$342-61401)$ Radhika Ghatani $11 / 14 / 231: 00 \mathrm{PM}$ Question Part 2 of 4 Completed: 1 of 2 My Score: 1/2 pts (50\%) Save The ability to determine the age of some individuals can be difficult if there are not quality government records of birth. Bone growth takes place at the growth plates at the end of long bones. Once all growth plates fuse, growth stops, and an individual is considered a biological adult. The age at which growth plates fuse for males is approximately normally distributed with a mean of 19.1 years and a standard deviation of 15.1 months. Complete parts (a) through (d) below. Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2) (a) What is the probability a male's growth plates fuse after age 21 ? The probability a male's growth plates fuse after age 21 is 0.0655 . (Round to four decimal places as needed.) (b) What is the probability a male's growth plates fuse before age 17 ? The probability a male's growth plates fuse before age 17 is (Round to four decimal places as needed.)
Solution
Step 1 ：The question is asking for the probability that a male's growth plates fuse before the age of 17. This is a problem of normal distribution. The mean is 19.1 years and the standard deviation is 15.1 months (or 1.2583 years).
Step 2 ：We need to calculate the z-score for the age of 17 and then find the corresponding probability from the standard normal distribution table. The z-score is calculated as \( (X - \mu) / \sigma \), where X is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：Substituting the given values into the z-score formula, we get \( z = (17 - 19.1) / 1.2583 = -1.6688741721854317 \).
Step 4 ：After calculating the z-score, we can use a standard normal distribution table to find the corresponding probability. The probability that corresponds to the z-score of -1.67 is approximately 0.0476.
Step 5 ：This is the area under the standard normal curve to the left of the z-score -1.67.
Step 6 ：Final Answer: The probability a male's growth plates fuse before age 17 is approximately \(\boxed{0.0476}\).
