- 00:00
[Finding Probability using the Normal Distribution, Part 4,MathTutorDVD.com]Hello.Welcome to this lesson.Here, we're going to continue working these problemswith the normal distribution, and we have one up hereon the board.We see that ACT scores, which is a standardized test,are normally distributed with a mean of 24.2-- that'sthe mean test score-- and a standard deviation of 4.2.

- 00:21
So that might be something else that surprisesyou-- is these standardized test scores are normallydistributed.Much like body temperatures and other things,you see how so many different things are normallydistributed.You're always going to have a peak wheremost of your students are kind of clustered,but then you're going to have outliers on both sides.They're going to fall away in the shapeof that normal distribution.The question is-- what is the probability

- 00:42
that a student-- a randomly-drawn studentscores greater than 31?Greater than 31?So as before, let's write down what we know.The mean is 24.2.The standard deviation is 4.2.So if we wanted to draw-- and we almost alwaysdo want to draw a quick picture of what's going on here--

- 01:02
then we can draw a normal distribution.Like this.All right.It falls off like this.And the mean value of this distribution is given at 24.2.Standard deviation is 4.2 points.So 4.2 points above and 4.2 points below the mean.So we want to find-- what is the probability that the score is

- 01:25
greater than 31 for a randomly-drawn student?So what we're after is this right here.Because it's greater than 31, we're after that shaded area.All right.So what we want to do is calculate the Z-scores--or the Z-score of this value that we're interested in so.The Z-score is x minus mu, which is the mean dividedby the standard deviation.

- 01:46
31 minus the mean.24.2 over the standard deviation, which is 4.2.And that gives us the Z-score.When you do the subtraction and the division,that gives us a Z-score of 1.62.1.62.So ultimately, what we're tasked with-- you can kind of think

- 02:09
of it as this Z-score corresponds to this numberright here.What we're tasked with is finding the probabilitythat a randomly chosen student-- so we say random variable xis greater than 31.Randomly chosen from a population of studentsthat have taken the test.Right?Which is exactly the same thing as saying, hey,what's the probability of a Z-score off that table beinggreater than 1.62?

- 02:31
So this is how we translate it from the real world where we'retalking about randomly-drawn student--our random variable being greater than 31,which is our test score.We translate it into a problem involving the Z-scores.Right?Now, we cannot read areas greater than Z off of thattable.We have to change it, as we've been doing in the past,

- 02:51
to Z being less than negative 1.62.And we've explained why in the past.And so whenever you look that up, youwill find that the probability-- when you look upnegative 1.62 in the chart, what you're going to get is 0.0526,That is the probability of a student

- 03:12
scoring greater than 31.Fairly low probability because 31 on the ACTis a very high score.All right.Part b says-- what is the probabilitythat a randomly-drawn student's score willbe between 25 and 32?So here, we're trying to find outnot exactly if it's greater than or less than a score.

- 03:33
We want to find out-- what is the probabilitythat the randomly-drawn student will be between this range?All right.So again, a picture is important.So let's go ahead and quickly draw this guy here.You'll get very fast at drawing these guys.The mean here-- we said the mean was 24.2.And we want to know-- what about 25 to 32?

- 03:56
So this is just kind of a rough hack here at drawing this guy,but we want to know what that shaded area is rightthere-- between 25 and 32.So we need to find a Z-score that corresponds to 32,and a Z-score that corresponds to 25.So for the first one, Z is going to beequal to x minus the mean over the standard deviation, which

- 04:18
is 32 minus the mean of 24.2 overthe standard deviation of 4.2.And that's the standard deviation.When you do that math, do the subtraction,and do the division, you'll get 1.86.That corresponds to this upper guy right here.And then the next one is 25 minus 24.2

- 04:44
over the standard deviation, 4.2.So you do that subtraction and that division.You'll get 0.19.So we have the two Z-scores of interest.Now, what do we do with them?Well, what we're ultimately trying to dois trying to find the area between these twovalues of our random variable.So I'm trying to show you different ways you might

- 05:04
see it written in your test.You might see something like this.We know we want the area between there.You might see it as the random variable being greater than 25but also less than 32.So you might see it like this.And when you look at that, you might say, wow,that looks complicated.This is coming back from basic algebra.

- 05:24
When you see the variable in the middle,you read it from the inside out.So you say x is less than 32.x is greater than-- because the big side'sover here-- greater than 25.It's a way of compactly writing a range.So it's basically saying from 25 to 32is basically what it's doing.How do you calculate that?

- 05:46
Well, if we plug in the Z-value for here,we're going to get the area of the entire curve.We plug in the Z value for here, we're going to get this curve.We subtract them.That's what we want to do.So we're going to say this is equal to the probabilitythat Z is less than 1.86, and then we'llsubtract from that the probability that Z

- 06:06
is less than 0.19.Because we can read this straight from the chartand we can read this straight from the chart,we'll subtract those guys and get the answer.So when we put 1.86 into the Z-score chart, we get 9-- well,I should say 0.9686.

- 06:27
This guy gives us-- when we put it in there-- 0.5753.So when we subtract them, the totalprobability-- when we subtract these two numbers, 0.3933.So remembering that probabilitiesare between 0 and 1 and 0.5 probability is 50-50,

- 06:49
right, then this is just slightly less than 50-50.It's almost 0.04.So this is just a little bit less than 50/50.If I had these guys take a test, pick a student at random,the score will be between 25 and 32.That's a pretty size-able area, not as much as halfof the curve.That would be 0.5.But still, it makes sense that this amountwould be corresponding to around 0.3, 0.4-- something like that.

- 07:12
You see, the process is the same in all of these cases.You identify what you have for the meanand the standard deviation, identify your endpoints,shade the area, find the Z-scores,calculate the answer using the chart in your book.

### Video Info

**Series Name:** Mastering Statistics, Vol 2

**Episode:** 15

**Publisher:** Math Tutor DVD

**Publication Year:** 2013

**Video Type:**Tutorial

**Methods:** Normal distribution, Probability

**Keywords:** mathematical concepts; mathematics; normal schools

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Instructor Jason Gibson presents a lesson on calculating probabilities. In this segment, he demonstrates how to acquire ranges of standardized test scores.