Does bread lose its vitamins when stored? Small loaves of bread were prepared with flour that was fortified with a fixed amount of vitamins. After baking, the vitamin C content of two loaves was measured. Another two loaves were baked at the same time, stored for three days, and then the vitamin C content was measured. The units are milligrams per hundred grams of flour (mg/100g). Here are the data (provided by Helen Park; see Helen Park et al., "Fortifying bread with each of three antioxidants", Cereal Chemistry, 74(1997), pp. 202-206).) (Example taken from Introduction to the Practice of Statistics, 3^rd Ed., David S. Moore and George P. McCabe.)

Immediately after baking: 47.62, 49.79

Three days after baking: 21.25, 22.34

When bread is stored, does it lose its vitamin C? We won't actually perform the formal statistical test here, but I'm interested in making sure you know what steps to do. For each answer, be sure to explain.

a) Would you use a paired test or a two-independent groups test?
When asked a question using a technical term (or two), you should ask yourself, what is the difference between these two things?  And explain your answer accordingly.  In this case, the difference is that one set of tests assumes that the data consist of two independent group, and the other assumes that the data are paired or matched.  In this case, we have four separate loaves, all cooked under similar conditions, and then two were selected for a special treatment.  The two groups are independent (and clearly there is no matching, which is to say there's no need to associated one of the 3-day loaves with one of the 0-day loaves.)  So you should use a two indendent groups test.

b) What test statistic would you use?

Please please please be aware that the word "statistic" is a technical term.  It means "a function of data", more or less.  So this question asks for a function of data.   A likely approach is to compare the mean vitaman C content of each group, and so you want to estimate the difference between the two means. So one good test statistic is Xbar - Ybar, where X represents the 0-day group and Y the 3-day group.  Of course, this only gets you so far, because you're goign to need the SD, too.  So another test  statistic is (Xbar - Ybar)/(sigma/sqrt(n)), although this isn't worth full credit because sigma is not known.  A full-credit answer would be
(Xbar - Ybar)/(s/sqrt(n))  where s is our usual estimate of the standard deviation.

Note that just saying "t-test" isn't good enough.  This is not a statistic, it is a test procedure.  But if you said this, and then wrote down what it was, that was good for full credit.

Another full-credit answer is to say that because you only have two observations, you should do a non-parametric approach, and so a test statistic would be, for example, the sum of the ranks of the 0-day group. (Or of the 3-day group.)

c) What assumptions are you making?
If you did a t-test, you assumed all observations were independent, the X group and the Y group were normally distributed, and the variances were equal.  (Many of you forgot this last point.)

If you did the non-parametric test, your only assumption was that the observations were all independent.

There are other assumptions that can be generally lumped into the category of "researcher honesty."  In other words, we assumed that conditions in the oven were not deliberately changed from loaf to loaf, and that before cooking everything started out the same.

Normally, this sort of thinking occurs before step (b), because you need to know your assumptions before you can carry out a test.

d) What is the sampling distribution of this test statistic?

If you did the t-test, the sampling distribution is a t-distribution with (n+m-2) = (2 + 2 - 2) degrees of freedom.  If you did the rank test, then the distribution doesn't really have a name that I'm aware of, at least.

e) Do the data suggest that the variances of the two populations are equal? (Don't do a formal test. Just give me an informed opinion.)
Each group has only two observations, so the range is a good quick look at the variance.  (The mean is precisely half-way in between the two numbers, so its not too hard to compute the sample variance.)  So the range of the zero-group is about 1.2 and the range of the 3-group about 1.1, so they're roughly equivalent.  If you went ahead and actually calculated the variances, you would get SDs that were a little further off, and so you might conclude that they were the same or different, depending on how far apart you thought they were.  The point was to get you to do a little "back of the envelope" verification of the assumption that the variances of each group are equal (and also to remind you of this assumption to help you in part (c).