Final answer:
The question involves finding the probability of a sample mean falling within a specific range using the Central Limit Theorem and the sample's standard error and population parameters, followed by the Z-score transformation and the use of a standard normal distribution table or calculator.
Step-by-step explanation:
The question relates to the field of probability and specifically to the Central Limit Theorem, which is often used to find the probability of obtaining a certain sample mean from a normally distributed population or a population that can be approximated to normal when dealing with large sample sizes. When a question asks about the probability that a random sample will yield a mean within a certain range, and provides information about the population mean and standard deviation along with the size of the sample, one must utilize the Central Limit Theorem to calculate the probability.
For instance, after determining the sampling distribution's standard deviation (the standard error), one would use the Z-score formula to convert the sample mean to Z-scores and then use a standard normal distribution table or a calculator with a normalcdf function to find the probabilities associated with those Z-scores. Calculations would involve finding the probability of the sample mean being less than the upper value and subtracting the probability of the sample mean being less than the lower value.