This is Module 2 of our course on experimental design. The general topic here is of comparative experiments and then some basic statistical methods that are used to analyze the data from those types of experiments. I would hope that some of the material in this module is a review for a lot of you taking this course because it is the sort of material that is covered in most basic introductory courses on statistics. We're going to talk a little bit about random samples and how to summarize data from those samples. Numerical measures like the sample average or sample mean, sample variance, the standard deviation, and then some graphical methods. We'll talk a bit about populations versus samples and parameters of populations like the population mean or population variance and standard deviation and then how we actually go about estimating those parameters with the sample data. Then the bulk of this presentation, really, we'll talk about how we analyze data from simple comparative experiments. Generally, the framework that we use for doing that is the hypothesis testing framework, which we'll talk about in some detail. The principle hypothesis testing technique that we're going to talk about is the two-sample t-test. Now, we are going to talk about a couple of variations of the two-sample t-test. But that's the primary analysis engine that we use for looking at data from these simple experiments. We'll also talk a little bit about checking assumptions and what the importance of those assumptions are, and how violations of those assumptions might be some threat to the validity of your experiment. Here's a very typical simple comparative experiment. We have a group of scientists or engineers who are trying to improve the performance of a product and this product is Portland cement. What they've done is they've taken the original recipe for the mortar and they've modified it by adding polymer latex materials in an effort to reduce the setup time or the drying time of the mortar. This has been very successful. They've observed a very dramatic change in the drying time. So that part of the experiment is over and now what they're looking at is the tension bond strength as adding this material to the recipe changed the bond strength of the cement. To test this, they have prepared 10 samples of the modified mortar and then they've prepared another 10 samples of the unmodified mortar, that's the original recipe and the data that we see here is the tension bond strength that they've observed in testing those 10 samples from each of these different recipes. So this is the sample data that we're going to be using to drive this discussion of the t-test. How do we visualize this data? Well, one way to do it is with something called a dot diagram. This is a diagram where we have a scale, either horizontal or vertical, it doesn't matter. Then we portray the sampled data as dots along that scale. In this case, I've stacked the two dot diagrams on top of each other with the modified mortar dot diagram on top and the unmodified mortar dot diagram on the bottom. These displays are very useful in giving you information about the sort of the middle of the data and the spread of the data. You can see that I've also calculated the average bond strength for both of these formulations. The tension bond strength, y bar 1 16.76, is the modified mortar, and 17.04 is for the modified mortar. When you look at these diagrams, a couple of things jump out at you. It appears that the average of the modified mortar is probably a little lower than the average of the unmodified mortar and in fact, the numbers reveal that 16.76 for the modified mortar and 17.04 for the original recipe. It also appears that the spread of the observations is about the same. That is if you look at the spread here and compare that to the spread here, they're very similar. So you might suspect that the averages or means might have been affected by this change in the recipe, but perhaps not the inherent variability. Now, dot Diagrams work very well with small samples. If you have no more than about 20 or maybe 30 observations, dot diagrams are a very nice way to display the information visually in the sample. On the other hand, dot diagrams can get a little busy and not very easy to construct or to interpret for larger samples. This is a large sample tool, the histogram. Now, there's some debate about when you switch from tools like a dot diagram or a stem-and-leaf plot or something like that to a histogram. But I think generally if you have between 50 and 100 observations, you really should be using a histogram. The way you construct a histogram is pretty simple. You simply divide the range of your variable into intervals or bins and usually these bins are of equal length or equal width and then we count the number of observations that fall in each one of these bins. Then we draw a diagram like the one you see here where the horizontal scale is the variable of interest and then either the frequency or relative frequency of the counts is the horizontal scale. Histograms also show spread, central tendency, and shape. This histogram is for 200 observations on metal recovery or yield from a smelting process. You can see that the average metal recovery is somewhere around 70 or 71 or 72 percent and that there's a fair amount of variability. It goes all the way from about the low 60s' up to almost 85 percent. But the shape of the distribution is relatively symmetric. So you get a lot of information from a histogram. But the reason you use histograms with large samples rather than small samples is that the shape of a histogram is dramatically impacted by the number of bins that you choose and the width of those bins. If you have small samples, small changes in those parameters can dramatically affect the shape of the histogram. So that's why they work better with large samples. Here's another display of sample data that I find extremely useful. This is something called a box plot. These are the box plots for the Portland cement data that we've looked at earlier. Now, the way a box plot is constructed is that we have either a vertical or a horizontal box. These are vertical boxes. The lower edge of the box corresponds to the 25th percentile of the sample data and the upper edge of the box corresponds to the 75th percentile. The land in the middle is the 50th percentile or the median. These lands that extend from the ends of the box are called whiskers. By the way, some people call these box whisker plots. The whiskers extend to the minimum and maximum values that were observed in the sample. Now there are some rules about drawing these things that incorporate outliers or unusual vantage. But we're not going to get into those. We're going to look at the simple box plot. When you look at these box plots for our two formulations of the mortar, what do you notice? Well, you notice that in both cases the median, LAN, is in about the middle of the box. Okay? All right, that tells you that the sample is probably drawn from a symmetric distribution. Also, if you look at the length of the boxes, including the whiskers, they're about the same on both of these displays. So that's an indication that the variability in the two populations are probably very similar. The other thing that you notice is that the central tendency of the unmodified mortar does appear to be higher than the central tendency of the modified recipe. That's kind of an important issue because if adding this material to the recipe really greatly improved the cure time, this was a victory. But if it has a negative impact on strength, this may affect the usability of the product. So probably what one would want to do after seeing this data is to investigate whether or not there is statistical evidence to support the claim that the main tension bond strength in these two recipes is the same and that's the problem that we'll start to address next.
