[SOUND] So far, we have looked at comparing two populations where the hypothesized mean difference has been 0. But what if we were looking for a minimum level of difference, how do we test then? Let me show you how by taking you through this example. A company's contract with its maintenance supplier is coming to an end, and it's considering another company as a possible replacement. Currently, there are some complaints about the time it takes the current supplier to fix the problems that get reported to them. The new company is more expensive, $200,000 additional annual cost. But it promises a faster response time. Should the company switch their supplier? Since here we are concerned about the mean time to fix the problem, we can frame the problem as the difference between the two populations. Let's call the current supplier they have, Supplier A, and the new supplier being considered, Supplier B. Then mu sub A represents the mean time taken by Supplier A to fix the problem and mu sub B represents the mean time it takes for Supplier B to fix the problem. In order to justify the additional cost for Supplier B, management wants a reduction of at least 5 hours in the repair time. This 5 hour reduction is calculated from the time the problem is reported to when the problem is resolved. Is that really the case? So, now we need to collect data to settle this question. Then we can answer this question with a hypothesis test about the mean differences between the two suppliers. This is step one. Here, just as before, the onus will be on the Supplier B to show that, indeed, they are that much better than Supplier A. If you look at the problem, this is what we want Supplier B to show. That is, their average time will be less than the average time of Supplier A by 5 hours or less than 5 hours. Now the equality sign has to be in the null. So we get the following as the null and the alternate hypotheses. Look at the null hypothesis closely. It states that the average time for fixing a problem for Supplier A, as compared to average time for Supplier B, will be 5 hours or more. Otherwise, we have no reason to believe that they will be faster by this much and thus worth the additional cost. Once you have the null and the alternate hypotheses will be its complement. And just as before, we classify this as a one-tailed test. If you test for equal versus not equal, then we would have a two-tailed test. The management wants to check on this at 5% level of significance. Before we can move on to step three, we need to have data for comparing these two suppliers, collecting data at random and independently for each supplier. One way would be if you have a log of elapsed times from your current supplier, and this log should be large enough, that includes some trivial problems as well as some complicated problems. Then selecting a large sample will make it more likely that you have a good representative sample. Then from the new supplier, you can ask for similar data. Maybe a data of their times for one of their customers with similar business and volume as yours. Then you take a large sample from that as well. While this step is really important issue, the technical aspects of this step is beyond the scope of our course. We will assume here that we have access to representative sample data that has been collected at random and independently from each population. Like before, the p-value is going to be the probability of a finding of sample results, like the one we found. This probability, which we call p-value, can be found by knowing how many standard errors separates our sample differences from the hypothesized difference, which we refer to as the test statistics, denoted by t. We have done this in earlier lessons. The difference in this lesson is the value of D0. Until now, all of our examples has this difference set to 0. Now we will set it to be a specific value. In this particular case, it will be 5. Again, we are not going to do this manually. We will use Excel. But now, we have to enter this number as the hypothesized difference. Look closely, and you will see that we have entered the value of 5 as the hypothesized mean difference. Based on this result, the mean time for Supplier A for fixing the problem is 23.42, and for Supplier B, is 22.05. So, Supplier B is faster than Supplier A, but are they fast enough for us? We were looking for differences of 5 or more. The t value for how far is the difference that we observed from the hypothesized difference of 5 is -6.29. You should just know that at 5% level significant, which is 95% confidence level, you can't be more than two standard errors away, but we are. So what is the probability of observing what we have? It is practically 0. So now let's go back to our formulation and make a decision. Looking at the p-value, which is extremely small, then we reject the null hypothesis. This means that the Supplier B has failed to show us that we can reduce the repair time by at least 5 hours, so that we can justify the additional cost of contract. So if you have a minimum value, we can enter it and test the difference against this value. [SOUND]