that a random sample of 13 time intervals between eruptions has a mean longer than 89 ​minutes?

The probability that the mean of a random sample of 13 time intervals is more than 89 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 89 ​minutes?

The probability that the mean of a random sample of 24 time intervals is more than 89 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 89 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 89 minutes _____▼ (Increase -or- decrease) because the variability in the sample mean _____▼(Increase -or- decrease) as the sample size _____▼ (decreases / increases).

(e) What might you conclude if a random sample of 24 time intervals between eruptions has a mean longer than 89 ​minutes? Select all that apply.

A.The population mean must be more than 79​, since the probability is so low.

B.The population mean may be less than 79.

C.The population mean must be less than 79​, since the probability is so low.

D.The population mean is 79​, and this is an example of a typical sampling result.

E.The population mean is 79​, and this is just a rare sampling.

F.The population mean may be greater than 79.

G. The population mean cannot be 79​, since the probability is so low.

Thank you!

Answer 1

Let X is a random variable shows the time interval between eruption. Given information:

(a)

The z-score for X=89 is

So the probability that a randomly selected interval is longer than 89 minutes approximately is

(b)

The z-score for is

So the probability that a randomly sample of 13 time intervals between eruptions has a mean longer than 89 minutes approximately is

(c)

The z-score for is

So the probability that a randomly sample of 24 time intervals between eruptions has a mean longer than 89 minutes approximately is

(d)

If the population mean is less than 89 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 89 minutes decrease because the variability in the sample mean decrease as the sample size increases.

(e)

F.The population mean may be greater than 79.