Finding the area under a Normal distribution curve using z-scores in StatCrunch

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 area under a normal distribution curve using z-scores in StatCrunch.  Here's our problem statement: Find the indicated area under the curve of the standard normal distribution, then convert it to a percentage and fill in the blank: About <blank> percent of the area is between z = -1 and z = 1 or within one standard deviation of the mean.

Solution

OK, there's three ways to work this problem.  The first is to use the normal distribution calculator in StatCrunch.  To do that, I need to call up StatCrunch.  I'm going to pop this window out, and then I'm going to resize it so that we can see everything a bit better here.  And now in StatCrunch, I go to Stat --> Calculators --> Normal.  Here's my Normal calculator in StatCrunch.  And notice here I'm looking for the standard normal distribution.  The standard normal distribution has a mean of 0 and a standard deviation of 1.  And that's the defaults that we see here in StatCrunch.  So we're gonna leave those alone.

We're looking for the area in between two z scores.  The standard normal distribution has z-scores already encoded into its horizontal --- horizontal axis here.  So this is a z-score of 1, and this is a z score of -1.  So we want the area in between, and that means we need to select this Between button up at top.  And notice the default values that I'm putting in for the boundaries of that area in the middle of my distribution, -1 and 1.  These are the very numbers that I wanted.  And so the area underneath the curve is 0.68268949.  This is the area underneath the curve.  So to convert from decimal to percent, notice that we have to move that decimal point over two places.  So now I've got 68, and then we want two decimal places beyond that, so 0.27.  I check my answer.  Well done!

And that's the first way to do it.  And it's actually pretty easy to do it this way.  A lot of students go this route.  The second way to solve this problem is with the z score tables.  And uh, yeah, that's a little but more involved. I show you how to do that in the lecture videos.  So if you want to go old school like that, you know, feel free.  It's a little more involved than what you see here in StatCrunch.  Notice in StatCrunch, I just pulled it up and there it was --- so much easier to do this in StatCrunch.

The third way to solve this problem is simply to recognize that what we're looking for here is part of the Empirical Rule.  The Empirical Rule says that the area or the amount of data in between 1 standard deviation of your mean is going to be 68.27%.  So if you recognize this as the Empirical Rule (and you do have that memorized, right?), then you should be able to just pop the number in without even using StatCrunch --- even easier!  So for those of you out there who haven't yet memorized the Empirical Rule, I highly recommend doing that because there's a lot of problems where it actually comes in pretty useful.

And so again, that's how we do it at Aspire Mountain Academy.  Be sure to leave your comments below and let us know how good a job we did or how we can improve.  And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see.  Thanks for watching!  We'll see you in the next video.