The probability model for the average height of your sample is a normal distribution with a mean of 70 inches and a standard deviation of approximately 0.63 inches.
Since heights of people of the same sex and similar ages follow a normal distribution, and you know the mean and standard deviation for the population of young men, we can determine the probability model for the average height of a sample of 20 men.
Here's how:
Central Limit Theorem: When taking random samples from a normally distributed population, the average of those samples will also be normally distributed, regardless of the sample size. This is known as the Central Limit Theorem.
Sample Mean Distribution: Therefore, the average height of your sample of 20 men will follow a normal distribution with:
Mean: equal to the population mean, which is 70 inches.
Standard deviation: reduced by the square root of the sample size. In this case, the standard deviation would be 2.8 inches / sqrt(20) ≈ 0.63 inches.
Therefore, the probability model for the average height of your sample is a normal distribution with a mean of 70 inches and a standard deviation of approximately 0.63 inches.