RANDOM VARLABLES AND DISTRIBUTIONS
Normal distribution: Finding a probability, basic
Suppose that the heights of adult men in the United States are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. What proportion of the adult men in the United States are at most 6 feet tall? (Hint: 6 feet $=72$ inches.) Round your answer to at least four decimal places.
Answer
Final Answer: The proportion of the adult men in the United States who are at most 6 feet tall is approximately \(\boxed{0.8413}\).
Step 1 ：The problem is asking for the proportion of adult men in the United States who are at most 6 feet tall. This is a problem of finding the cumulative probability of a normal distribution up to a certain point.
Step 2 ：The normal distribution is defined by two parameters: the mean and the standard deviation. In this case, the mean is 69 inches and the standard deviation is 3 inches.
Step 3 ：The point up to which we want to find the cumulative probability is 72 inches (6 feet).
Step 4 ：To find the cumulative probability up to this point, we need to calculate the z-score, which is the number of standard deviations the point is away from the mean. The formula for the z-score is: \(z = \frac{X - \mu}{\sigma}\) where X is the point of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 5 ：Substituting the given values into the z-score formula, we get \(z = \frac{72 - 69}{3} = 1.0\).
Step 6 ：After calculating the z-score, we can use a z-table to find the cumulative probability up to this point. The cumulative probability corresponding to a z-score of 1.0 is approximately 0.8413.
Step 7 ：Final Answer: The proportion of the adult men in the United States who are at most 6 feet tall is approximately \(\boxed{0.8413}\).
