Answer to Question #15911 in Statistics and Probability for lim
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 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