Constructing and using a proportion sampling distribution table

Intro

Howdy!  I'm Professor Curtis of Aspire Mount Academy here with more statistics homework help.  Today we're going to learn how to construct and use a proportion sampling distribution table.  Here's our problem statement: A genetics experiment involves a population of fruit flies consisting of three males named Alex, Bart, and Christopher and one female named Debbie.  Assume that two fruit flies are randomly selected with replacement.

Part A

OK, Part A of this problem says, "After listing the possible samples and finding the proportion of females in the sample, use a table to describe a sampling distribution of the proportion of females."  OK, so the first thing we need to do here is make a table of the possible outcomes in our sample space and then look at the proportions of females in each of those individual outcomes.  We can then sort that information in order to produce our probability table.  It would probably be easier to do this in Excel because Excel has much better sorting functionality then StatCrunch.  So let's just go ahead and use Excel for that.

And let's just list the different possible outcomes in our sample space.  So we're selecting two fruit flies from among the four that are in the population, and we're selecting with replacement.  So the first one we select goes back in to possibly be selected again.  So let's just say the first one we pick out is Alex, we put them back in, and then we pick him again.  Or we could pick Alex for the first one and then Bart for the second one, or Alex for the first one and then Christopher for the second one, or we could pick Alex for the first one, and Debbie for the fourth one.

Our second one --- really fourth one here in the series.  So ... that's the possible subset.  And I look at the pattern here.  So I've got the first one repeated four times, and then I've got each one listed in sequence.  So I could actually, if I wanted to, I could just repeat that pattern three more times.  Whoops.  One for each of the individual fruit flies.  And then here I'm just going to put in B four times, and then C four times, and then D four times.  This is the pattern that we've got established here, right?

So now I look at each row.  Each row is a sampling, and I'm going to say, "OK, what's the proportion of females?"  That's what we're looking for here, the proportion of females in each one of these samples.  Well, Alex is male. In fact, the only female we've got, here is Debbie.  So the proportion of females here is going to be 0%.  Zero.  Zero.  Here 50% is female --- Alex is male, Debbie's female, one half is 50%.

And I just go through and mark the others the same fashion.  So whoops, that's 100%.  So now I've got all my, oh, I've got all my proportions out.  So now I'm going to select everything and I'm going to come up here to Data and select Sort. I'm going to sort on column C, smallest to largest.  Boom, baby!  So now it's really easy to get what I need because all I got to do is --- notice zero is the first number, zero here, the first number in our table.  We've got nine zeros out of 16 total.  So the probability is the part over the whole; the part is 9, the whole is 16.  I do the same thing for each of the numbers in sequence. So I've got six .. for 50%.  6 over 16 can actually reduce to 3 over 8.  And then of course there's only one value for that last option there.  So I check my answer.  Fantastic!

Part B

Now Part B says, "Find the mean of the sampling distribution."  The mean is actually best found in StatCrunch.  We could actually do it here in Excel, but it's --- it's just that I'm lazy.  So let me go ahead and open up StatCrunch, pop the window out so we can actually move around and do something with it.  Alright, so here in StatCrunch, I'm going to actually transfer these numbers over here.  So it's 0, 5, 1, and then I'm going to actually label these.  We could label these Proportion --- oh, I got my caps lock on --- Proportion and Probability.  Right.  The probability here is 9 over 16, so ... will this calculate it for me?  Oh damn.  Ugh!  Stupid computer.  Where's my calculator?  Hey, there's my calculator.  The actual number here --- 0.5625.  And three eighths?  I should probably know that one.  But again, I'm lazy.  And of course, one sixteenth.  Gotcha.

OK, now I've got my probability table here in StatCrunch, so now I can just go up to Stat --> Calculators --> Custom.  The values are the proportions, and the weights are the probabilities.  E voila! 25%.  It's so easy.  Now, nice work!

Of course, you know, I can actually get the same thing in Excel.  I mean, if you really wanted, I can come back here in Excel, I could actually put in that probability.  This would actually calculate it for me.  Ooh, yeah, I like that, 9 divided by 16.  Oh yeah, baby!  1 divided by 8, and 1 divided by 16.  Oh yeah, baby!  So here in Excel, what we would do is I'm going to take, and I'm going to multiply each of those proportions by its corresponding probability, and then copy of that down so I can get the same thing there.  Then I'm going to sum that up.  And there's my mean.  Again, a little calculation intensive.  I'm lazy.  I like StatCrunch to kind of do everything for me.  But if you wanted to stay in Excel, there's the way you would actually get the same answer.

Part C

Now Part C asks, "Is the mean of the sampling distribution from Part B equal to the proportion of --- population proportion of females? If so, does the mean of the sampling distribution of proportions always equal the population proportion?"  Well, you should recall from lecture --- well, if you're watching, you know, Aspire Mountain Academy lecture videos, you'll know you should know this.  If you're not, well, I hope that your instructor pointed this out to you.  It is in the textbook.

There is, you know, a listing of population parameters that are biased and a listing that are unbiased.  And proportion is one of those population parameters that is an unbiased estimator.  So we would expect that the mean of the sampling distribution would be the same as the population parameter, in this case, proportion.  And we see that that's so; the population has only for members to it.  There's four fruit flies in the population.  Three of them are male, one of them is female.  1 out of 4 is 25%.  So yeah, the mean of the sampling distribution does equal the population proportion, and that's because the proportion is an unbiased estimator.  So it looks like this answer option is the one we want.  Excellent!

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