Final answer:
The question relates to using the Central Limit Theorem to find probabilities concerning sample means. To solve such problems, calculate z-scores of the given values, look up the corresponding probabilities in the standard normal distribution, and find the interval probability.
Step-by-step explanation:
Understanding the Central Limit Theorem and Sampling Distributions
The student's question involves the concept of the Central Limit Theorem (CLT) and its applications to sampling distributions. The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed, regardless of the distribution of the population, provided the sample size is sufficiently large. When we draw samples of a certain size from a population with a known mean (μ) and standard deviation (σ), we can determine probabilities regarding the sample means using the normal distribution.
In the student's question, they’re asked to find the probability that a single value is between two numbers and that the sample mean of a sample of size n=81 is within a range. However, it's worth noting that details provided in the reference Information are inconsistent with the original question. Nonetheless, I will illustrate the general approach using the relevant data.
For example, an unknown distribution with a mean (μ) of 90 and a standard deviation (σ) of 15 is given, and we want to find the probability that the sample mean of size n=25 is between 85 and 92. To do so, we would calculate the z-scores for 85 and 92 using the formula for the standard deviation of the sample mean, which is σ/√n. After finding the z-scores, we would look up these values in the standard normal distribution table to find the probabilities and then find the probability for the interval by subtracting the smaller probability from the larger one.