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

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 equality.  Here's our problem statement: The test statistic of z = 2.45 is obtained when testing the claim that the population proportion is greater than 30%.  Part A: Identify the hypothesis test as being two-tailed left held or right tailed.  Part B: Find the P-value.  Part C: Using a significance level of alpha (α) = 5%, should we reject HO or should we fail to reject HO?

Part A

OK, Part A wants us to identify if the hypothesis test is being two-tailed, left-tailed or right-tailed.  To do that, we simply look at the claim that we're making.  Remember that this often comes from the alternative hypothesis that we create, and very often the alternative hypothesis reflects the claim that's being made.  Here the claim that's being made is that the population proportion is greater than 30%.  If you look at the inequality sign — the greater-than sign — it's like an arrow pointing to the right that indicates that this is a right tailed test.  So I'm gonna put that here in my answer field.  Well done!

Part B

Now Part B asks us to find the P-value.  To do this, I'm going to pull up StatCrunch and get into my Normal calculator.  The only thing we have to go on is the z-score, and so that's why I'm pulling up the Normal calculator.  So to do that, I go to Stat –> Calculator –> Normal.  And here in my Normal calculator, I'm going to reflect the alternative hypothesis, or the claim here in my problem statement.  But in order to do that, I have to do that through the z-score.

Notice the default values that come up in my Normal calculator are for the standard normal distribution.  This is what we want, so I'm going to leave those values alone.  Then here are my probability fields down below.  I'm going to make sure that this inequality sign matches the tests that I'm running.  In Part A we concluded that we're doing a right tailed test, so that means this inequality sign needs to be greater-than-or-equal-to here.  In the next field, I'm going to put the z-score that they give me.  And now I press Compute! and this area underneath the curve that's shaded in red on your Normal calculator is this number here — 0.007.  And that is the P-value.  So I'm going to put that here in my answer field.  Fantastic!

Part C

And last but not least, Part C asks us to choose the correct conclusion below.  We're either going to fail to reject HO or we're going to reject HO.  And the way we do that is by comparing the P-value to our α level.  The α level that we’re given here is 5%.  The P-value that we're given is seven tenths of 1%, so the P-value here is less than our α value.  Therefore, we are in the region of rejection, and we're going to go ahead and reject HO and say that there is sufficient evidence to support the claim.

Rejecting HO and saying there's not sufficient evidence doesn't add up; it's not consistent.  So if we reject HO, there's going to be sufficient evidence to support the claim.  If we fail to reject HO, there's going to be insufficient evidence to support the claim.  Here we're actually going to reject HO because our P-value is less than our α value.  Well done!

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