Suppose a geyser has a mean time between eruptions of 89 minutes . Let the inter

Question

Suppose a geyser has a mean time between eruptions of 89 minutes . Let the interval of time between the eruptions be normally distributed with standard deviation 25 minutes . Complete parts (a) through (e) below.

(a) What is the probability that a randomly selected time interval between eruptions is longer than 99 minutes? The probability that a randomly selected time interval is longer than 99 minutes is approximately nothing . (Round to four decimal places as needed.)

(b) What is the probability that a random sample of 14 time intervals between eruptions has a mean longer than 99 minutes? The probability that the mean of a random sample of 14 time intervals is more than 99 minutes is approximately nothing . (Round to four decimal places as needed.)

(c) What is the probability that a random sample of 40 time intervals between eruptions has a mean longer than 99 minutes? The probability that the mean of a random sample of 40 time intervals is more than 99 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 99 minutes, then the probability that the sample mean of the time between eruptions is greater than 99 minutes increases decreases because the variability in the sample mean increases decreases as the sample size increases. decreases.

(e) What might you conclude if a random sample of 40 time intervals between eruptions has a mean longer than 99 minutes? Select all that apply.

A. The population mean may be greater than 89 .

B. The population mean is 89 , and this is an example of a typical sampling result.

C. The population mean must be more than 89 , since the probability is so low.

D. The population mean cannot be 89 , since the probability is so low.

E. The population mean is 89 , and this is just a rare sampling.

F. The population mean must be less than 89 , since the probability is so low.

G. The population mean may be less than 89 .

Explanation / Answer

Given: µ = 89, = 25

To find the probability, we need to find the Z scores first.

Z = (X - µ)/ [/n]

(a) n = 1, To calculate P( X > 99) = 1 - P (X < 99), as the normal tables give us the left tailed probability only.

For P( X < 99)

Z = (99 – 89)/[25/1] = 0.40

The probability for P(X < 99) from the normal distribution tables is = 0.6554

Therefore the required probability = 1 – 0.6554 = 0.3446

(b) n = 14, To calculate P( X > 99) = 1 - P (X < 99), as the normal tables give us the left tailed probability only.

For P( X < 99)

Z = (99 – 89)/[25/14] = 1.5

The probability for P(X < 99) from the normal distribution tables is = 0.9328

Therefore the required probability = 1 – 0.9328 = 0.0672

(c) n = 40, To calculate P( X > 99) = 1 - P (X < 99), as the normal tables give us the left tailed probability only.

For P( X < 99)

Z = (99 – 89)/[25/40] = 2.53

The probability for P(X < 99) from the normal distribution tables is = 0.9943

Therefore the required probability = 1 – 0.9943 = 0.0057

(d) If the population mean is less than 99 minutes, then the probability that the sample mean of the time between eruptions is greater than 99 minutes decreases because the variability in the sample mean decreases as the sample size increases.

(e) Option A : The population mean may be greater than 89.

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