Performing mean hypothesis testing on the number of words per page

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 mean hypothesis testing on the number of words per page.  Here's our problem statement: A simple random sample of 10 pages from a dictionary is obtained.  The number of words defined on those pages are found with the results: n = 10, x-bar = 55.3 words, s = 16.6 words.  Given that this dictionary has 1,456 pages with defined words, the claim that there are more than 70,000 defined words is equivalent to the claim that the mean number of words per page is greater than 48.1 words.  Use a 0.05 significance level to test the claim that the mean number of words per page is greater than 48.1 words.  What does the results suggest about the claim that there are more than 70,000 defined words, identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.  Assume that the population is normally distributed.

Part 1

OK, the first part of this problem asks us for the null and alternative hypotheses.  The null hypothesis is by definition a statement of equality, so Answer option C is not going to be the right answer because this null hypothesis is not a statement of equality.  Of the three answer options that remain, let's look at our alternative hypothesis.  And normally we get that from the claim.  The claim here from our problem statement --- well, there's two claims: There's an original claim, and then there's a modified claim.  And it looks like we're going to be using the modified claim to define our null and alternative hypothesis.  And that is that the mean number of words per page is greater than 48.1 words.  So we take the one that says greater than 48.1.  And here it is right here, Answer option B.  Nice work!

Part 2

OK, the second part of this problem asks us for the test statistic . And to find the test statistic, we're going to pop out StatCrunch.  So I'll take this window, and I'm going to resize it so that we can see better what's going on here.  OK, inside StatCrunch, I go to Stat --> T Stats (because I don't know the population standard deviation) --> One Sample (because we have just the one sample) --> With Summary (because we don't have any actual data). Here in the options window, I need to put in my sample statistics from the problem statement.  We have those right here, so I'm just going to take that information and stick it in here.  The sample mean is x-bar; that's the 53 --- excuse me, 55.3.  And then the sample standard deviation is going to be s; that's the 16.6.  Sample size is going to be n, and that's 10.

Check for this radio button for Hypothesis test.  That's the default selection.  We want to keep that because we're performing a hypothesis test.  We want to make sure this area matches what we selected here in the previous part of the problem.  So we need to change this claimed value from zero to 48.1.  And then I need to make sure that this inequality sign matches what we have over here for our alternative hypothesis.  And now I've got everything I need. I press Compute!, and here in my results window, the second to last value is the test statistic.  I'm asked to round to two decimal places.  Excellent!

Part 3

Now the next part of the problem asks for the P-value.  We've already done all the work to calculate it.  Look back here at the results window.  It's that last value there in the table, right next door to our test statistic.  We're asked to round to three decimal places.  Nice work!

Part 4

And now the last part of this problem asks us to state the final conclusion.  To do this, we're going to compare our P-value with our significance level we have here in the problem statement.  It says, "Use a 5% significance level." Here, we've got a P-value of over 10%. 10% is well above 5%, so we're outside the region of rejection.  And when you're outside the region of rejection, you fail to reject the null hypothesis.  Every time you fail to reject, there is not sufficient evidence.  And here our original claim was that there was more than 70,000 defined words.  Excellent!

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