Final answer:
To calculate the probability that a sample mean falls within a specific range for a normally distributed sample mean with known sample size, population mean, and standard deviation, we first determine the standard error and use it to calculate z-scores for the desired range. Then, using the standard normal distribution, we find the probabilities for these z-scores to determine the probability that the sample mean lies within the range.
Step-by-step explanation:
The specific question "If the sampling distribution of the sample mean is normally distributed with n = 12, what is the probability that the sample mean falls between 51 and 53?" involves concepts from statistics such as sampling distribution, normal distribution, and probability. However, to answer this question, we need additional information such as the mean (μ) and standard deviation (σ) of the population from which the samples are drawn. Since none was provided for this specific example, I will illustrate using related information provided from another example.
In a similar example, let's consider a population with a mean of 45 and a standard deviation of eight. When samples of size n = 30 are drawn and we wish to find the probability that the sample mean is between 42 and 50, we first calculate the standard error of the mean by dividing the population standard deviation by the square root of the sample size. This gives us the standard deviation of the sampling distribution. We then use the standard normal distribution (z-distribution) since the sample size is sufficiently large to apply the Central Limit Theorem, which allows us to approximate the sampling distribution as normally distributed regardless of the population's distribution.
The z-scores for the sample means can be calculated, and then we use a standard normal distribution table or z-score calculator to find the probabilities corresponding to these z-scores. The probability that the sample mean falls within the desired range is found by subtracting the probability associated with the lower z-score from the probability associated with the higher z-score.