The distribution of annual profit at a chain of stores was approximately normal with a mean of $66,000 and standard deviation of $21,000. the executives conducted an audit of the stores with the lowest 20% of profits, what is closest to the maximum annual profit at a store where the executives conducted an audit? round your answer to the nearest thousand dollar.

So the value of height that separates the bottom 20% of data from the top 80% is $484318.  And this value represent the answer for our question.

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solutio to the problem

Let X the random variable that represent the annual profits of a population, and for this case we know the distribution for X is given by:

Where and

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

Since we want the lowest 20% of profits

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.20 of the area on the left and 0.80 of the area on the right it's z=-0.842. On this case P(Z<-0.842)=0.2 and P(Z>-0.842)=0.80

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 20% of data from the top 80% is $484318.  And this value represent the answer for our question.

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