Finding Areas and Quantiles for Z Curve

Our method for finding probabilities for normal distribution problems will be as follows:

• Convert any normal distribution problem to a standard normal.
• Look up the probability in the standard normal (or Z) table (or use StataQuest).

EXAMPLE:\ Let X be the blood platelet count (measured in thousands per cc of blood) in humans, and suppose . What is the probability that an individual has a platelet count between 185.4 and 360.2? It is a good idea to always draw a picture. The solution is as follows. We have

is given by

which, in turn is P(-1.96<Z<1.96), which is

We might also ask what the value of X is equivalent to the 99% percentile (99% of all individuals will have a platelet count below this value). To solve this we reverse the process used above:

• Look up the Z value corresponding to our probability in the standard normal table.
• Convert this value of Z to X by .

From the Z table, we find that z=2.33 has area to the left of 0.99. Now using the equation for Z we get

which means that

Thus, someone with a blood-platelet count of 376.72 has a higher count than 99% of the population.