Converting z-scores into real-world values 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 convert z-scores into real-world values with StatCrunch.  Here's our problem statement: Consider a value to be significantly low if its z-score is less than or equal to -2, or consider a value to be significantly high if its z-score is greater than or equal to 2.  A test is used to assess readiness for college.  In a recent year, the mean test score was 22.4 and the standard deviation was 5.1.  Identify the test scores that are significantly low or significantly high.

Part 1

OK, this first part is asking us for the test scores that are significantly low.  Now, if I wanted to, I could go “old school” and use the formula that I talked about in the lecture to calculate — with my calculator, punch the numbers out on my calculator and use that formula to convert from a z-score to a real-world value.  But I love the 21st century because it allows us to use technology like StatCrunch.  So I'm gonna use StatCrunch, and I'm gonna show you how easy this is in StatCrunch.

First, since we're dealing with z-scores, we need to get our normal distribution calculator out.  To do that, we're going to go to Stat –> Calculators –> Normal.  Here's my normal calculator.  You'll note that the default values here are for the standard normal distribution.  So what we want to do first is, since we're dealing with scores that are significantly low and scores that are significantly high, I'm gonna take this Between option up here at the top of the calculator.  That way I get the scores here that are significantly low here in the left tail of my distribution and the ones that are significantly high will be in the right tail of my distribution.

The bounds for these tails you can see here in the problem statement.  It says anything less than -2 and anything more than 2 is going to be low or high, respectively.  So I'm just gonna put those values in here.  I hit Compute!, and this is the number that I want to select.  And I want to copy it, the reason being is that now what I'm going to do is take the mean value and the standard deviation value that was given to me in the problem statement, and I'm just going to put that here in my distribution so StatCrunch can do all of the calculation for me.  I'm going to clear out these values in the probability fields, and then here I'm gonna paste that value that we had before.  Now when I hit Compute!, I've got my distribution.

And you can see the bounds here.  So anything less than this left bound is going to be significantly low, and anything more than this right bound is going to be significantly high.  And those numbers are given to me right here in the probability field.  So what test scores are significantly low?  Well, test scores are going to be less than, in this case, 12.2.  Good job!

Part 2

And what test scores are significantly high?  Well, we've already done the work here.  So all I got to do is just take the number and put it in — anything greater than 32.6.  Nice work!

And that's how we do it in 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.