The probability a​ male's growth plates fuse before age 17

Question

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 18.6 years and a standard deviation of 15.7 months.

​(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.)

Answer 1

For this problem we have to know how to correctly use the normal distribution to find the probability that any X falls between any interval on the distribution (where here X is going to be an age at which growth plates fuse). The area under the curve of the normal distribution is the total probability that the growth plates fuse at some age (we are assuming they always will, so the total area is equal two one). So for example we should think of a normal distribution with the mean centered at 18.6 years, the probability that the growth plates would fuse before 18.6 would be represented as the area under the curve including the leftmost tail up to the mean which is the dead center of the curve. We can visualize this as the left half of the curve being 'filled-in' and the probability would be .5.

So hopefully, now we know how to use the normal distribution to find the probability. Well. . . actually I lied, we need to know one more thing, and that's how to standardize our normal distribution. The standard normal distribution (sometimes called the z-distribution) just allows us to think of our probabilities in terms of how many standard deviations away from the mean the values we use to construct our interval are. The formula for any z along the curve of the standard normal distribution that corresponds to our unstandardized normal distribution is z = (x - mean) / (standard deviation). The z of the mean is thus always 0; 1 is always one standard deviation above the mean; -1 one standard deviation below the mean.

To calculate the probability that the age at which the growth plates fuse is less than (or equal to?) 17 we have to take the entire area under the standardized curve from and including the left tail up to the z-score for 17. So now, some easy math (17 - 18.6 years) / (1.308333 years)* = (-1.6) / 1.308333 = -1.22293 = approx. 1.22. Now, either in your course material or somewhere online (like those on this page: https://ask.learncbse.in/t/usage-of-z-table-1-a-find-the-area-under-the-normal-curve-between-z-0-71-and-z-1-28/22155) you should find a table of the areas under the normal distribution for some z. Sometimes you'll have to do a little arithmetic to get the answer. The table I have gives me the area from the mean to the z-score for positive z. The area from the mean to positive or negative 1.22 is .3888, we subtract this from the area from the tail to the mean (.5000) to get the area from the tail to 1.22. Our answer is thus a probability of .5000-.3888 = 0.1112 that the growth plates will fuse before age 17.

*15.7 months = 1.308333 years