Using StatCrunch to perform hypothesis testing on two proportions of referee calls

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use StatCrunch to perform hypothesis testing on two proportions of referee calls.   Here's our problem statement: Since an instant replay system for tennis was introduced at a major tournament, men challenged 1429 referee calls with the result that 417 of the calls were overturned.  Women challenged 747 referee calls, and 230 of the calls were overturned.  Use a 1% significance level to test the claim that men and women have equal success in challenging calls.  Complete Parts A through C below.

Part A

OK, Part A says, “Test the claim using a hypothesis test,” and the first part asks us to determine the null and alternative hypotheses.  To do this, we need to consider the claim that's being made.  From the problem statement, the claim is that men and women have equal success in challenging calls.  That means that the proportion of men who challenge calls and get them overturned is the same as for the women.  So normally we would adopt that claim as our alternative hypothesis.  However, equality by definition belongs with the null hypothesis.  So the null hypothesis has to say that the two proportions are equal.

Well, we can't then use the claim for our alternative hypothesis because that is the null hypothesis.  So that means we have to take the compliment of the claim.  The complement of being equal to is being not equal to, and so we're looking for the option where the null hypothesis has the two proportions equal and the alternative hypothesis has the two proportions not equal.  And that's going to be answer option A.  I check my answer.  Nice work!

Now we’re asked to identify the test statistic.  And to do that, I'm going to pull up StatCrunch because it's easiest to get this information by using statistical software like StatCrunch.  Inside StatCrunch, I go to Stat –> Proportion Stats (because I'm dealing with proportions) –> Two Sample (because I'm comparing two different samples: one for men and one for women) –> With Summary (because I don't have actual data, just summary statistics).

In the options window, there are fields to put in summary statistics for both of the samples.  Ideally, it doesn't really matter which one you label Sample 1 and Sample 2 because in theory as long as you're consistent everything should work out.  However, my experience in working these homework problems from Pearson teaches me that it's often best practice to use Sample 1 for whatever of our samples is mentioned first.  So here in the problem statement, the men are mentioned first.  And so we're going to make that Sample 1.

So the number of successes here would be the number of calls that is overturned; that’s 417.  The total number of observations — that's the sample size.  And I'm going to put in the same information for the women.  My radio button for Hypothesis Test is already selected.  Notice we're doing a test on the difference.  So whereas the hypotheses that were listed here in our answer fields from the first part of the problem — they're not listed as differences; it's just one is on one side of the equals or inequality sign and the other is on the other — here it's actually organized to where it's a difference, and that's OK.

If the null hypothesis is that both of the proportions are equal, if I subtract p2 from each side (so I bring it over to the left and get this quantity here on the left side), I'm left with zero on the right.  So I'm just gonna leave this first field alone.  And then the inequality sign for my alternative hypothesis needs to match, and it does.  I press Compute!, and out comes my results window.  If I scroll over here to get the right side of this table, the next-to-last value in that table is always my test statistic.  So I'm going to put that here in my answer field.  Fantastic!

The P-value comes from that same table.  It's the last value listed.  Excellent!

Now we’re  asked to evaluate the P-value with respect to the significance level.  We're asked to use a 1% significance level.  43.6% is definitely greater than 1%, so the P-value is going to be greater than the significance level.  That means the area of the P-value is larger than the area of the significance level, so we can't fit the P-value area inside the area for the significance level.  So we're outside that critical region; therefore, we're outside the region of rejection, and we fail to reject the null hypothesis.  Because we fail to reject the null hypothesis, there's not sufficient evidence.  I check my answer.  Well done!

Part B

Now Part B asks us to “test the claim by constructing an appropriate confidence interval.”  Back here in StatCrunch, I could go back through all the menu options, or I could just go up here to the top left-hand corner of my results window and press the Options button.  In the drop down menu that appears, I press Edit, and I'm back to my options window.  I don't have to re-input all of these values that I put in earlier.  All I have to do is come down here, select the radio button for Confidence Interval, make sure my confidence level matches, and I press Compute!  Out comes my results window.

If I scroll over here to the right, I can see my upper and lower limits for my confidence interval.  So I'm going to put those here my answer field.  Excellent!

Now, what do we conclude from the confidence interval?  Well, when you're evaluating confidence intervals on two samples, typically you're going to be looking to see whether or not zero is included in the confidence interval.  Here in the confidence interval we've constructed, zero is inside the confidence interval.  Because zero is inside the confidence interval, that means that the two proportions that we're comparing could be the same.  If the difference between these two proportions is zero, that means they're equal to each other.  If they're equal to each other, that means they’re the same.

So because the confidence interval limits include zero, there does not appear to be a significant difference.  If they weren't equal, then there would be a significant difference, but zero is inside the confidence interval.  Therefore, they could be the same.  So then if they're the same, there's no significant difference between the two proportions, and therefore if we're going to ask about rejecting the claim that men and women have equal success, well, there's not sufficient evidence to warrant rejection of that claim because we actually have evidence that suggests they could be the same.  I check my answer.  Well done!

Part C

Now Part C asks, “Based on the results, does it appear that men and women have equal success in challenging calls?”  Well, from what we concluded right here with our confidence interval, we concluded that they're probably the same.  So that's what I'm going to select here.  Nice work!

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