Using a z-score to complete hypothesis testing for a claim of inequality

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 a z-score to complete hypothesis testing for a claim of inequality.  Here’s our problem statement: The test statistic of z = 1.25 is obtained when testing the claim that p does not equal 0.2978.  A: Identify the hypothesis test as being two-tailed, left tailed, or right tailed.  B: Find the P-value.  C: Using a significance level of alpha equals 10%, should we reject H-naught, or should we fail to reject H-naught?

Part A

OK, so Part A is asking for the type of hypothesis testing we’re conducting.  So to do that, we look at our claim.  And we see that the inequality for our claim is not-equal-to.  That’s the telltale sign that this is going to be a two-tailed hypothesis test.  If this had said that p is less than, then that would be like an arrow pointing to the left, and we would select the left tail test.  Or if it had said p is greater than, and that would be like an arrow pointing the right, and we would select right tailed.  But here we have not-equal-to, so that's the sign we have a two-tailed test.  Well done!

Part B

And now, Part B asks us for the P-value.  To get the P-value, I’m going to go back to StatCrunch and call up the Normal calculator.  You can use the z-score tables if you want — you’ve got the links here to that in your problem — but I find it's much easier to just use StatCrunch.  So here in StatCrunch, I go to Stat –> Calculators –> Normal.  And I want the standard Normal distribution, and that’s the default here in the calculator, so I don't need to adjust the mean and standard deviation.

But I do need to select this Between option here because we do have a two-tailed test.  So I’m going to select the Between option.  So now I’m going to get — I want the area for my tails.  The area that’s going to come out of the calculators is going to be the area between the tails, but that’s OK; we can just take the complement to get the area of the tails.  And that's were looking for.  This z-score is actually bounding the area of the tails, which is the P-value that we’re looking for.

So all I need to do here is stick in my z-score on the left, z-score on the right, hit Compute!, and this is the area in between the tails.  So to get the actual area of the tails, I’m going to take my calculator here and I'm going to subtract 1 from this number — or rather this number from 1.  And there's my area in the tails, which is my P-value.  So I’m going to put that here rounded to three decimal places.  Well done!

Part C

And now the last part, Part C, asks us to choose the correct conclusion below.  In the problem statement we’re asked to use a significance level of alpha equals 10%.  So we compare our P-value with the alpha level, and our P-value is greater than the alpha level.  So therefore we’re not in the critical region; we’re actually outside the critical region in between the tails.  So we’re not in the tails of distribution.  We’re actually in the center of our distribution, and so therefore we do not have sufficient evidence to support the claim.  We’re going to fail to reject the null hypothesis.  I check my answer.  Nice work!

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