**Answer:**

And if we solve for a we got

So the value of height that separates the bottom 9% of data from the top 91% is 58.

And the answer for this case would be:

a) 58

**Step-by-step 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".

**Solution to the problem**

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:

Where and

For this case the figure attached illustrate the situation for this case.

We know from the figure that the lower limti for D accumulates 9% or 0.09 of the area below and 0.91 or 91% of the area above.

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

(a)

(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 0.09 of the area on the left and 0.91 of the area on the right it's z=-1.34. On this case P(Z<-1.34)=0.09 and P(z>-1.34)=0.91

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 9% of data from the top 91% is 58.

And the answer for this case would be:

a) 58