A segment on tonight's MythBusters addresed the question of "whether buttered toast falls buttered side up or down more often?" This is one of my favorite daily puzzles that can be addressed by a basic understanding of experimentation and statistics. My own curiosity on this question was satisfied by a segment of Newton's Apple -- if my memory is correct -- which found that it is the typical height of the table surface which determines the, originally, upward facing side falling on the floor. Pushing the toast from a ladder completely reversed this trend as the toast could tumble a full 360° and land in its original orientation: buttered side up.
Yet, notice the MythBusters question: it asks if toast being buttered effects how it ends up -- regardless of its original orientation, even if that is buttered up in most all daily cases. So first they had to find a way to drop toast in an unbiased way independent of the original orientation. Not surprisingly, Adam found that pushing it from the table was not satisfactory on this note. Eventually they developed a machine that dropped unbuttered toast landing up 11 times, and down 13 times -- orientation was determined by a magic marker X which we must assume is unbiasing. It is reasonable to conclude that 11 up and 13 down is indicative of a "fair" mechanism. Now when they buttered a side of 24 slices of toast they also found 12 up and 12 down. These sample sizes are too small, but roughly, it does not appear that the butter had any effect!
However, when they drop the toast from a two-story building (27'5") and find that the dry toast side X lands up 26 out of 48 drops (54%) and the buttered side X lands up 29 out of 48 drops (60%), Jamie posits that the 6% discrepancy is because he could see that the buttered side had a concave impression, and like a leaf, the convex non-buttered side tended to fall face down. Adam concludes, "if you really want to ensure, in general, you're toast landing buttered side up or down, we can tell you, you should butter with a good vigor and that the resultant bowl will make your toast generally fall butter side up." However, though he "generally" qualified his statement, strictly speaking, it is not statistically supported and when Jamie is offering a mechanism for a perceived statistical finding, he is premature. (However, if he is offering a simple observation, that's all it is.)
In this case, the null hypothesis is that the difference between the dry 54% and the buttered 60% is just due to chance. (Or, if we were to repeat the experiment, it's probable that a similar skew would happen.) The alternate theory is that there is some causal mechanism (i.e. the bowl shaped impression) that affects the outcome. If we can show that there is a low probability of repeating the experiment and observing a similar significance of difference (6%), that implies support for the alternative hypothesis. Unfortunately, neither test alone is statistically significant. For example, the probability of getting 29 out of 48 drops buttered side up even on a fair coin is 8.5 %.
z = (observed - expected) / StandardError
z = (29 - 24) / Sqrt(48)*Sqrt(.5*.5) = 1.445
=> P = 8.5%
The random chance of getting 26 buttered side up his 27%.
The probability that the difference between getting 26 in the "dry" control case, and 29 in the buttered case also is 27% and not significant.
z = (observed - expected) / StandardErrorofDifference
z= ((60%-54%) - 0%) / Sqrt((SEdry)\^2 + (SEbuttered)\^2)
z=6% / Sqrt(7.19\^2 + 7.07\^2)% = .5950
=> P = 27%