Suppose a geyser has a mean time between eruptions of 82 minutes. Let the interv

Question

Suppose a geyser has a mean time between eruptions of 82 minutes.

Let the interval of time between the eruptions be normally distributed with standard deviation 19 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 10 time intervals between eruptions has a mean longer than

90 minutes?The probability that the mean of a random sample of 10 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 34 time intervals between eruptions has a mean longer than

90 minutes?The probability that the mean of a random sample of

34 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

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

90 minutes? Select all that apply.

A.

The population mean must be less than 82, since the probability is so low.

B.The population mean must be more than

82, since the probability is so low.

CThe population mean may be greater than 82.

D.The population mean cannot be 82,since the probability is so low.

E.The population mean is 82, and this is just a rare sampling.

F.The population mean may be less than 82.

G. The population mean is 82, and this is an example of a typical sampling result.

Explanation / Answer

a) here P(X>90)=1-P(X<90)=1-P(Z<(90-82)/19)=1-P(Z<0.4211)=1-0.6631 =0.3369

b) for samle size n=10 ; std error =std deviation/(n)1/2 =6.008

hence P(X>90)=1-P(X<90)=1-P(Z<(90-82)/6.008)=1-P(Z<1.3315)=1-0.9085 =0.0915

c)

for samle size n=34 ; std error =std deviation/(n)1/2 =3.2585

hence P(X>90)=1-P(X<90)=1-P(Z<(90-82)/3.2585)=1-P(Z<2.4551)=1-0.9930 =0.0070

d)decreases because the variability in the sample mean decreases as the sample size increases

e)CThe population mean may be greater than 82.

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