Understanding How Hypothesis Testing Works TwoSample Z Tests

Understanding How Hypothesis Testing Works Two-Sample Z Tests Each slide has its own narration in an audio file. For the explanation of any slide click on the audio icon to start it. Professor Friedman's Statistics Course by H & L Friedman is licensed under a Creative Commons Attribution-Non. Commercial-Share. Alike 3. 0 Unported License.

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 2

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: Women Men 83. 7 74. 3 X s 16. 0 18. 0 n 64 54 Is there a difference between men and women? Test at α = 0. 05. Understanding Two-sample Hypothesis Testing 3

Example [continued] Here’s what we’ve been doing. � � � H 0: µ 1 = µ 2 H 1: µ 1 ≠ µ 2 Computed The computed Z value of 2. 97 is in the region of rejection since it is > 1. 96. Thus, we Reject H 0 at p <. 05. Men and women are different with regard to leadership aptitude. Understanding Two-sample Hypothesis Testing 4

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 H 0. � 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 onesample test, we start by assuming that the claim is correct. � We try to Reject H 0, the “straw man, ” using the sample evidence. Understanding Two-sample Hypothesis Testing 5

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 6

Understanding Hypothesis Testing [continued] � In this case, the sample evidence resulted in a Z value of 2. 97. 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 -2. 97 and then the area from 2. 97 to + ∞. For the right side of the distribution we compute 1. 00 – p. Understanding Two-sample Hypothesis Testing 7

Understanding Hypothesis Testing [continued] . 0015 � This . 0015 -2. 97 represents the likelihood of getting the sample evidence - a difference of 9. 4 or greater. find this combined area is a total of. 003 or, in other words, only 3 chances in 1, 000. � We Understanding Two-sample Hypothesis Testing 8

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 (. 0015+. 0015). This is a lot less than. 05. Understanding Two-sample Hypothesis Testing 9

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 H 0, 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 10

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 11