Statistics: Find the probability that a randomly selected individual earned less than $24,000.

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The average annual salary in Pennsylvania was $25,000 in 1992. Assume that salaries were normally distributed for a certain group of wage earners, and the standard deviation of this group was $3,000. 

Find the probability that a randomly selected individual earned less than $24,000.

Nov 6th, 2015

Thank you for the opportunity to help you with your question!

We know  the mean  u = 25,000  s = 3000.  Where  X is a random variable representing annual salary and is normally distributed.

We want to find  Pr( X < 24000).

To do this, we Standardize this normal distribution:

Pr( X < 24000) = Pr( (X- u)/s < (24000 - u)/s ) =  Pr( Z <  (24000 - 25000)/3000) = Pr(Z < -1/3) = 0.3707 (approximately)

According to the Standard Normal graph  The probability that corresponds to  Z < -0.33  is 0.3707

Therefore the probability that a random person earns less than $24,000 is 37%

If we want more accuracy, the probability is 37.1%

Please let me know if you need any clarification. I'm always happy to answer your questions.
Nov 6th, 2015

what about randomly selected sample of 10 individuals, the mean salary was less than $24,000.

Nov 6th, 2015

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Nov 6th, 2015
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