Question

Suppose a geyser has a mean time between eruptions of 74 minutes74 minutes. Let the interval of time between the eruptions be normally distributed with standard deviation 28 minutes28 minutes. What is the probability that a randomly selected time interval between eruptions is longer than 86 minutes? (b) What is the probability that a random sample of 15 time intervals between eruptions has a mean longer than 86 minutes? (c) What is the probability that a random sample of 32 time intervals between eruptions has a mean longer than 86 minutes? 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 8686 minutes, then the probability that the sample mean of the time between eruptions is greater than 8686 minutes ▼ decreases increases because the variability in the sample mean ▼ decreases increases as the sample size ▼ decreases. increases. (e) What might you conclude if a random sample of 32 time intervals between eruptions has a mean longer than 86 minutes? Select all that apply. A. The population mean is 74 , and this is just a rare sampling. B. The population mean must be more than 74 , since the probability is so low. C. The population mean may be less than 74. D. The population mean is 74 , and this is an example of a typical sampling result. E. The population mean may be greater than 74. F. The population mean cannot be 74 , since the probability is so low. G. The population mean must be less than 74, since the probability is so low.

Answer 1

We are given,

(a)

Probability that a randomly selected time interval between eruptions is longer than 86 minutes = P(X>86)

We use Excel function " NORMSDIST()" as :

Probability that a randomly selected time interval between eruptions is longer than 86 minutes = 0.334

(b)

Probability that a random sample of 15 time intervals between eruptions has a mean longer than 86 minutes =

We use Excel function " NORMSDIST()" as :

Probability that a random sample of 15 time intervals between eruptions has a mean longer than 86 minutes = 0.0485

(c)

Probability that a random sample of 32 time intervals between eruptions has a mean longer than 86 minutes =

We use Excel function " NORMSDIST()" as :

Probability that a random sample of 32 time intervals between eruptions has a mean longer than 86 minutes = 0.0077

If the population mean is less than 86 minutes, then the probability that the sample mean of the time between eruptions is greater than 86 minutes decreases because the variability in the sample mean decreases as the sample size increases.