Finding the positive predictive value of a polygraph test

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 positive predictive value of a polygraph test.  Here's our problem statement: The table below displays results from experiments with polygraph instruments.  Find the positive predictive value for the test.  That is, find the probability that the subject lied given that the test yielded a positive result.

Solution

OK, so we've got our table of data here, and we're asked to find that probability.  It's a conditional probability, the probability that something happens given that something else is happening.  To find the conditional probability that they want here, we're going to have to do a little calculation.  And typically what we do is we take our data here and we dump it into StatCrunch.  But I find that this type of problem is easier to do in Excel.

So I'm going to put my data in Excel.  And in fact, I've already got the data loaded up here in Excel, so we can go to town on it.  So here's our table that we see here in the problem statement.  And the thing I like about Excel is that --- remember that probability is just the part over the whole.  Well, I can get the numbers I need for the part and the whole really quickly here in Excel.  The conditional probability that we need is really dividing two different probabilities.  The first is going to be the probability that both of the events are occurring, and then we divide that by just the event that the one event occurred.

So the first probability we want is the probability that the subject lied.  That's from the problem statement here.  So that's this column here.  And the probability that the results are positive --- so that's this cell here where you see the 40.  So 40 is the cell there, and I don't know why that's not showing up differently, but yeah.  So 40 is the cell that we're looking for.  That's the part.

The whole that we want is everything all together.  This is where Excel comes in really handy.  So if I select everything, now here's my sum right here.  It's summed everything automatically for me right down here.  That's going to give me 107. So if I take 40 and divide it by the 107 --- you can do this in your calculator, but hey, Excel is a spreadsheet, and spreadsheets are made for calculation.  So I'm just going to put that there.  And then that's the first probability.

Now the second probability that we want is the probability of just the one event happening, which is the given event that the test shows a positive result.  Well, the positive result is going to be the 20 plus the 40, which gives us 60.  So I take the 60 and divide it by the whole (the 107).

And so now we've got the first probability here and the second probability here.  So now it's just the one divided by the other.  So if I take this first probability calculated and divide it by the second, that's the probability that we're looking for.  And it looks like it's an even two thirds.  Round to three decimal places.  Well done!

And 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.