The size of a coin test
To test whether a coin is fair, you toss it times and reject fairness if the number of heads is at most or at least .
If the coin is in fact fair, what is the probability that you reject (the size of the test)?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Under fairness, the number of heads is , and every one of the sequences has the same probability.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The size of a test is the probability of the rejection region under the null hypothesis. Here the null distribution is binomial and the region is two small tails, so it is a short count.
Step 1: count. Out of equally likely sequences:
- heads: sequence; head: .
- heads: ; heads: .
That is sequences in the rejection region.
Step 2.
Comment. With discrete data you usually cannot hit a size of exactly . Widening the region to or heads would give , so this test is the largest symmetric one with size below .