Using StatCrunch to perform hypothesis testing on two matched pair means of acting award ages

Intro

Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework hep, Today we’re going to learn how to use StatCrunch to perform hypothesis testing on two matched pair means of acting award ages.  Here's our problem statement: The following data list the ages of a random selection of actresses when they won an award in the category of Best Actress along with the ages of actors when they won in the category of Best Actor.  The ages are matched according to the year that the awards were presented. Complete Parts A and B below.

Part A

OK, the first part of Part A asks us to find the null and alternative hypothesis for a hypothesis test.  We’re asked to use a 5% significance level to test the claim that, for the population of ages of best actresses and best actors, the differences have a mean less than zero, indicating that the best actresses are generally younger than the best actors.  OK, so the null hypothesis by definition is a statement of equality.  And the claimed value here is going to be zero; we get that from the problem statement.  And then the claim is just going to be that the main difference is less than zero, which is the claimed value.  I check the answer.  Well done!

Now the next part of Part A asks me to identify the test statistic.  To do this, I’m going to take this data, and I’m going to dump it into StatCrunch.  Here’s my data in StatCrunch.  I’m going to resize this window so we can see everything a bit better.  OK, here's my data in StatCrunch.

So now we’re looking at matched pairs here.  And the reason why we know they’re matched pairs is that there's a unique value for the actor age for every value of the actress age, and they’re uniquely paired together.  So that tells me I have matched pairs.

So in StatCrunch I’m going to go to Stat –> T Stats –> Paired.  And here in my options window, I need to tell StatCrunch where my data is located.  So which goes in Sample 1, and which goes in Sample 2?  Well, generally it's going to be the order in which they are listed in the problem statement.  Here the actress is listed first, the actor is listed second, so that’s the order I’m going to put them in here in my options window.

We want the hypothesis test to make sure these values match what we have earlier.  And now they do.  I hit Compute! and here is my test statistic in the results window.  The second to last number in that table in the results window is always your test statistic.  I’m asked to round to two decimal places.  Nice work!

And now the next part in Part A asks me to identify the P-value.  I also get that from the table here in my results window.  It's the last value in that table that you see right here in the end.  I’m asked to round that number to three decimal places.  Well done!

And now the final part of Part A asks me to make a conclusion on the hypothesis test. We’re going to compare the P-value here with the significance level.  Our significance level, if you remember, here is 5%.  So we compare that with our P-value, which is about 1.6%., so definitely less than 5%.  So we’re less than our significance level.  That means we’re inside the region of rejection, and therefore we’re going to reject the null hypothesis.  Every time you reject the null hypothesis, “there is sufficient evidence.”  I check my answer.  Nice work!

Part B

Now Part B has two parts to it.  The first part of Part B asks me to construct a confidence interval based on the hypothesis test that we just conducted.  I could go through the menu options in StatCrunch again, or I could just go up here to Options –> Edit and I'm back into my options window.  I click the radio button for Confidence Interval, and now I need to make sure I have the right confidence level.  So back here in StatCrunch, our significance level is 5%, but we’re looking at matched pairs.

So normally we would just take the confidence level as the complement of our significance level.  But because we have matched pairs, we have to take the complement of two alpha — so I’m taking the complement of 10%, not the complement of 5%. The complement of 10% is 90%.  And I could put the extra zero here, but I don’t have to; it’s all the same number.  I hit Compute! and here in my results window the lower and upper limits of my confidence interval.  So I come back down here.  I put those numbers here in my answer fields.  Nice work!

And the final piece asks, “What feature of the confidence interval leads to the same conclusion reached in Part A?  Well, here we’re going to go back and look at our null hypothesis.  Remember, our null hypothesis is that the mean of the differences is zero.  So we look to see if zero is part of our confidence interval.  Zero is outside of the confidence interval; in fact, zero is up here ahead of our confidence interval.  So all the values that we get in our confidence interval are going to be less than zero.

So the confidence interval contains only negative numbers and, because zero's not a part of that, that means we’re, you know, 90% confident that zero is not going to be the mean of the differences.  Therefore, we could say that the null hypothesis is going to be rejected, because that’s what the null hypothesis says.  The null hypothesis says that it’s zero, but here zero’s not a  part of our confidence interval, so we’re pretty confident it's not zero.  And therefore were going to reject the null hypothesis.  Nice work!

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