Identifying statistical values related to hypothesis testing of two population proportions

Intro

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to identify statistical values related to hypothesis testing of two population proportions.  Here's our problem statement: In a large clinical trial, 390,245 children are randomly assigned to two groups.  The treatment group consisted of 193,395 children given a vaccine for a certain disease, and 34 of those children developed the disease.  The other 196,850 children were given a placebo, and 136 of those children developed the disease.  Consider the vaccine treatment group to be the first sample.  Identify the values of n1, p1, p-hat1, q-hat1, n2, p-hat2, q-hat2, p-bar, and q-bar.

Part 1

OK, the first thing we're asked to provide is the sample size for the first sample.  The problem statement says that we're considering the vaccine treatment group as the first sample, so the sample size is given there in the problem statement.  That's the total number in that first group that we're given the vaccine, which here is the 193,395.  So I'm just going to put that here in my answer field.  Well done!

Part 2

Next we're asked to provide p-hat1.  p-hat1 is the proportion of those in the first sample --- that's the subscript 1 --- that is a probability of quote unquote success.  And here success is going to be defined as actually developing the disease.  I know that sounds backwards.  It's like, we want to --- we're developing a vaccine, right?  You want to have success be not getting the disease, but it actually is easier to work the problem statistically if you just flip flop your definition of success.  So here we're going to say success is going to be actually developing the disease.

Well, the probability --- that's all we're looking at here is a proportion or probability, which would then be the part over the whole.  The part that developed the disease is 34 in that first sample, and over the whole, which is the 193,395.  So I'm going to whip out my calculator here and just do a simple little division here.  Take the 34, which is the part, divided by the whole, 193395.

And notice here the scientific notation that's given here in my calculator, this "e minus 4."  This is really as a notation here.  It's saying this number that's out listed here is being multiplied by 10 to the -4.  So if you were to write this out in decimal form, this decimal point would really need to be moved 4 places to the left.  That's the -4 here.

So it's asking here in the problem statement for us to provide an answer rounded to eight decimal places.  So I'm going to put in one, two, three, and then that first number --- see, then my decimal place has been moved over four places.  And then I finish putting the number in.  This is going to get tricky.  Let's count the numbers here to make sure we've got --- one, two, three, four, six, seven, eight.  Oh, look at that.  Actually I need to come up one because the zero here, which is my eighth character, the 5 next to it means I'm going to be rounding up.  So I'll put that 1 there on the end.  I check my answer.  Well done!

Part 3

q-hat is just the compliment of p-hat.  So to get that, we're just going to subtract this value from 1.  Instead of typing this number in again, I could say 1 minus and then type this number in again, but I don't want to have to do the typing.  I'm a little lazy.  So I'm just going to make that number negative and then add it to 1.  It gives me the same effect.  So now I need to take this number.  That's eight decimal places.

Part 4

And now I need to do the same thing with the second sample.  So the size of the second sample, those who were given the placebo, is listed here in the problem statement.  That's the 196850.

Part 5

And then the proportion of success, which again is going to be developing the disease --- so I got to take the 136 that actually develop the disease there in the second sample and divide by that sample size.  So again I get a really small number.  Well done!

Part 6

And now we're looking for q-hat2, which of course is going to be the complement of p-hat2.  So again, I'm just going to make this number, which is p-hat2 --- make that negative.  Add it to 1.  Boom.  There's my q-hat2.  Excellent!

Part 7

Now I'm looking for p-bar.  p-bar is going to be the pooled proportion.  So remember when we were doing the individual p-hat.  So p-hat1, I just take the 30 (the part which developed the disease, the 34) divided by the whole. I did the same thing with p-hat2; I took the part that developed the disease, the 136, and divided by its whole. And now we're going to have to combine those together.  So now I'm going to have to take both the parts and add them together to get a pooled part.  And then I take that pooled part and divide it by a pooled whole.  So I'm going to take both of these holes and add them together.  And here I get my pooled proportion, the p-bar.  Well done!

Part 8

And last but certainly not least, I'm asked for q-bar, which again, is just going to be the complement of p-bar.  I'm going to take this value here for p-bar in my calculator and make it negative, add the 1, and there's my value for q-bar.  I'll put that here in my answer field.  And it's easy to get lost with the number.  It's got eight decimal places, so it's really easy to get lost.  I got eight.  I check my answer.  Oh, it did not like that!  I mistyped it somewhere.  Where did I mistype this?  Nine, nine, six, four, three, eight.  That's what I hate about these, this eight decimal place thing, is it's hard to keep track of all these little numbers.  Excellent!

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