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.
What happens if a patient dies? If you want to do a paired t test you have to drop the odd data point and conduct your paired t test on 11 patients. You could also conduct a two-sample t test between the 12 "before" measurements and the 11 "after" measurements. You would lose the advantage of working with paired data, though.
Case studies 1 and 2 should have both been 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.
Embryonic cells (stem cells) from a single human blastula are genetically equivalent and any of them has the potential to form any kind of tissue that is normally found in an adult human body. Your stem cell lines have very complex cultural requirements. They require fetal bovine serum in 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 the hormone extends the life of your cultures, you will set up twenty cultures from ten original cell lines, growing them in complex medium to the point at which each culture contains about 100 cells. You will then feed the cultures from then on with synthetic medium. One culture from each cell line will receive the hormone. 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 treated and untreated cultures. Alternative hypotheses, of course, are that the hormone increases or decreases survival time.
What statistical test will you run on the two sets of 10 data points each?