Finding the sample size needed to estimate a population proportion

Intro

Howdy! I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we’re going to learn how to find the sample size needed to estimate a population proportion.  Here's our problem statement: The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition.  He wants to first determine the percentage of adults who have heard of the brand.  How many adults must he survey in order to be 80% confident that his estimate is within six percentage points of the true population percentage?  Complete Parts A through C below.

Part A

OK, Part A says we should assume that nothing is known about the percentage of adults who heard the brand.  So the first step we’re going to have to take to calculate the sample size that we need is to find the critical value.  To do that, I’m going to open up StatCrunch so I can access the calculator inside StatCrunch.  I could also do this with the z-score tables, but I'm just going to use StatCrunch since it's my preference.

I’m going to pull up the Normal calculator. And we want the standard Normal distribution; that's the default here in the Normal calculator.  So then I want the two-tailed critical value, so I’m going to click the Between option.  And then here in the percentage [field], I’m just going to put in I want 80% confidence.  So that’s 80%.  I hit Compute!.  There are my two critical values; I really only need the positive one (1.28), so that's what I'm going to use.

Now that I’ve found the critical value, I can actually substitute into my equation.  Here's my equation for sample size, and I just substitute in what I have.  So 1.28 was the critical value we just found.  This is where we know it's a two-tailed area — z-alpha-over-2.  Alpha over two says we want to split alpha amongst the two tails of our distribution.  So that’s how I know it's two-tailed.

So that's my critical value 1.28.  We don't know anything about the percentage of adults who part of the brand.  In that case, the most conservative percentage that we can collect for p-hat is going to be one half, which then means q-hat is also one half.  ½ times ½ gives us 0.25.  And then we want to be within six percentage points.  So notice how we write that here — 6%, six percentage points of the true population percentage.

Now that I've got my numbers substituted into my equation, all I need to do now is just calculate that out.  We get 113.7 and the seven is repeated off into infinity.   We’re asked to round up to the nearest integer so we get this partial person counted for.  Since we can’t really count partial people — we’re only counting whole people — we’re going to round up to the nearest integer.  That gives us 114, so I put that here in my answer field.  Fantastic!

Part B

Now the second part, Part B, asks that we repeat the calculation, but now we are going to assume that 79% of adults have heard the brand.   We can easily repeat the calculation with the new numbers.  So I’m going to go back here and start over with my original equation for determining sample size, and now I plug in new values.

We’re still 80% confident of the estimate, so our critical value doesn't change.  And we still want to be within six percentage points of the true population percentage, so this value for E doesn’t change.  The only thing that changes are values for p-hat and q-hat.  We want to be — a survey suggests that 79% of adults of brand, so that's our proportion of success, which then means the proportion of failures is going to be the complement of that.  So we’ll just subtract 0.79 from 1 to get 0.21.  And now we’ve got new numbers to put into our equation.  We crunch those out of our calculator, and we get 75 with a bunch of decimals behind it.  So we’re just going to round up to the next nearest integer to give us 76.  So I put that here my answer field.  Nice work!

Part C

And now the last part, Part C, asks, “Given that the required sample size is relatively small, could he simply survey the adults at the nearest college?”  Well, what kind of sample do you have when you sample what's nearby?  I hope you said, “A convenience sample,” because that's what we've got.  And of course, convenience samples are biased samples, so we don't want to be sampling with that methodology.

So we going to say, “No, you shouldn't just simply survey the adults of the nearest college because it's a convenience sample.”  So we look at our answer options and find the one that corresponds with that.  I check my answer.  Nice work!

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