Using Function Notation Try it
The values in the table represent a function.
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2
f(x)
-2
1
The ordered pair given in the first row in the table can
be written using function notation as
f(x) = -2 when .r is
1
3
4
-2.
-3
0
0
4
Done
Intro

a)

And we can find this probability using the complement rule and the normal standard table or excel and we got:

b)

And if we solve for a we got

So the value of height that separates the bottom 15% of data from the top 85% is 103.35.  

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".  

Part a

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

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the complement rule and the normal standard table or excel and we got:

Part b

For this part 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 0.15 of the area on the left and 0.85 of the area on the right it's z=-1.036. On this case P(Z<-1.036)=0.15 and P(z>-1.036)=0.85

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 15% of data from the top 85% is 103.35.