Question

Suppose a geyser has a mean time between eruptions of 65 minutes . Let the interval of time between the eruptions be normally distributed with standard deviation 26 minutes . Complete parts ?(a) through ?(e) below.

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

76 ?minutes?

The probability that a randomly selected time interval is longer than

76

minutes is approximately

nothing

.

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

?(b) What is the probability that a random sample of

16

time intervals between eruptions has a mean longer than

76

?minutes?

The probability that the mean of a random sample of

16

time intervals is more than

76

minutes is approximately

nothing

.

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

?(c) What is the probability that a random sample of

31

time intervals between eruptions has a mean longer than

76

?minutes?

The probability that the mean of a random sample of

31

time intervals is more than

76

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

76

?minutes, then the probability that the sample mean of the time between eruptions is greater than

76

minutes

?

increases

decreases

because the variability in the sample mean

?

increases

decreases

as the sample size

?

decreases.

increases.

?(e) What might you conclude if a random sample of

31

time intervals between eruptions has a mean longer than

76

?minutes? Select all that apply.

A.

The population mean must be more than

65

?,

since the probability is so low.

B.

The population mean is

65

?,

and this is just a rare sampling.

C.

The population mean is

65

?,

and this is an example of a typical sampling result.

D.

The population mean may be less than

65

.

E.

The population mean may be greater than

65

.

F.

The population mean cannot be

65

?,

since the probability is so low.

G.

The population mean must be less than

65

?,

since the probability is so low.

Answer 1

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

(a)

The z-score for X=76 is

So the probability that a randomly selected time interval between eruptions is longer than 76 minutes is

(b)

The z-score for is

So the probability that a random sample of 16 time intervals between eruptions has a mean longer than 76 minutes is

(c)

The z-score for is

So the probability that a random sample of 31 time intervals between eruptions has a mean longer than 76 minutes is

(d)

If the population mean is less than 76 minutes, then the probability that the sample mean of the time between eruptions is greater than 76 minutes decreases because the variability in the sample mean decrease as the sample size increases.

(e)

E. The population mean may be greater than 65.