1. Understanding How Hypothesis Testing Works
Two-Sample Z Tests
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Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

2. Understanding Hypothesis Testing
In previous lectures, we were content to learn how to perform a statistical test of hypothesis, following prescribed steps. This is more-or-less a cookbook type of approach and will certainly get the job done.
In this lecture what we really want to do us understand the process of hypothesis testing and the conclusion(s) we might reach thereby.
Understanding Two-sample Hypothesis Testing

3. Example: Two-sample test
A researcher compares men and women with regard to leadership aptitude. Test scores range from 0 (absolutely no aptitude) to 100.
Results are as follows:
Is there a difference between men and women? Test at α = 0.05. 
Women
Men X̅
83.7
74.3 s
16.0
18.0 n 64 54
Understanding Two-sample Hypothesis Testing

4. Here’s what we’ve been doing.
Example [continued]
Here’s what we’ve been doing.
H0: µ1 = µ2 H1: µ1 ≠ µ2
Computed
The computed Z value of 2.97 is in the region of rejection since it is > Thus, we Reject H0 at p < Men and women are different with regard to leadership aptitude.
Understanding Two-sample Hypothesis Testing

5. Understanding Two-sample Hypothesis Testing
The null hypothesis is a “straw man”: We have set it up only because we wish to knock it down. We are trying to reject H0.
The “claim” in this case is that there is NO difference between men and women when it comes to leadership aptitude. Just as with a one- sample test, we start by assuming that the claim is correct.
We try to Reject H0, the “straw man,” using the sample evidence.
Understanding Two-sample Hypothesis Testing

6. Understanding Hypothesis Testing [continued]
We want to determine whether the sample evidence of a 9.4 difference in leadership aptitude scores is likely to have occurred given the claim that men and women have the same aptitude.
There are two possibilities:
1) Men and women in the population are the same when it comes to leadership aptitude. The difference of 9.4 that we observed in the samples was just sampling error, i.e., chance.
2) The 9.4 difference between the two samples is too great to be explained by chance. Rather, the populations of men and women are different with regard to leadership aptitude.
Understanding Two-sample Hypothesis Testing

7. Understanding Hypothesis Testing [continued]
In this case, the sample evidence resulted in a Z value of Since this is a two-tail test, we have to double the probability to cover both sides of the distribution.
Using the cumulative Z distribution table, we look up the area from –∞ to and then the area from to + ∞. For the right side of the distribution we compute 1.00 – p.
Understanding Two-sample Hypothesis Testing

8. Understanding Hypothesis Testing [continued]
This represents the likelihood of getting the sample evidence - a difference of 9.4 or greater.
We find this combined area is a total of or, in other words, only 3 chances in 1,000.
.0015
.0015
-2.97
2.97
Understanding Two-sample Hypothesis Testing

9. Understanding Hypothesis Testing [continued]
So what we have found is that the sample evidence of a difference of 9.4 in aptitude scores, which resulted in a computed Z of 2.97, is not a likely outcome if the two population means are the same.
Note that the probability (the p-value) for this computed Z value is .003 ( ). This is a lot less than .05.
Understanding Two-sample Hypothesis Testing

10. p-value vs. α
We see that if you have the p-value, the probability of getting the sample evidence from the distribution set forth in H0, you do not need to first establish the critical value(s) in order to conduct these hypothesis tests.
There are many computer programs that give you the p-value so that you can compare that probability with the significance level α at which you are working.
Understanding Two-sample Hypothesis Testing

11. Do Your Homework Practice, practice, practice.
Do lots and lots of problems. You can find these in the online lecture notes.
Understanding Two-sample Hypothesis Testing