[SOUND] Up to now, we have discussed drawing two independent samples from two different populations. We will now look at pairing up the samples. This is also known as dependent samples or match samples. So, what would be the difference in these experiments? In a paired difference experiment, we draw a single sample, and then we can apply two different treatments to the same sample and record the results from each treatment. For example, ask subjects to taste two products and rate them both. Or we can apply a single treatment and compare the subjects before and after. For instance to assess the effectiveness of training ask subjects to go through training and compare their productivity before and after the training. Or we can draw two samples and match them on certain dimensions and for a given treatment, compare the results between these two matched samples. For example, to test the effectiveness of a reading program, select two groups of children where every member of group one is matched with a person with similar reading abilities to be included in group two. And then teach each group with a different method, and analyze their differences. What you see under these three scenarios presented here, is that the two data samples being compared were not developed independently of one another. Thus, we wouldn't need to do our analysis differently than what we have done so far. Where we only had independent populations and samples. Let's begin with an example. A company's considering two designs for its product. Focus group across the nation are used to see if one design ranks higher than the other design. The participants are asked to rate two products on many dimensions. The results of the ratings are aggregated as a score. 100 is the highest and 0 is the lowest. The analysis is to be done at 1% level of significance to see if one design outperforms the other. So the scenario just presented is an example of a pair testing. If we were doing independent tests we would give one product to one sample. And other product with different sample which would have been selected randomly and independently of the first sample. By doing a paired sample we are controlling for a possible variations that we may get in each sample if they were independently chosen. For instance, if we have people that were easy to please in one sample as compared to the other sample. In paired sample, we will be looking for differences of score in each participant and that is why our way of analyzing the data will be different than before. So as I just said, in paired testing we're looking for difference of score in each participant. So the hypothesis is about this difference. So every participant we recorded difference for the first and the second variable. So the possible sets of hypotheses will be all about this difference, mu-sub-d, versus some hypothesized difference denoted by the capital D. Now back to our example. We were testing two products. Let's call them product A and product B. We start off with no preconceived idea about superiority of one versus the other. So we can go ahead and assume that they will score about the same. Thus, the null hypothesis will be that the mean difference in rating for both products will be zero. And the alternate will say that it's not. We are testing these hypothesis at .05 level of significance. Fortunately we can do the analysis with Excel, which will give the following results. As you can see, we ran the data through a paired test. The process of rejecting or not rejecting the now hypothesis is done, just as we did before. However, because the testing is done using a paired test, the t statistics from which we will find the p-value is found by using the equation you see here. Now we can move on to the decision making. Since we're testing for equality then this is a two-tail test and we'll look at the p value for the two-tail. Which is about 0.04 and then is less than 0.05 and thus we will reject the null hypothesis. Rejecting the null hypothesis in plain English will be that at 5% level of significance the mean ratings for the two products are different. Well great, can we tell from the output which product did better? The answer to that is yes. Let's look back at the Excel output now. Focusing on the mean scores of product A and product B, we can see that the product A scored better. And we just concluded that the difference is significant. That's why we reject the null hypothesis that they were about the same. If we had started our analysis with the belief that product A is the better product. Then we would have expressed our alternate hypothesis, such that it would show the disbelief. And this means expressing it as such. The average rating difference between product A and product B is greater than zero. And the compliment for the null In this case, we would have used the p-value of one-tail test, which is 0.019, and that is less than alpha, and we will reject the null hypothesis. Again, meaning that product A is a better product as compared to B. Now that you know A is better, you may want to know by how much. To answer this question we need to develop the confidence interval. In this case, that would be based on this new equation. Again, it is different from before because it is for a pair test. From the Excel output we will get the mean difference to be 4.44. The t of alpha over 2 is 1.65. We need to calculate the standard deviation of the differences separately. This is not done in Excel automatically. I have done this and found it to be 33.901. You can watch the Excel video demonstration to know exactly how I got this value. Now we can calculate the standard error which is 2.144. Now we are ready to calculate the margin of error and then confidence interval for the mean differences. Margin of error is found by multiplying the critical value with the standard error which gives us 3.54. Then the 90% confidence interval would be 0.90 to about 7.98. This means we can be 90% confident that the true difference is between 0.90 and 7.98 for the ratings of these two products. Now let's practice. To know whether or not students will perform better at a standardized test after completing a preparatory class, a group of 65 students are given test before and then again after having gone through the training. Test the claim at 5% level of significance. What are the null and alternate hypotheses? Since we are testing the same student pre and post training this is a paired example. The test claims that students will score, will increase and that is alternate hypothesis. So this is the non hypothesis, and that will be that the difference will be 0 or less. Testing at 5% of o significance we run the data and output shows the following. What is your conclusion? Will the training help students do better on the test? The P value of point 0.005 is less than 0.5, which will result in rejecting the null hypothesis. Based on this data, we can conclude that the mean test scores after the training will be greater than pre-training at 5% level of significance. Whenever possible, one should use the pair tests. Using the same individuals eliminates any differences in the individuals themselves and allows for comparison of the results from the two processes. We can also conduct pair tests by using different individuals as long as the subjects being compared are matched on some characteristics. The advantage that paired sample has is the ability to control for variability and in comparison to independent samples, this will result in two major advantages. One, is that we can use smaller sample sizes for our studies. And second, the results are more reliable. So whenever possible, opt for paired tests. [SOUND]