Question 12
The average height for an NCAA Division 1 men's basketball player is 77 inches. Assume this data is normally distributed with a standard deviation of 8 inches. Use this information to answer the following question.
What is the probability that a randomly selected NCAA D1 men's basketball player is more than 72 inches tall? Make sure to round your answer to 2 decimal places; i.e. if your answer was 0.654321 then you would type in 0.65 .
Answer
Final Answer: The probability that a randomly selected NCAA D1 men's basketball player is more than 72 inches tall is \(\boxed{0.73}\).
Step 1 ：We are given that the average height for an NCAA Division 1 men's basketball player is 77 inches. This data is normally distributed with a standard deviation of 8 inches. We are asked to find the probability that a randomly selected NCAA D1 men's basketball player is more than 72 inches tall.
Step 2 ：To solve this problem, we need to use the concept of Z-score in statistics. The Z-score is a measure of how many standard deviations an element is from the mean. In this case, we want to find the probability that a player is more than 72 inches tall. This means we need to find the Z-score for 72 inches and then find the area to the right of this Z-score under the standard normal curve.
Step 3 ：The formula for Z-score is: \(Z = \frac{X - \mu}{\sigma}\) where: \(X\) is the value (72 inches in this case), \(\mu\) is the mean (77 inches in this case), and \(\sigma\) is the standard deviation (8 inches in this case).
Step 4 ：Substituting the given values into the formula, we get: \(Z = \frac{72 - 77}{8} = -0.625\).
Step 5 ：Now that we have the Z-score, we can use it to find the probability that a randomly selected NCAA D1 men's basketball player is more than 72 inches tall. We need to find the area to the right of this Z-score under the standard normal curve. This is also known as the survival function (1 - CDF).
Step 6 ：Using the standard normal distribution table or a function, we find that the probability corresponding to a Z-score of -0.625 is approximately 0.73.
Step 7 ：Final Answer: The probability that a randomly selected NCAA D1 men's basketball player is more than 72 inches tall is \(\boxed{0.73}\).
