HW Score: $59.52 \%, 8.33$ of 14 points Points: 0 of 1 Determine whether the finite correction factor should be used. If so, use it in your calculations when you find the probability: In a sample of 110 eruptions of the Old Faithful geyser at Yellowstone National Park, the mean interval between eruptions was 129.58 minules and ihe standard deviation was 108.54 minutes. A random sample of size 35 is selected from this population. What is the probability that the mean interval between enuptions is belwach 120 minutes and 139 minutes? The probability that the mean interval between eruptions is between 120 minutes and 139 minutes is (Round to four decimal places as needed.)

Step 1 ：The problem is asking for the probability that the mean interval between eruptions is between 120 minutes and 139 minutes. This is a problem of finding the probability of a range for a sample mean from a population.

Step 2 ：We know the population mean (\(\mu = 129.58\) minutes), the population standard deviation (\(\sigma = 108.54\) minutes), the sample size (\(n = 35\)), and we are asked to find the probability that the sample mean (\(\bar{x}\)) is between 120 minutes and 139 minutes.

Step 3 ：We can use the Central Limit Theorem (CLT) to solve this problem. The CLT states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Step 4 ：First, we need to check if we can use the CLT. The rule of thumb is that the sample size should be at least 30 for the CLT to be used. In this case, the sample size is 35, which is greater than 30, so we can use the CLT.

Step 5 ：Next, we need to standardize the scores to the standard normal distribution. This involves subtracting the population mean from the sample mean and then dividing by the standard error (which is the population standard deviation divided by the square root of the sample size).

Step 6 ：Finally, we can use the standard normal distribution to find the probability that the z-score is between the z-scores corresponding to 120 minutes and 139 minutes.

Step 7 ：The calculated z-scores are \(z1 = -0.5221673514104882\) and \(z2 = 0.5134463935581196\).

Step 8 ：The calculated probability that the mean interval between eruptions is between 120 minutes and 139 minutes is approximately 0.3954.

Step 9 ：Final Answer: The probability that the mean interval between eruptions is between 120 minutes and 139 minutes is approximately \(\boxed{0.3954}\).