Using the Poisson distribution to evaluate malignant tumor data

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use the Poisson distribution to evaluate malignant tumor data.  Here's our problem statement: A rare form of malignant tumor occurs in 11 children in a million, so it's probability is 0.0000114.  Cases of this tumor occurred in a certain town which had 10,787 children.  Part (a) Assuming that this tumor occurs as usual, find the mean number of cases in groups of 10,787 children.  (b) Using the unrounded mean from Part A, find the probability that the number of tumor cases in a group of 10,787 children is zero or one.  (c) What is the probability of more than one case?  (d) Does the cluster of four cases appear to be attributable to random chance?  Why or why not?

Part A

OK, Part A asks for the mean number of cases.  So to calculate the mean, I'm just going to whip out my little calculator here.  And the mean is just going to be total number of children that we're evaluating, which is 10,787, multiplied by the probability that these children are going to actually have one of these malignant tumors (so 0.000011).  There's the mean number of cases; you can see it's pretty small.  I'm asked around to three decimal places, so I'm going to do that.  Well done!

Part B

Now Part B asks for the probability that the number of cases is exactly zero or one.  Notice here in the problem statement it asked us to use the unrounded mean from Part A.  So the probability I'm going to get with the Poisson distribution.  And to do that I'm going to pull up StatCrunch.  And then the unrounded mean we're just going to take directly from the calculator here.  So the first step is to load up StatCrunch.  And I'll pop that out here.  Then we get to resize this window so that it's more visible to see what's going on.  Excellent!

Now I'm going to pull up the Poisson calculator by going to Stat --> Calculators --> Poisson.  Here's my Poisson calculator.  Notice I have to put the mean value in, and that's what the calculated value from Part A is all about.  I could just type this in, but I'm prone to transcription error.  So I'm just going to make it easy on myself and just right click on there and copy that value.  Come over here, and right click again, and Paste, and there's my mean value.  Now we want to get the probability the number of cases is exactly zero or one.  So that's the same as being one or less.  So less than or equal to one gives me --- oh, 99%.  So I round to three decimal places, and put that in here.  Well done!

Part C

Part C asks for the probability of more than one case, so what I need to do is switch this around.  That's the same thing as saying two or more, so greater than or equal to two.  And there's my value there.  Notice that the number I'm typing in here is the complement of the probability from the first part.  And that actually makes sense, because this part here is for the number that's zero or one, and this is for anything that's greater than one.  So it makes sense that the probabilities here are complements.  So instead of using the calculator here in StatCrunch, I could have just taken this number and subtracted it from 1.  Excellent!

Part D

And now Part D says, "Let a probability of 5% or less be very small and a probability of 95% or more be very large. Does the cluster of four cases appear to be attributable to random chance? Why or why not?"  Well, we can see that for the four cases out of the 10,787, you know the probability of that happening is going to be less than seven tenths of 1%, which is the probability for more than one case.  And it says here anything 5% or less is considered very small.  So the probability that we're getting these four cases is going to be very, very small.  That said, it actually happened.  And when something that is highly unlikely to happen because it has a very small percentage of probability of happening actually happens, then you got to ask yourself, What's going on here?  Because that doesn't occur by random chance.  That's not attributable to the random nature of life; that's attributable to some machination somebody or something is doing something to create this unlikely event from hap---to happen.  So we would say no, it's not random chance because the probability of it happening is really, really small.  And that's going to be this answer option here.  Fantastic!

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