Question

To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed.

Please explain!

Answer 1

z = 1.28 < a - 200/20

And if we solve for a we got

a = 200 + 1.28 * 20 = 225.6

So the value of height that separates the bottom 90% of data from the top 10% is 225.6.

Explanation:

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

X ~ N (200,20)

For u = 200 and o = 20

For this case we can use the z score in order to solve this problem, given by this formula:

Z = x-u/o

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

P (X > a) = 0.1 (a)

P (X < a) = 0.9 (b)

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 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1

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

P ( X < a) = P (X-u/o < a - u/o) = 0.9

P (z < a-u/o) = 0.9

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

z = 1.28 < a - 200/20

And if we solve for a we got

a = 200 + 1.28 * 20 = 225.6

So the value of height that separates the bottom 90% of data from the top 10% is 225.6.