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]

Solutions to T Test Problems Problem #1 You'll run a t test for independent samples. It doesn't matter that the number of animals in each data set is the same, nor that they are all the same type of animal. You sampled 12 treated individuals and 12 different untreated individuals. There is no special relationship between a data point from one group and any particular data point from a second. The sampling method was independent. Problem #2 This study calls for running a paired t test. The same individuals were sampled (weights measured) at the beginning and at the end of the study. Thus each data point in the first set can be paired with a data point from the same individual in the second set. Variability among distinct individuals contributes considerable experimental error to many experiments. Such error can mask effects, especially small effects, even if the null hypothesis is indeed false. For example, if the average individual lost 10 pounds but the standard deviation at the beginning of the experiment was 55 pounds, the loss might not show up as a significant difference. By controlling for individual variability the paired t test can focus on the average change in weight. Problems 1 and 2 were both easy to call. If you had trouble with either case, then you really should review the criteria for selecting an appropriate t test and try to clear up any misconceptions. Problem #3 This case can be thought of as a set of replicate experiments. In each experiment one culture from a single source was fed one medium and a second culture from the same source was fed the other medium. The experiment was replicated 10 times, using 10 different sources. Since each replicate experiment consists of a pair of data points linked by the common origin of the respective cultures, you have a set of 10 pairs of data (two sets of paired data). A paired t test is appropriate for the same reasons it was appropriate for problem #2. The paired method controls for experimental error that might be contributed by the 10 different sources. Why not conduct all of the replicate experiments on cultures from a single source, eliminating all experimental error that is contributed by individual variability? Then we run the risk that the result won't hold for cultures from other embryos. We want to know if the medium we are testing will work for most or all cultures, not only for cultures from one particular embryo. Problem #4 This time your samples are all coming from the same population of cultures, presumably all identical except that half of them were sampled at one time and half at the other time. All of the data points are linked by the fact that they were obtained from cultures from a common source. However, there is no special one to one correspondence between any one data point in one set and a unique data point in the other. There is no basis for a paired t test, so we must run a test for independent samples. The assay itself is the variable in this example. If the assay was 100% accurate and reliable, we would only have needed one sample at each time. On the other hand, any significant difference should be considered preliminary until the experiment can be repeated on at least one or two more sets of cultures. Problem #5 You lost one animal, but because each data set represents an independent sample it is not necessary that the numbers of data points be equal. You conduct the t test for independent samples, comparing a set of 11 data points with a set of 12. Problem #6 This time you lost both data points that were to be contributed by the deceased individual. You now have 11 data points in each set. It shouldn't be a problem unless the others start dropping off as well. Problem #7 The p value is the probability that the null hypothesis is true. The higher the p value, the greater is the probability that there is no significant difference between means. A probability of 0.05 (1 in 20 chance) or less that the null hypothesis is true is considered sufficient evidence on which to reject it. Rejecting a null hypothesis means we accept an alternative. The result with the Zucker rats was that the treated group weighed less, so we accept the alternative hypothesis that the drug reduced weight gain. The other alternative, which was not supported by the data, would be that the drug caused additional weight gain. Of course, working with probabilities there is always a chance that the results of an experiment are simply wrong. Realistically, though, experimental results are seldom wrong due to an improbable distribution of samples. They are usually wrong because of a bad experiment, especially when an experiment is not well controlled. Problem #8 The difference in sample means may have been 12 hours, but apparently there was enough variability among cultures that the difference was not significant. With p > 0.3 it is unlikely that you will get a significant difference even by testing more cultures. Unless a difference is supported by probability it is not considered significant at all. Think about it. Maybe the range over which cultures lasted was quite wide. Perhaps the difference in sample means would be reduced to zero or even reduced if you just switched two data points. You would not be correct to say that the result is insignificant. The result is that medium 2 has no apparent effect on longevity of a culture, and that finding is indeed significant. The difference in means was insignificant, not the result itself. Problem #9 With a p value of 0.07 you are so close to finding a significant difference that it is sorely tempting to drop a data point in favor of your hypothesis, or maybe "round off" to 0.05. Scientific integrity requires that you treat the data as they stand, however. You do not have sufficient evidence with which to reject the null hypothesis. How to proceed? Because it is so important to you that you come to a conclusion, an appropriate course of action is to repeat the experiment. Analysis will be more complicated because the second experiment will be conducted on a different set of cultures. To keep it simple you could repeat the experiment twice and if the results are consistent, pool all of the data points. By the way, with 45 data points in each set you probably would no longer need a t test. You could base your analysis on the normal distribution and simply look at overlap between standard deviations in order to determine a p value. If the results are not consistent, well, welcome to the real world of science. Getting to the truth can be quite a struggle.