Finding the percentile for a standard normal distribution

Intro

Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the percentile for a standard normal distribution.  Here's our problem statement: Assume that a randomly selected subject is given a bone density test.  Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1.  Draw a graph and find P15, the 15th percentile.  This is the bone density score separating the bottom 15% from the top 85%.

Part 1

OK, the first part of this problem is asking us which of the graphs represents P15.  To do this, we need to remember that percentiles are the area of the curve that are below the given percentile value.  So here, the given percentile value is 15.  Therefore, we want the area of the curve that is below P15.  And that area will correspond to 15% of the total.

Well, answer option A is not going to work for us, because this is not the area below P15.  Answer option B looks like the one that we want to select, but let's look at the other answer options just to make sure.   Answer option C has the area shaded to the left, which is good, but this area is no way 15% of the total because the area shaded here is clearly more than half and 15 is not more than 50.  So there's no way this is right.  And answer option D has the area under the curve shaded to the right, which is not what a percentile is, so that's also incorrect answer.  Option B is the one we want.  Good job!

Part 2

Now the second part of this problem asks us to identify the bone density score.  This is actually the z-score that corresponds with P15.  To do this, you can actually use a standard normal distribution table — or what I'm going to do is use StatCrunch to do this.  So here I have StatCrunch open.   And to do this, I'm going to go to Stat –> Calculators –> Normal.  The reason why I'm using the Normal calculator is because it says in the promise statement that our scores are normally distributed.

So here's my standard normal distribution; the mean is zero, the standard deviation is one.  Those are the default values here in the calculator, so I don't need to make any adjustments there.  I want the area to the left, and that's the area that's shaded here, so I don't need to do anything with the inequality sign here.  I want to make sure that's to the left and it is.   This field here will show me my z-score once I put in the corresponding percentile here, which is in this case 15%.  So I put in 15% in decimal form and I hit Compute!  And there's my z-score.  I'm asked to round to two decimal places.  Well done!

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