Finding the P-value given the test statistic

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today, we're going to learn how to find the P-value given the test statistic.  Here's our problem statement: Use technology to find the P-value for the hypothesis test described below.  The claim is that, for a smartphone carrier’s data speeds at airports, the mean is mu = 18.00 Mbps.  The sample size is n = 29, and the test statistic is t = 2.074.

Solution

OK, finding the P-value here is really easy if you understand one simple concept.  And that is that the P-value is the area in the tail of the distribution bounded by the test statistic.  And that’s why the test statistic is given here, because it provides the boundary for that area in our distribution.

But which distribution are we going to be using?  Well, look at your test statistic.  Your test statistic is a t-score.  That means we’re going to be using the Student-t distribution.  So I’m going to call up StatCrunch here, and inside StatCrunch, I’m going to go to Stat –> Calculators –> T because I want the Student-t distribution.

Now here’s my Student-t calculator.  The degrees of freedom is one less than the sample size, and that’s why they gave us the sample size here.  In this case, it’s 29, so our degrees of freedom will be one less than that, which is 28.

And then we need to get this inequality sign here right, and that’s got to match our alternative hypothesis.  Well, to get the alternative hypothesis, we have to look at the claim.  The claim here is that mean value equals 18.  Well, equality by definition belongs with the null hypothesis, so we can’t adopt the claim as the alternative hypothesis, which means we have to take the complement of this.  The complement of being equal to is being not equal to, and not equal to means we’re going to have a two-tailed test.  So I’m going to come up here in my distribution calculator and select the Between option, because the P-value is actually split between the left and right tail of my distribution.  And now I’ve got two test statistics: One is going to be positive, and one is going to be negative.  So I’m going to put those values in here.

And now I’ve got everything that I need, I go ahead and hit Compute! and out comes the area in between the tails.  Remember that in StatCrunch, this Between option is calculating the area in between the tails.  But the P-value is the area of the tails, so I have to take the complement of this area that’s between the tails to get the area of the tails.  So I call up my little calculator here, take 1 minus this value here, and there is my P-value.  I’m asked to round to three decimal places.  Good job!

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