Final answer:
The sampling distribution of the mean is the distribution of sample means taken from a population. The Central Limit Theorem states that if samples of sufficient size are drawn from a population, the distribution of sample means will be normal, even if the population distribution is not normal. The theorem also explains that the mean of the sample means will equal the population mean and the standard deviation of the sample means will be equal to the population standard deviation divided by the square root of the sample size.
Step-by-step explanation:
The sampling distribution of the mean refers to the distribution of sample means taken from a population. It is the distribution of the average values of different samples taken from the same population. The sampling distribution of the mean is approximately normal, regardless of the shape of the population distribution, as long as the sample size is large enough.
The Central Limit Theorem states that if samples of sufficient size are drawn from a population, the distribution of sample means will be normal, even if the distribution of the population is not normal. In other words, when the sample size is large, the distribution of the sample means approaches a normal distribution. The mean of the sample means will be equal to the population mean, and the standard deviation of the sample means will be equal to the population standard deviation divided by the square root of the sample size.
In this particular case, random samples of size 81 are taken from a population with a mean of 5.75 and a standard deviation (not provided). To fully answer the question, the standard deviation of the population is needed to calculate the standard error of the mean, which is equal to the population standard deviation divided by the square root of the sample size.