Download Working with the Normal Model

Working with the Normal Model
The normal model is commonly used in statistics. It is used to model data that is
unimodal, bell-shaped and symmetric. In this class we will be working a lot with
the Normal model. We can use STATA to calculate values similar to those found
in the Normal Table in the back of the book.
Suppose we want to find the proportion of the area under the normal curve that
lies below z  1 , i.e. one standard deviation above the mean.
To find this area type
display normprob(1)
in the command window.
This gives the following output:
You can verify that the result .84134475 coincides with the value given in the
back of the book.
If, instead, we want to find the proportion of the area under the normal curve that
lies above z  1 , i.e. more than one standard deviation above the mean, we
would need to type
display 1-normprob(1)
Suppose we instead want to know how many standard deviations above the
mean we need to be in order to lie in the 90th percentile of the normal curve. To
find this value type
display invnorm(0.9)
in the command window.
This gives the following output:
Using this command, we find that that the corresponding z-value is equal to
1.2815516. In other words, we need to be 1.28 standard deviations above the
mean to be in the 90th percentile.
Example - The height of U.S. men (in inches) approximately follows a normal
model with mean 69.1 and standard deviation 2.9. Let X be the height of a
randomly sampled man.
Suppose we want to estimate the probability that a man is between 5'6" and 6'.
Then we can simply type
display normprob((72-69.1)/2.9)-normprob((66-69.1)/2.9)
which gives the following output:
This implies that .69880214, or approximately 70% of all U.S. men are between
5'6" and 6'. Note that the command normprob takes the z-score as input rather
than the actual observation.
Suppose we want to know what height a man must be in order to be in the top
10% of all U.S. males. We can calculate this value by first finding how many
standard deviations, z, above the mean we need to be in order to be in the top
10%, and thereafter using the formula x  z   to find the proper value.
Doing these two tasks together we can write,
display invnorm(0.9)*2.9+69.1
This gives the output:
This means that a man must be at least 72.8165 inches to be in the top 10% of
all U.S. males.
Exercise 1: The height of U.S. men (in inches) approximately follows a normal
model with mean 69.1 and standard deviation 2.9. Let X be the height of a
randomly sampled man.
(a) Find the probability that a man is shorter than 60 inches.
(b) Find the probability that a man is between 60 and 72 inches.
(c) What is the shortest a man can be and still be in the top 20% of all U.S.
males?