Final answer:
The mean of the distribution of sample mean scores is equal to the population mean, which would be used if it were known. The provided examples illustrate the application of the central limit theorem in calculating probabilities and conducting hypothesis tests related to sample means.
Step-by-step explanation:
The student in the question is asking about the properties of a sampling distribution. Specifically, they are inquiring about the mean of the distribution of sample mean scores (often referred to as the sampling distribution of the sample mean) when taking samples of size n = 15 from a population. The mean of the sampling distribution of the sample mean is always equal to the mean of the population from which the samples are drawn. Therefore, if the population mean of the GMAT scores is not specified, we would use the actual population mean if it were known.
The examples provided all illustrate various scenarios involving sampling distributions and probabilities related to sample means. For example, a situation where with a sample size of 80 from a population with a known mean and standard deviation, we could estimate probabilities related to the sample mean using the central limit theorem.
It is essential to understand that larger sample sizes tend to produce a sampling distribution that is approximately normal, even if the population distribution is not perfectly normal. This is a consequence of the central limit theorem. The examples provided illustrate how these concepts are applied in different statistical analyses, such as hypothesis testing and calculating probabilities.