Mens' heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in

the u. s. air force requires that pilots have heights between 64 in. and 77 in.

a. find the percentage of men meeting the height requirement

b. if the air force height requirements are changed to exclude only the tallest 2.5% of men and the shortest 2.5% of men, what are the new height requirements?

a)

b) The new requirement would be that the height would be between b=63.112 and a=74.088

Step-by-step explanation:

1) 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".  

2) Part a

Let X the random variable that represent the heights 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 on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

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 0.975 of the area on the left and 0.025 of the area on the right it's z=1.96. On this case P(Z<1.96)=0.975 and P(z>1.96)=0.025

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 97.5% of data from the top 2.5% is 74.088.

In order to find the lower limit we just need to do a similar operation but with a change of sign:

The new requirement would be that the height would be between b=63.112 and a=74.088

6/20

step-by-step explanation: