The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If convenient, use technology to find the probability.
For a sample of $n=39$, find the probability of a sample mean being less than 12,751 or greater than 12,754 when $\mu=12,751$ and $\sigma=2.1$.
For the given sample, the probability of a sample mean being less than 12,751 or greater than 12,754 is (Round to four decimal places as needed.)

Step 1 ：We are given the population mean (\(\mu\)), standard deviation (\(\sigma\)), and the sample size (\(n\)). We are asked to find the probability of the sample mean being less than 12,751 or greater than 12,754. This is a problem of finding the probability of a sample mean in a normal distribution.

Step 2 ：The first step is to calculate the standard error of the mean (SEM), which is the standard deviation divided by the square root of the sample size. In this case, \(\mu = 12751\), \(\sigma = 2.1\), and \(n = 39\). Therefore, the SEM is \(\frac{\sigma}{\sqrt{n}} = 0.336269122990683\).

Step 3 ：Next, we calculate the z-scores for the two values (12,751 and 12,754). The z-score is a measure of how many standard deviations an element is from the mean. It is calculated as \((X - \mu) / SEM\), where X is the value for which we want to find the z-score. For 12,751, the z-score is \(\frac{12751 - \mu}{SEM} = 0.0\). For 12,754, the z-score is \(\frac{12754 - \mu}{SEM} = 8.921425711997712\).

Step 4 ：We can use the z-scores to find the probabilities. The probability of a value being less than a certain value is given by the cumulative distribution function (CDF) of the normal distribution at the z-score of that value. The probability of a value being greater than a certain value is given by 1 - CDF at the z-score of that value. For a z-score of 0.0, the probability is 0.5. For a z-score of 8.921425711997712, the probability is essentially 0.

Step 5 ：Since we want the probability of the sample mean being less than 12,751 or greater than 12,754, we need to add the two probabilities together. Therefore, the total probability is \(0.5 + 0 = 0.5\).

Step 6 ：The question also asks whether a sample mean in this range would be considered unusual. In statistics, a result is often considered unusual if its probability is less than 0.05. Since 0.5 is much greater than 0.05, a sample mean in this range would not be considered unusual.

Step 7 ：Final Answer: The probability of the sample mean being less than 12,751 or greater than 12,754 is \(\boxed{0.5}\). A sample mean in this range would not be considered unusual.