Performing hypothesis testing on means of right and left hand reaction times

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 means of right and left hand reaction times.  Here's our problem statement: Several students were tested for reaction times (in thousands of a second) using their right and left hands.  Each value is the elapsed time between the release of a strip of paper and the instant that it is caught by the subject.  Results from five of the students are included in the graph to the right.  Use a 5% significance level to test the claim that there is no difference between the reaction times of the right and left hands.

Part 1

OK, the first part of this problem is asking us for the hypotheses for the test.  So in this first dropdown blank here, it says let μ-d be the blank of the right and left hand reaction times.  μ-d is going to be the mean of the differences.  That's how μ-d is defined.  The null hypothesis will always carry an equal sign.  So we select that there.  And then for the alternative hypothesis, we look to our claim.  The claim that we're testing here, it says, "Test the claim that there is no difference" --- in other words, that they're the same.  Well, we can't really set this to be equal, because equality by definition belongs to the null hypothesis.  So then we have to take the compliment of that.  And the complement of being equal to is being not equal to.  So now we check our answer.  Good job!

Part 2

Now the next part asks us for the test statistic.  And to do that, we're going to access the data here, and we're going to dump the data into StatCrunch.  I resize the window here so we can get a better look at what's going on.  And now with the data here in StatCrunch, I'm going to go up to Stat --> T Stats --> Paired (because for every left hand there, there's a corresponding right hand for the same student).

Here in my options window, the first sample is just going to be the variable that's listed first.  And the second sample is going to be the variable that's listed second, I look down here, and the default selection is for hypothesis testing, so I don't have to change that.  And then I make sure these fields match what we got here earlier for our null and alternative hypothesis, and they do.  So now I press Compute!, and here in my results window is my test statistic, the second to last value listed there in the results window.  So I'm just going to go ahead and type that in.  It says round to three decimal places --- 72.418.  Well done!

Part 3

Now the next part of the problem asks for the critical values.  Notice the critical values are not listed here on the results window from the hypothesis testing that we just did.  And so we're going to have to go to our distribution calculator in order to calculate the critical values.  And the distribution that you want to look at is the t value --- is the t distribution.  Because look at the values that were listed here.  So your test statistic was a t score, and the critical value they're asked for is a t score.

So we're going to go up to Stat --> Calculators --> T.  Here in my distribution calculator, the first thing I want to look at is the alternative hypothesis because this tells me, "Do I have a one-tail test or a two-tail test?"  And the alternative hypothesis is not equal to, so not equal to means we have a two-tail test.  So I'm going to click the Between option, and I'm going to have two critical values: One on the left, one on the right.

Now to get the correct critical value, I need first to enter in the corresponding value for degrees of freedom.  We've got five data points you can see here, or you could actually --- if you wanted to, you could go back and count them here in your StatCrunch window.  So we've got five pairs.  Degrees of freedom is one less than that.  So I'm going to put 4 here for my degrees of freedom.  And then for --- remember the Between option here in StatCrunch lists the area in between the tails.  So the area of the tails, the significance level, which here it says we want to use 5% --- so the area in between has to be the complement of 5%.  So I subtract that from 1 and get 95%.  So there's 95% in between the tails.  Then I just press Compute!, and here are my critical values which I can now put here in my answer field.  And it wants three decimal places.  Nice work!

Part 4

Now this last part of the problem is asking us for a conclusion.  To conclude our hypothesis test, notice there's nothing here with P-values.  So we're going to have to do it straight with the test statistic and the critical value, which is easy enough to do here in StatCrunch in the results window.  Notice we have our critical values listed here.  So those values are marking the boundaries of our tails there in our distribution.  And what we're going to compare it with is the test statistic.

Here's the test statistic: -2.4.  So -2.4 is going to put me here in the central region, because notice the boundary here is negative 2.7.  So -2.4 is gonna put me just inside that central region in between the tails.  So since I'm not in the tail, I am therefore not in the critical region.  I'm not in the region of rejection, and therefore there's not sufficient evidence.  We're going to fail to reject the null hypothesis.  And whenever you fail to reject the null hypothesis, there's not sufficient evidence.  Of course, the claim here is that there is no difference between the reaction times.  Excellent!

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