Question

Suppose a geyser has a mean time between eruptions of 62 minutes. Let the interval of time between the eruptions be normally distributed with standard deviation 14 minutes. Complete parts ​(a) through ​(e) below. ​(a) What is the probability that a randomly selected time interval between eruptions is longer than 68 ​minutes? The probability that a randomly selected time interval is longer than 68 minutes is approximately nothing. ​(Round to four decimal places as​ needed.) ​(b) What is the probability that a random sample of 9 time intervals between eruptions has a mean longer than 68 ​minutes? The probability that the mean of a random sample of 9 time intervals is more than 68 minutes is approximately nothing. ​(Round to four decimal places as​ needed.) ​(c) What is the probability that a random sample of 34 time intervals between eruptions has a mean longer than 68 ​minutes? The probability that the mean of a random sample of 34 time intervals is more than 68 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 68 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 68 minutes ▼ decreases increases 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 34 time intervals between eruptions has a mean longer than 68 ​minutes? Select all that apply. A. The population mean is 62​, and this is just a rare sampling. B. The population mean may be less than 62. C. The population mean cannot be 62​, since the probability is so low. D. The population mean may be greater than 62. E. The population mean must be more than 62​, since the probability is so low. F. The population mean is 62​, and this is an example of a typical sampling result. G. The population mean must be less than 62​, since the probability is so low. Click to select your answer(s).

Answer 1

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

(a)

The z-score for X=68 is

So the probability that a randomly selected interval is longer than 68 minutes approximately is

(b)

The z-score for is

So the probability that a randomly sample of 9 time intervals between eruptions has a mean longer than 68 minutes approximately is

(c)

The z-score for is

So the probability that a randomly sample of 34 time intervals between eruptions has a mean longer than 68 minutes approximately is

(d)

If the population mean is less than 62 ​minutes, then the probability that the sample mean of the time between eruptions is greater than 68 minutes decrease because the variability in the sample mean decrease as the sample size increases.

(e)

D.The population mean may be greater than 62.