To test whether there is a tendency for cancer patients to hold on to life long enough to survive for a holiday or birthday, researchers gathered data from thousands of cancer deaths and focused on the 2-week periods surrounding Christmas, Thanksgiving, and each patient's birthday. The null hypothesis states that the overall proportion of cancer patients who die in the week before a special day is 0.5 (the same as the proportion who die in the week following) and the (two-sided) alternative hypothesis says the proportion dying in the week before differs from 0.5. They looked at various sex, race, and age groups separately for each of the three special days, and calculated P-values for 18 tests. (These are precise binomial probabilities, which tend to differ slightly from normal approximations.)
a. In general, if a null hypothesis is true, what is the probability that, by chance, sample data produce evidence to reject it at the 0.05 level?
b. If we tested a true null hypothesis 100 times, about how many times would it be rejected, just by chance?
c. If we test a true null hypothesis about 20 times, about how many times would it be rejected, just by chance?
d. How many of the P-values in the table above are small enough (less than 0.05) to reject the null hypothesis that the proportion of cancer patients dying the week before a special day (as opposed to the week after) equals 0.5?
e. Which is the best conclusion to draw from the study's results?
1. There is no convincing evidence that cancer patients die any more or less often before a special day as opposed to after.
2. Special days have no effect on cancer patients' time of death, except that blacks in general tend to die in the week before Thanksgiving, instead of the week after.