Final answer:
The mean value theorem is not directly related to the Central Limit Theorem, but the provided intervals and probabilities refer to the application of the Central Limit Theorem for confidence intervals, and for calculating probabilities of sums in normally distributed samples.
Step-by-step explanation:
The mean value theorem (MVT) in mathematics states that for a continuous function that is differentiable on the open interval, there exists at least one point where the instantaneous rate of change (derivative) is equal to the average rate of change over the entire interval. Now, regarding the specific conditions mentioned in the provided information, we need to focus on the Central Limit Theorem (CLT) for means and sums, which helps in understanding the distribution of sample means and sums. The Central Limit Theorem for means implies that the distribution of the sample means will approach a normal distribution regardless of the shape of the population distribution, given a sufficiently large sample size. Similarly, the Central Limit Theorem for sums also indicates that the sum of a large number of random variables will tend to be normally distributed, regardless of the underlying distribution.
Concerning confidence intervals and their relation to repeated samples, the statement that about 90 percent of the confidence intervals calculated from those samples would contain the true population mean (indicated by μ) is an illustration of the confidence level of an interval estimate.
To answer specific example questions using the CLT: if we consider an unknown distribution with a mean of 80 and a standard deviation of 12, the probability calculations for sums greater than or less than a specific value require the use of the normal distribution as an approximation for the distribution of sums, applying the CLT