Final answer:
The Central Limit Theorem is applied to calculate probabilities of sample means; for large samples from a normally distributed population, the sample means themselves are normally distributed. When testing hypothesis for smaller samples or unknown population standard deviation, a t-test is used. The probability of a sample mean being unusual is determined by how low the calculated probability is.
Step-by-step explanation:
The question pertains to finding probabilities related to the means of samples taken from populations with a known mean (mu) and standard deviation (sigma), and concerns the use of the Central Limit Theorem which allows us to treat the distribution of sample means as a normal distribution when the sample size is large (typically n > 30). The use of this theorem is essential when working with sample means and sums from a normally distributed population, or even a non-normally distributed population if the sample size is large enough.
For a sample of n = 39, to find the probability of the sample mean being less than 12,752 or greater than 12,755, we would calculate the z-scores for both values and use the standard normal distribution to find the corresponding probabilities. Since we are dealing with sample means, we need to adjust the standard deviation by the square root of the sample size (n), which is called the standard error. If the calculated probability is very low, the event would be considered unusual.
When performing hypothesis tests on sample means where the sample size is smaller, or the population standard deviation is unknown and the population is normally distributed, one should use the t-distribution instead of the normal distribution. For example, when testing whether a sample mean of 12.8 significantly differs from a population mean of 13 with a sample standard deviation of 2 and a sample size of 20, a t-test would be appropriate. The Central Limit Theorem being used in different scenarios shows that as n increases, the sample mean tends to approach the population mean.