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

​(a) What is the probability that a randomly selected time interval between eruptions is longer than 90 ​minutes? The probability that a randomly selected time interval is longer than 90 minutes is approximately nothing. ​(Round to four decimal places as​ needed.)

​(b) What is the probability that a random sample of 11 time intervals between eruptions has a mean longer than 90 ​minutes? The probability that the mean of a random sample of 11 time intervals is more than 90 minutes is approximately nothing. ​(Round to four decimal places as​ needed.)

​(c) What is the probability that a random sample of 16 time intervals between eruptions has a mean longer than 90 ​minutes? The probability that the mean of a random sample of 16 time intervals is more than 90 minutes is approximately nothing. ​(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 90 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 90 minutes ▼ increases decreases because the variability in the sample mean ▼ decreases increases as the sample size ▼ increases. decreases.

​(e) What might you conclude if a random sample of 16 time intervals between eruptions has a mean longer than 90 ​minutes? Select all that apply.

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

B. The population mean may be less than 79.

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

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

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

F. The population mean may be greater than 79.

G. The population mean is 79​, and this is an example of a typical sampling result. Click to select your answer(s).

Answer 1

a) Probability of a randomly selected time interval is longer than 90 minutes, P(X >90) =

b) Probability that the mean of a random sample of 11 time intervals is more than 90 minutes =

​(c) Probability that the mean of a random sample of 16 time intervals is more than 90 =

​(d) If the population mean is less than 90 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 90 minutes decreases because the variability in the sample mean decreases as the sample size increases.

​(e) Options:

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

• F. The population mean may be greater than 79.