Performing hypothesis testing on the variances of skull measurements

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 the variances of skull measurements.  Here's our problem statement: Researchers measured skulls from different time periods in an attempt to determine whether interbreeding of cultures occur.  Results are given below.  Assume that both samples are independent simple random samples from populations having normal distributions.  Use a 5% significance level to test the claim that the variation of maximal skull breaths in 4,000 BC is the same as the variation in 150 AD.

Part 1

OK, we've got our two samples here.  Sample size is n.  The sample mean is x-bar, and the sample standard deviation is s.  The first part of the problem asks us, "What are the null and alternative hypotheses?"  Well, the null hypothesis is by definition a statement of equality, so we're not going to select Answer option C.  Of the three answers that remain, we're going to be looking to see what is the alternative hypothesis so we can select the correct answer.

Typically, the alternative hypothesis is going to reflect the claim.  What's the claim here?  Well, here we're testing the claim that the variation of maximal skull breaths is the same for both of those years.  So we're looking for something that says they're equal.  Well, equality by definition belongs to the null hypothesis.  So we have to take the compliment of the claim as our alternative and say that they're not equal to.  And if I look at the answer options that are left, Answer option B is going to be the one we want because here it says not equal to.  Fantastic!

Part 2

The next part of this problem asks for the test statistic.  Notice the test statistic is an F-score.  The F-score is calculated very easily.  We can do that inside or outside of StatCrunch.  I'm going to use StatCrunch just because I'm a little lazy.  We could do it well enough in our calculator.  But like I said, I'm just a little bit lazy, so I'm just going to let the computer do it for me.  Hey, I love living in the 21st century.

So here we have StatCrunch. I'm going to go to Stat --> Variance Stats (because we're dealing in variance) --> Two Sample (because we have two samples we're comparing) --> With Summary (because we don't have actual data, just summary stats).  Here in the options window, we're asked for the variances.

But keep in mind that when you're looking for an F-score, there's one little caveat you've got to remember.  And that is typically when we're putting in our summary stats here in the option window, the sample that's listed first is normally Sample 1, and the sample that's listed next is Sample number 2.  But when you're dealing with an F-score, the sample with the greater variance or the greater standard deviation is always going to be Sample number 1.  So notice here the sample listed second has the higher standard deviation.  So even though the samples is listed second, it's actually going to be Sample number 1.

Notice also we're asked here for the variance, but we're given standard deviation.  So that means we've got to take this number and square it to get the variance. So here in my calculator, I'm going to put in the standard deviation that I'm given, I square it, and that gives me the sample variance.  Notice I'm putting all of the numbers in. I do the same thing for the other sample.

And now I perform a hypothesis test. I need to make sure that these values are correct.  This is typically always going to be 1 here for your claimed value.  And that's what we have here.  When you take any number and divide it by itself, if they're the same, you're going to get 1 out.  So we're just gonna leave that alone.  And then here we need to make sure that this inequality sign matches, and it does.  So now I hit Compute!, and here's my F-score right here in the table.  If you ever see an F-score that's less than one, that means you flip-flopped your samples.  So go back and reverse.  You've got Sample 1 and Sample 2 inappropriately labeled.  You've got to flip those around and put them in the right order.  That's a little caveat you got to remember when you're dealing with your F-score.  Fantastic!

Part 3

Now it asks for the P-value, and the P-value is there in my results window, that last value there at the end of the table.  I'm asked to round to three decimal places.  Fantastic!

Part 4

And now I'm asked for a conclusion for the hypothesis test.  Well, with a P-value of almost 80%, it doesn't it matter what significance level we would compare that with; anything that would be reasonable to use, we're going to be greater than.  So we're definitely outside the region of rejection.  Therefore we're going to fail to reject the null hypothesis.  Whenever you fail to reject the null hypothesis, there is always insufficient evidence.  So we want to select Answer option C.  Nice work!

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