Questions: (c) What is the probability that a random sample of 23 time intervals between eruptions has a mean longer than 94 minutes? The probability that the mean of a random sample of 23 time intervals is more than 94 minutes is approximately (Round to four decimal places as needed.)

Solution Steps

We want to find the probability that a random sample of 23 time intervals between eruptions has a mean longer than 94 minutes. We assume that the time intervals are normally distributed with a population mean \( \mu = 90 \) minutes and a population standard deviation \( \sigma = 10 \) minutes.

To find the probability, we first calculate the Z-score for the sample mean of 94 minutes using the formula:

\[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]

where:

• \( \bar{x} = 94 \) (the sample mean),
• \( \mu = 90 \) (the population mean),
• \( \sigma = 10 \) (the population standard deviation),
• \( n = 23 \) (the sample size).

Substituting the values, we get:

\[ Z = \frac{94 - 90}{10 / \sqrt{23}} \approx 1.9183 \]

Next, we calculate the probability that the sample mean is greater than 94 minutes. This is given by:

\[ P(\bar{X} > 94) = 1 - P(\bar{X} \leq 94) = 1 - \Phi(Z) \]

Using the Z-score calculated, we find:

\[ P(\bar{X} > 94) = \Phi(\infty) - \Phi(1.9183) \approx 0.0275 \]

Final Answer