1 Where we are going : a graphic: Hypothesis Testing. 1 2 Paired 2 or more Means Variances Proportions Categories Slopes Ho: / CI Samples Ho: / CI / CI Ho: / CI Ho: Ho: / CI Ho: Ho: / CI

2 Two Sample: Hypothesis testing. Population 1 Sample 1 Gather data Make comparisons Population 2 Sample 2

3 Hypothesis testing: Same 8 steps (briefly): 1. Research hypothesis (Ha:). 2. Null hypothesis (Ho:). 3. A. State type I error. B. State type II error. 4. State , (the probability of type I error). 5. Draw the rejection region; state rejection criterion. 6. Compute the test statistic. 7. Make a decision about Ho:. 8. State a conclusion.

4 Hypotheses about two proportions. Chapter 8. (Pulse rate example, again). Consider that we are interested in knowing if the proportion of people with elevated pulse is different between the two groups (exercise and resting). An increased pulse is considered to be 21 or above. The population of interest is all the graduate students at Clemson.

5 Hypotheses about two proportions.: continued, 1. So: are the proportions different? 1. Research (alternate) hypothesis: Ha:  e   r 2. Null hypotheses: Ho:  e =  r 3. A. Type I error: Claim the proportions with high pulse rate are different between groups when in fact they are equal. B. Type II error: Claim the proportions with high pulse rate are equal when in fact they are different.

6 Hypotheses about two proportions.: continued, Type I error tolerated: let  = Rejection Region: t  /2,large = t.025,large = Reject Ho if t c is less than or greater than Test statistic:

7 Hypotheses about two proportions.: continued, 3. At this point, the experiment is conducted and data are collected. The following were the results: (large samples only) 128 measurements on 15 second pulse rate were made. Group Number Number >20 Proportion > 20 exercise rest total

8 Hypotheses about two proportions.: continued, 4 So: p =.38 ; p1 =.52; p2 = Decision: Since 2.91 > 1.96, Reject Ho: (plot 2.91 on the drawing in 5.) in favor of Ha:.

9 Hypotheses about two proportions.: continued, 5 8. Conclusion. A recent study of 128 randomly selected students showed that the proportion with elevated 15 second pulse rate was greater among students who had moderate exercise than students who rested (.52 and.26 respectively). We are 95% certain our answer is correct; or, there is a 5% chance we have rejected incorrectly. If the null is in fact correct, 5 out of 100 studies done in this way will lead to rejection.

10 There you are: Hypothesis tests for the mean, proportion, and variance when there are 2 samples. We have mentioned 8 steps and worked three examples showing the steps. You should notice the similarity. The only thing different is the distribution and the test statistic. This will be true for the remainder of the conditions we will consider - that is, the concept of hypothesis testing is basically the same, the distributions used and the test statistics computed will change. In some cases, the test statistic will be tedious, but we have a computer to do that work for us !