Final answer:
The probability that the average shoe size is exactly 10 is zero, as the normal distribution is continuous and the probability of any single value is zero. However, we can determine probabilities for ranges of values using z-scores.
Step-by-step explanation:
The question refers to the concept of normal distribution in statistics, specifically within the context of men's shoe sizes, with a sample mean and standard deviation provided. When dealing with the probability of obtaining an exact sample mean from a normal distribution, it's essential to understand that the probability of any specific point on a continuous distribution is zero. Instead, we would look for the probability that the sample mean falls within a range. Since the sample size is 43 (a sufficiently large sample), we can use the central limit theorem to find the sampling distribution of the sample mean which would also be normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (standard error).
If tasked with finding the probability of the sample mean being exactly 10, we would state that due to continuous nature of the normal distribution this probability is effectively zero. However, if the question were rephrased to ask for the probability that the sample mean is at most or at least 10, we could calculate those probabilities using a z-score formula and looking up the value on a standard normal distribution table or using a statistical software.
Nevertheless, the fact that an exact value on a continuous distribution has a probability of zero is a fundamental concept in statistics and serves as a critical learning point in understanding how probability distributions function.