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# Answer on Statistics and Probability Question for lim

Question #15911
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?
For any normal random variable X with mean &mu; and standard deviation &sigma; , X ~
Normal( &mu; , &sigma; ), (note that in most textbooks and literature the notation is
with the variance, i.e., X ~ Normal( &mu; , &sigma;&sup2; ). Most software denotes the normal
with just the standard deviation.)

You can translate into standard normal
units by:
Z = ( X - &mu; ) / &sigma;

Moving from the standard normal back to
the original distribuiton using:
X = &mu; + Z * &sigma;

Where Z ~ Normal( &mu; =
0, &sigma; = 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 &mu; and standard deviation &sigma; then the
sample average xBar is normally distributed

with mean &mu; and standard

In this question we have
X ~ Normal( &mu;x = 490 , &sigma;x&sup2;
= 2025 )
X ~ Normal( &mu;x = 490 , &sigma;x = 45 )

Find P( X &gt; 525 )
P(
( X - &mu; ) / &sigma; &gt; ( 525 - 490 ) / 45 )
= P( Z &gt; 0.7777778 )
= P( Z
&lt; -0.7777778 )
= 0.21835

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