In adults vitamin D levels are approximately normally distri

In adults, vitamin D levels are approximately normally distributed with a mean of 40.0 ng/mL and a standard deviation of 16.6 ng/mL. Vitamin D insufficiency is defined as <30 ng/mL What is the probabililty that...

i. What is the probability that an adult has a Vitamin D level of less than 30ng/mL?

ii. If an adult has a Vitamin D level of 75, what percentile is this?

iii. If a study measures Vitamin D levels in 20 adults. What is the probability that mean Vitamin D level exceeds 50.0 ng/mL?

Solution

Normal Distribution
Mean ( u ) =40
Standard Deviation ( sd )=16.6
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 30) = (30-40)/16.6
= -10/16.6= -0.6024
= P ( Z <-0.6024) From Standard Normal Table
= 0.2735                  
c)
P(X > 50) = (50-40)/16.6/ Sqrt ( 20 )
= 10/3.712= 2.6941
= P ( Z >2.6941) From Standard Normal Table
= 0.0035