Performing hypothesis testing on two proportions of common attributes

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform hypothesis testing on two proportions of common attributes.  Here's our problem statement: Two different simple random samples were drawn from two different populations.  The first sample consists of 40 people with 19 having a common attribute.  The second sample consists of 1800 people with 1,272 of them having the same common attribute.  Compare the results from a hypothesis test of p1 = p2 where the 1% significance level and a 99% confidence interval estimate of p1 - p2.

Part 1

OK, the first part of this problem asks us for the null and alternative hypotheses for our hypothesis test.  The null hypothesis is by definition a statement of equality, so we know we're going to be selecting Answer option B, Answer option C, or Answer option D.  To distinguish between these three options to find the correct answer, we need to look at the alternative hypothesis.  Notice they're all different in each case.

The alternative hypothesis typically reflects the claim unless the claim has any semblance of equality to it, in which case we're going to take the compliment for our alternative hypothesis.  If you read the problem statement up here, you'll notice that the typical words testing the claim that don't appear here in the problem statement, so we have to kind of infer what's going on from the words that were given.  Here it says, "Compare the results from our hypothesis test of p1 = p2."  Right here, what we're saying here is that the claim is that the population proportions are equal to each other.  That's the claim.  Well, equality by definition belongs to the null hypothesis, so therefore we want to take the compliment of this.  And that's going to be not equal to, so it looks like C is going to be our correct answer here.  Excellent!

Part 2

The next part asks for the test statistic. To get the test statistic, we need to run a hypothesis test and so I'm going to open StatCrunch.  And we're going to move this window here so we can see better everything that's going on.

OK, here in StatCrunch, I don't need any actual data because I've got summary statistics here in the problem statement.  So to run a hypothesis test, I'm first going to go to Stat --> Proportion Stats (because we're dealing with proportions) --> Two Sample (because we have two samples we're comparing) --> With Summary (because we don't have actual data, just summary statistics).

Here in the options window, I need to provide the summary stats for each of my samples.  In the problem statement, it says the first sample consists of 40 people with 19 having the common attribute.  So the success here is having the common attribute. So that's going to be 19.  And there's 40 total in that sample.  And then I do the same thing for the second sample.  The radio button for our hypothesis test is already selected, so I'm good there.

Notice how StatCrunch writes the hypothesis test a little differently than what you see in your assignment.  So you have to do a little bit of algebraic equivalence here.  If p1 is equal to p2 and I subtract p2 from each side, then that's going to leave zero on the right side.  Or looked at another way, if I were to say that these two are equal and I subtract them, any number minus itself is going to be zero.  So this is the number that we want for our claimed value here in our options window.  Make sure that this inequality sign for your alternative hypothesis matches, and it does.

So now we've got everything we need. I press Compute!, e voila!  Here at the very end of the table, second to last value, I see my test statistic, which is a z-score.  Nice work!

Part 3

The next part asks us to identify the critical values.  The critical values are going to come from the distribution that we're using.  And because we're dealing with proportions and we have a z-score for our test statistic, we're going to need to use the standard normal distribution, because z-scores come from the standard normal distribution.  So back here in StatCrunch, I go to Stat --> Calculators --> Normal.

To know whether we use a one-tail or two-tail test, we need to go back here and look at our alternative hypothesis here.  The alternative hypothesis has an inequality sign of not equal to; that means we have a right tail test --- excuse me, a two tail test --- one tail on the right and one tail on the left.  And so therefore we want to use the Between option here in our calculator.  So we've got two tails that we're looking at for our distribution.

For our critical region, the significance level is 1%.  That means there's 1% of the area under the distribution curve found the tails of the distribution.  That means the area in between, which is what StatCrunch is measuring here, is going to be 99%.  So I just stick 99% here, hit Compute!, e voila!  Here are my critical values that I need to enter here in my answer field.  And I could enter in the negative and then a comma and then the positive one, but I'm a little lazy.  So I just like to use this plus or minus sign, and then that way I don't have to type the number twice.  I only type the number once.  Excellent!

Part 4

Now the next part asks for a conclusion based on the hypothesis test.  Look here, we're asked to use the test statistic to evaluate the hypothesis test.  And that's easy enough to do.  So here we have our critical values plus or minus 2.57, so that's going to be the edge here of this red area.  The test statistic itself, -3.17.

Well, -3.17 is located here on the number line, which is inside the left tail of our critical region.  Since we're inside the critical region, we're inside the region of rejection, and therefore we fail to reject --- excuse me, we reject the null hypothesis because we're in the reason of rejection.  So here the test statistic is in the critical region, so we reject the null hypothesis.  And whenever we reject the null hypothesis, there is always sufficient evidence.  Good job!

Part 5

Now the next part asks for a 99% confidence interval.  We could go through the menu options again, but yeah, I'm a little bit lazy.  So I'm just going to go back here to this previous window that we had.  Click on the Options button, and in the drop down menu, select Edit.  That takes me back to the options window where all I do is select the radio button for confidence interval, make sure my level is at the right level, and hit Compute!, e voila — the lower and upper limits for my confidence interval, which I can stick here in my answer field.  I'm asked to round to three decimal places. Nice work!

Part 6

And now this next part asks for a conclusion based on the confidence interval.  Well, we have to look and see where is zero. Is it inside or outside of the confidence interval?  And there's the lower limit.  There's the upper limit.  Zero is going to be outside.  If we look at this as a number line, zero is going to be over here to the right of our interval where zero is outside the interval.  So it's not included in the interval.  And that means there's going to be a difference between the two parameters that we're evaluating.

And so that means we're going to reject the null hypothesis, because obviously the null hypothesis here are saying that they're the same.  Well, our confidence interval says that can't be.  So therefore we're going to reject this because it's a false statement.  Nice work!

Part 7

And now the last part of the problem asks, "How do the results from the hypothesis test and the confidence interval compare?"  Well, notice here in both cases, we rejected the null hypothesis.  So the results are going to be the same.  In each case, we suggested that the population parameters are not equal to each other, because we rejected the null hypothesis which says that they are equal to each other.  I check my answer.  Nice work!

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