Okay, so we've seen how to randomize block design is structured and we've had an overview how the analysis of variance actually is applied to the are RCBD. And so now we're ready to take a look at an example, and this example for one from the textbook. And it's about a medical device manufacturer that produces vascular grafts, now, these are essentially artificial veins. And these graphs are produced by extruding a material, PTFE, a resin, along with a lubricant into tubes, and these tubes become the artificial veins. Frequently some of the tubes in a production run contains defects, and these defects are small hard external protrusions on the material. And the surgeons don't like those, they consider them to be to defects. The manufacturer calls them flicks, and if the unit has flicks, then it's going to be rejected. The product developer who's responsible for this particular product suspects that extrusion pressure is a primary cause of the occurrence of flicks and wants to conduct an experiment to investigate this. The resin material that's used for this is produced externally, and they obtained this material from their external supplier in batches. And the engineer also suspects that there may be significant batch to batch variability. Because this material is supposed to be consistent with respect to things like the molecular weight, the mean particle size, retention, peak height ratio, there's a bunch of characteristics that should all be similar. But it probably isn't really exactly the same due to the manufacturing variation when the material is produced. So the product developer decides that this is a nuisance factor, and we're going to block on this. We're going to use a randomized complete block design to do this. There are four levels of extrusion pressure that we want to test and there are six batches of resin material that are available. So that would lead us to the randomized complete block design that you would see in table 4.3. So, notice that the way this is done is that every extrusion pressure, 8,500, 8,700, 8,900 and 9,100 PSI, they're all assigned to each one of the blocks. So we do block one first and we test those for extrusion pressures in random order, then we go to block to and do the same thing. And all the way out until we finish with block six. Here, each batch of resin is the block, it's a more homogeneous environment or experimental unit on which to test these extrusion pressures. Let's do the analysis of variance. Now, this slide is just showing you some of the details of the manual calculations in case you want to try to do this. And the manual calculations here are not terribly difficult, as I say, they are a little bit tedious and it's easy to make mistakes. A simple mistake such as failing to add the treatment totals up correctly can lead to really a catastrophically [LAUGH] bad answer. So you need to exercise some care if you're going to do this. It turns out that the calculations here are actually really fairly simple in this case. But we're going to see designs later on where it does get more complicated and the computer would be highly useful. The analysis of variance table is shown in table 4.4 in the book, and I'll show you that table in just a minute. Well, here it is. This is actually the computer output from one of the software packages, Design Expert. And you'll notice that the source of variability for the tips is called model here. And the p-value is 0.0019, a strong indication that there is a difference in the tips. If you wanted to use the p-value approach,that would be 0.0019. On the other hand, if you wanted to use a fixed significance level approach, you could look up the critical value. It's, say, 5% for an F random variable with 3 and 15 degrees of freedom. And that critical value is 3.29. Well, the computed value of F, and I'll show you that table again in just a minute, the computed value of F is 8.11. And because the computed value of the F statistic, F naught, is greater than 8.1, is greater than 3.29, it's 8.11. We would conclude at the 5% level of significance that the tips are significantly different. You might also conclude that the batches of resin material seem to differ rather significantly too. Because the mean square for blocks is relatively large compared to the mean square for error. And you can see that in this computer output that you see on the next page. Here's my F statistic, 8.11. But notice that the that the block mean square, 38.45, is pretty large relative to the residual or error mean square. So there is definitely an indication that the blocks had a real impact on this experiment. Now, if you had not run this as a randomized complete block, what would have happened? Well, all of that variability due to blocks would actually have ended up in the residual error and it would have inflated the error. Perhaps inflating it to the point that you wouldn't be able to actually detect the differences that exist between the tips. Here are the tip means and then here is the Fisher's LSD method applied to the tip means, and we'll take a look at that in just a minute. But here's the remains of the computer output, notice that the R square is about 77%. In other words, we've explained about 77% of the variability in this data. And is this is pretty good, anytime you explain 70 or 80% of the variability, you're usually pretty comfortable with what you've seen. And we should probably also take a look at the residuals from our experiment. This is a graph or a couple of graphs that tell you some things about the residuals. The plot on the left is a normal probability plot of the residuals, there seems to be no indication of problems with normality. And on the right is a plot of residuals versus the predicted Y, and what we see there is essentially random scatter. So there's no indication of any quality of variance present in the data. So we have a satisfactory residual analysis and we have a clear result in that there is a statistically significant difference between the tips. Here are a couple of other residual plots that could be of interest. Here's a graph of residuals versus the extrusion pressure. There's no indication that scatter is increasing as the pressure changes. And here is a scatter plot on the right of residuals versus the batches. And there's no indication that there is variability, huge amounts of difference in variability between the batches except for one. There's one batch, batch number 6, which seem to exhibit a little bit less variability than the others and perhaps that's just an anomaly. I don't think it's a real problem if we had several batches that look like that then that might be an indication that there's some inconsistency in the material that we get from suppliers. And that might be a source of actually another engineering study to find out why we've got that kind of inconsistency in material performance. So, basically, our residual plots seem to indicate that normality is not an issue and constant variance is not an issue. There are no obvious problems with randomization. No really serious patterns in the residuals versus blocks. We could also plot residuals versus the extrusion pressure, these plots give more information about the constant variance assumption. And one of the things you might be alert for in some of these plots would be possible outliers. One observation or two observations that are very, very different from the others, that will typically produce an outlier. That is a residual that's really large relative to those. Didn't see any indication of that in the analysis of this experiment.