Final answer:
To find the probability that the sample mean is within a specified range, calculate the z-scores for the range and look up the corresponding probabilities in the standard normal distribution. This involves using the Central Limit Theorem and knowledge of the population mean and standard deviation.
Step-by-step explanation:
The question involves finding the probability that the sample mean will fall between two specific values when sampling from a population with a given mean and standard deviation. This is a problem in inferential statistics, particularly involving the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough, regardless of the shape of the population distribution, provided the population standard deviation is known.
When a sample of 25 observations is taken from a population with a mean of 92 and a standard deviation of 15, we can calculate the z-scores for the given sample mean range (86.87 and 99.95) and use those z-scores to determine the corresponding probabilities from the standard normal distribution table.
To calculate the z-score, we use the formula z = (X - μ) / (σ/√n), where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Once we have the z-scores, we can look up the corresponding cumulative probabilities in a z-table or use statistical software to calculate the probability of being between these z-scores, which then gives us the probability that the sample mean is within the specified range.