Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today were going to learn how to create a probability distribution table for a proportion sampling distribution. Here’s our problem statement: Three randomly selected households are surveyed. The number of peoples in the households are 3, 5, and 10. Assume that samples of size n = 2 are randomly selected with replacement from the population of 3, 5, and 10. Construct a probability distribution table that describes the sampling distribution of the proportion of odd numbers when samples of sizes n = 2 are randomly selected. Does the mean of the sample proportions equal the proportion of odd numbers in the population? Do the same proportions target the value of the population proportion? Does the sample proportion make a good estimator of the population proportion? Listed below are the nine possible samples.
OK, we see here the list of nine possible samples, and the first part asks us to construct a probability distribution table. Many students see this type of problem and it's really intimidating because they just have no clue how to proceed to solve this problem. But once you understand how to do it, it's really pretty simple. I find that this type of problem is actually easier to work in Excel than in StatCrunch. You can work it inside StatCrunch, but for this type of problem StatCrunch is really clumsy.
So I'm going to work this problem in Excel because Excel does it much quicker, and it’s much easier to manipulate what we need to do in Excel than it is in StatCrunch. So here's my data in Excel. Now we’re going to enable editing. And I’m going to adjust my window here so we can actually see what is going on. OK, now we have the data here in Excel.
So what we’re going to do first is look for the proportion of odd numbers in each of the samples because here the problem statement we’re looking for the sampling distribution of the proportion of odd numbers. So all we do is to look and see how many odd numbers do we have in each sample. So here I’ve got two, and there’s two numbers total, so that's 100%, which is 1. Here I’ve got again two odd numbers, so that's also 100%. Here I've got one odd number, so that's 50%. And here we’ve got two odd numbers, so that’s 100%. And I’m going to keep going down each individual row here. Alright, there's my proportions.
And this border here drives me nuts, so I’m going to get rid of it here real quick. Let me blow this up so I can see what I’m doing. Whoops. Here we go. Now we’re all set. So here I got all the proportions. And now what I want to do is copy this column, and I’m going to stick it over here. We’re interested in just the numbers only. And I’m going to sort the column by going to Data and then I can go to Sort, or if you want you can just click this little quick button here because I want to sort from smallest to largest.
So now we see here that we've got three numbers in our distribution. The first one is zero, so I’m going to set that here. The next one is one-half, and the last one is one. Probabilities! The probabilities are just the “part” over the “whole.” How many zeros do I have? I’ve got one, and there’s nine numbers total, so therefore my probability is the “part” over the “whole” — 1 over 9. I’m going to do the same thing for each of the other numbers. Here I’ve got four out of nine total, so 4 over 9. The same thing with the 1; there’s 4 of them over 9 total. I check my answer. Fantastic!
Now, the next part of this problem asks me to choose the correct answer below. There’s four different options here, so let’s see what we’re looking at. Here we’re comparing the proportion of odd numbers with the mean of the sample proportions. So let's go ahead and calculate that here in our cell. How many odd numbers do we have in our data set? We’ve got one, two, three, four, and I’m going to go through here and select all the odd numbers. Oh, I’m holding down the Shift key; no wonder it doesn't work. I have to hold down the Ctrl key to select individual cells. That’s why that wasn't working. And it looks like — yep, that’s all the odd numbers. Down here my count is 12 out of 18 total. So I can calculate that here; 12 over 18 is going to be two thirds.
I can do the same thing with the mean of the sample proportions. Here are my sample proportions. All I have to do is take the average of that column. And it looks like they're the same. So the proportion of odd numbers in the population is equal to the mean of the sample proportions of odd numbers. So I select the answer option tells me that. Whoops! I selected the wrong one. This says “proportion of even numbers” and I want the proportion of odd odd numbers. Good job!
And now, the last part of this problem says, “Choose the correct answer below.” Again, we got four answer options here, and it looks like these answer options are talking about how good of an estimator sample proportions are. Well, sample proportions are unbiased estimators, so they're going to make a good estimator of the population proportion. And we can see an example of that here with these two numbers that we calculated from the previous part of the problem. This number represents the population, and this number represents the sample. And we can see that the two are equal, so the sample statistic is targeting the population parameter. So it makes a good estimator, so I’m going to select the answer option that tells me that. So they do target, so we don't want A or B; we want C or D. And it does make a good estimator, so you want answer option C. Nice work!
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Frustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help.