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 19.4 years and a standard deviation of 16.3 months. complete parts (a) through (d). (a) what is the probability a males growth plates fuse after age 20? the probability a males growth plates fuse after age 20 is 0.3300. (round to four decimal places as needed.) (b) what is the probability a males growth plates fuse before age 17? the probability a males growth plates fuse before age 17 is 0.0384. (round to four decimal places as needed.) (c) what proportion of male growth plates fuse between 16 and 17 years of age? the proportion of male growth plates that fuse between 16 and 17 years of age is \square. (round to four decimal places as needed.)

Explanation:

Step1: Convert standard deviation to years

Since \(1\) year has \(12\) months, the standard deviation \(\sigma=\frac{16.3}{12}\approx1.3583\) years. The mean \(\mu = 19.4\) years.

Step2: Calculate \(z -\)scores

For \(x_1 = 16\):
\(z_1=\frac{x_1-\mu}{\sigma}=\frac{16 - 19.4}{1.3583}=\frac{- 3.4}{1.3583}\approx - 2.50\)
For \(x_2 = 17\):
\(z_2=\frac{x_2-\mu}{\sigma}=\frac{17 - 19.4}{1.3583}=\frac{-2.4}{1.3583}\approx - 1.77\)

Step3: Find probabilities using \(z -\)table

\(P(X<16)=\Phi(-2.50)\), from the standard normal table \(\Phi(-2.50)=0.0062\)
\(P(X<17)=\Phi(-1.77)\), from the standard normal table \(\Phi(-1.77) = 0.0384\)

Step4: Calculate the proportion

\(P(16<X<17)=P(X < 17)-P(X < 16)\)
\(P(16<X<17)=0.0384 - 0.0062=0.0322\)

Answer:

\(0.0322\)