 Hi, this is Dr. Don. This problem is about estimating the minimum sample size for a proportion. We're given a statement that a researcher wishes to estimate with 99% confidence, which is our C. The population proportion of adults who think the present of their country can't control the price of gasoline. Her estimate must be accurate within 2% of the true proportion. 2% then is the margin of error, which would be .02 in decimal form. No preliminary estimate of the proportion is available. Find the minimum sample size needed. Well, if you do the math, and I'll show you an example in a minute, if we assume that the proportion is .5, 50%, that will give us the maximum minimum sample size. So if we are able to refine our estimate of the true proportion, we can reduce the minimum sample size. So let's look at that in stat crunch. Okay, I've opened stat crunch. And to solve this, we'll go to stat, which is where we go for most things. We want to go down to proportion stats, which we're dealing with proportion. We have a single sample. And now we want to get the sample size and we're given the width or the margin of error, which enables us to find the width. So I'll click on that. And it brings up this dialogue box and just ignore that fancy title up there. We need the confidence level C and our case is .99. We don't know the target proportion. So we're going to assume it's .5. As I said, that will give us the maximum minimum, if you can get around that mind bender there. Our width is twice the margin of error. Our margin of error was .2%, which would be .02. So the width is going to be .04. And we just click compute. And we get 41, 47 for our minimum sample size, assuming we do not know the target population. And I'm going to drag this down and we can check and see that the answer is 41, 47. The next part is what is the minimum sample size if we know or think that the proportion is 44%. So let's go back here. And as I said, if you don't know the proportion and you assume .05, that will give you the maximum minimum call up step crunch. Again, we want to change this to .44. And I want to round this off again to our width of .04 and click compute. And now our minimum sample size drops to 4.088. And let's drag that down. And we'll see, yes, that's the right answer for Part B. Part C is how do results from Part A and Part B compare? And the answer is having an estimate of the population reduces the minimum sample size needed. And again, I'll show you that. If we don't know the proportion, if we just put in .5 and .04 for the width, remember we got 41, 47. If I put in .4 and .04 there for my width again, you can see it drops down to 39.81. If I had .6, that's on the other side of the curve, that's 39.81. If I go down to .25, .25, compute 31.11. Just change here for a second. I've sketched out a graph here showing n sample size on the y-axis and the proportion p on the x-axis. And what we have is a curve or shape, something like this. The maximum minimum sample size n occurs at .5, which is up here. As p gets smaller, our estimate of p gets better, and it's smaller than .5 in decreases. And also as our estimate of p greater than .6, you can see also .5, excuse me, that in decreases. So the worst case, which would be the maximum minimum size is when we don't know how to estimate the proportion. We use .5 and any estimate that we have will give us a smaller minimum size than .5. So I hope that helps.