Experimental error: Under "Analytical resources," review "error analysis and significant figures" and "error representation and curvefiting."
Statistical tests: [Unpaired (independent) t test] [paired t test] [Chi-squared test] [sample problems] [solutions to sample problems]
Tables: ['Student's' t distribution] [Chi-squared critical values]

Sample Problems Choosing between paired and unpaired t tests Sometimes the choice of which type of t test to run is obvious, and sometimes the choice requires some careful reflection. In all of the cases below the numbers of data points are low enough that the statistical analysis should be based upon a t distribution rather than a normal distribution. Assume that the variances are equal for all data sets. Problem #1 As a biochemist working for a pharmaceutical company, your job is to test new drugs for possible side effects, both deterimental and beneficial. The chemistry of agent TFK-05W suggests that it may have the side effect of reducing the tendency toward obesity. The Zucker rat is an established genetic model for both obesity and hypertension. As rats of the obese strain age they gain weight much more rapidly than do the so-called "lean" Zucker strain. You plan to treat a group of animals over a period of one year and compare their average weight with that of a group of untreated animals. The odds of the drug actually showing this side effect are small and maintenance of rats for a year is expensive, so you limited the scope of the study to twelve animals in each group. The null hypothesis is that treated animals will show an average weight gain over one year that is no different from the average weight gain of untreated animals. What statistical test will you use to compare the two means? Reference TW Kurtz, RC Morris and HA Pershadsingh, The Zucker fatty rat as a genetic model of obesity and hypertension, Hypertension, Vol 13, 896-901, 1989 Problem #2 This study follows from problem #1. Not only did your study suggest that the agent TFK-05W indeed does affect weight gain, but it also proved effective and safe (so far) and it is in clinical trials. Because the drug was designed to treat symptoms that have nothing to do with obesity, the clinical trials do not focus on that problem and won't answer the question of whether or not the agent is a potential weight loss drug. The company, however, has permitted you to test the agent on a group of 12 people with morbid obesity, who have signed the appropriate consent forms. This time the plan is to treat the 12 obese individuals for a year, having measured their weights on the day treatment was started. The paid participants will be monitoring their weight regularly, taking the drug, and are required to keep a daily log of activity and eating habits so that the experiment can be properly controlled. Nevertheless, the simplest initial test of the hypothesis that obese individuals treated with TFK-05W for a year will show an average weight loss is to compare average weight at the beginning and end of the experiment. Thus, as with problem #1, you will have two sets of 12 data points each to compare. What statistical test will you use to compare the two means? Problem #3 Embryonic cells (stem cells) from a single human blastula are genetically equivalent. Any of them has the potential to form any kind of tissue that is normally found in an adult human body. Exploitation of stem cells for therapeutic purposes has the potential to revolutionize medicine and expand the average human lifespan considerably. Stem cells have very complex cultural requirements. So far your stem cell lines require the addition of fetal bovine serum to the culture medium, and the exact composition (and efficacy) of animal sera varies from lot to lot, To exercise the greatest control over your experiments, it would be valuable to be able to culture your cells in a synthetic medium that includes only those components that are essential to support survival and growth of your cultures. You have developed a synthetic medium that keeps your cells going for several days, but not indefinitely. You think that you can extend the life of your cultures by adding an expensive hormone to the medium. To test the hypothesis that stem cells cultured with medium 2 will survive longer than stem cells cultured with medium 1, you will set up twenty cultures from ten original embryos, growing them in complex medium to the point at which each culture contains about 100 cells. You will then remove the original medium and feed the cultures from now on with synthetic medium. One culture from each original embryo will receive medium 1 and the other medium 2. For your data you will record the time at which each culture declines to the point of having only 50% of its original viable cells remaining. The null hypothesis is that this average "survival time" will be the same for cultures treated with either synthetic medium. Alternative hypotheses, of course, are that feeding with one or the other medium will enhance survival time by comparison. What statistical test will you run on the two sets of 10 data points each? Problem #4 You suspect that a cause of decline of your stem cell cultures is a failure to produce sufficient superoxide dismutase to rid the cells of oxygen free radicals. You have an assay for the enzyme, but to conduct the assay you must destroy the culture. From a single source of stem cells you can prepare about thirty cultures that remain healthy about 10 days in your synthetic medium after growing to a sufficient number of cells to permit you to run your assay. From then on, they decline rapidly. You prepared thirty cultures from the same source and sampled half of them when alll thirty cultures had reached the point at which the assay was feasible. You then sampled the other half 10 days later. Your null hypothesis is that superoxide dismutase activity will not be significantly different between the two sets of cultures. What statistical test will you run to determine whether or not the difference is significant? Normally, this would be a rather poor experimental design, because all of the cultures are identical. Why conduct replicate sampling on the exact same culture? The issue is that enzyme assays are notoriously inaccurate. The chances of mulitple comparisons yielding dubious results are much smaller than for a single comparison. The p value that we obtain will give us a fairly accurate estimate of the level of confidence with which we can interpret the result. Interpreting t test results Problem #5 Referring to problem #1, one of your rats died of natural causes during the study, leaving 11 animals in one group while you still had 12 animals in the other. Does this turn of events ruin the experiment? If you do conduct the analysis, how will you modify it, if at all? Problem #6 Referring to problem #2, one of your human subjects died of a massive heart attack halfway through the study. The death was clearly not related to the drug treatments. Please anwer the same questions as for problem #5. Problem #7 In problem #2 it was stated that Zucker rats treated with the agent TFK-05W were significantly less obese than untreated rats. Does this mean that when the t test was run it returned a probability (p) value of > 0.05 or a p value of < 0.05? Problem #8 For the third problem the difference between means was 12 hours, that is, the average half life of one culture was 12 hours longer than the average half life of the other one. The t test returned a p value of 0.33. What is your conclusion regarding the original hypothesis? Would you be correct in stating that the result is significant or insignificant? Problem #9 In problem #4, average superoxide dismutase activity was 30% lower after 10 days than it was in the beginning. The t test gave a p value of 0.07. What result do you report? It turns out that this study is very important, and if you indeed find that a decline in superoxide dismutase activity is a primary cause of cell death then good things will happen to your career. How will you proceed from this point?