Answer to Question #15911 in Statistics and Probability for lim

Question

Find the indicated probability Weekly salaries of teachers in one state are normally distrubuted with a mean of $490 and a standard deviation of $45. What is the probblity that a randomly selected teacher earns more than $525 a week?

Answer 1

For any normal random variable X with mean μ and standard deviation σ , X ~
Normal( μ , σ ), (note that in most textbooks and literature the notation is
with the variance, i.e., X ~ Normal( μ , σ² ). Most software denotes the normal
with just the standard deviation.)

You can translate into standard normal
units by:
Z = ( X - μ ) / σ

Moving from the standard normal back to
the original distribuiton using:
X = μ + Z * σ

Where Z ~ Normal( μ =
0, σ = 1). You can then use the standard normal cdf tables to get
probabilities.

If you are looking at the mean of a sample, then remember
that for any sample with a large enough sample size the mean will be normally
distributed. This is called the Central Limit Theorem.

If a sample of
size is is drawn from a population with mean μ and standard deviation σ then the
sample average xBar is normally distributed

with mean μ and standard
deviation σ /√(n)

In this question we have
X ~ Normal( μx = 490 , σx²
= 2025 )
X ~ Normal( μx = 490 , σx = 45 )

Find P( X > 525 )
P(
( X - μ ) / σ > ( 525 - 490 ) / 45 )
= P( Z > 0.7777778 )
= P( Z
< -0.7777778 )
= 0.21835