# Answer to Question #15911 in Statistics and Probability 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?

Expert's answer

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

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

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