Question

If you get a sneak peak at the salary list of your job and realize you are in the lowest bracket, which is the lowest 1%, what is your salary if you know that the mean salary is $48,000 and it has a standard deviation of $5500?

Answer 1

`z=-2.33<(a-48000)/(5500)`

And if we solve for a we got

`a=48000 -2.33*5500=35185`

And for this case the answer would be 35185 the lowest 1% for the salary

Explanation:

Let X the random variable that represent the salary, and for this case we can assume that the distribution for X is given by:

`X \sim N(48000,5500)`

Where
`\mu=48000` and
`\sigma=5500`

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

`P(X>a)=0.99` (a)

`P(X<a)=0.01` (b)

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 0.01 of the area on the left and 0.99 of the area on the right it's z=-2.33. On this case P(Z<-2.33)=0.01 and P(z>-2.33)=0.99

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

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.01`

`P(z<(a-\mu)/(\sigma))=0.01`

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

`z=-2.33<(a-48000)/(5500)`

And if we solve for a we got

`a=48000 -2.33*5500=35185`

And for this case the answer would be 35185 the lowest 1% for the salary