EXAMPLE 2 Count Buffon’s coin
The French naturalist Count Buffon (1707–1788) considered questions ranging from evolution to estimating the number “pi’’ and made it his goal to answer them. One question he explored was whether a “balanced’’ coin would come up heads half of the time when tossed. To investigate, he tossed a coin 4040 times. He got 2048 heads. The sample proportion of heads is
That’s a bit more than one-half. Is this evidence that Buffon’s coin was not balanced? This is a job for a significance test.
The hypotheses. The null hypothesis says that the coin is balanced (p = 0.5). We did not suspect a bias in a specific direction before we saw the data, so the alternative hypothesis is just “the coin is not balanced.’’ The two hypotheses are
The sampling distribution. If the null hypothesis is true, the sample proportion of heads has approximately the Normal distribution with
The data. Figure 22.2 shows this sampling distribution with Buffon’s sample outcome marked. The picture already suggests that this is not an unlikely outcome that would give strong evidence against the claim that p = 0.5.
The P-value. How unlikely is an outcome as far from 0.5 as Buffon’s ? Because the alternative hypothesis allows p to lie on either side of 0.5, values of far from 0.5 in either direction provide evidence against H0 and in favor of Ha. The P-value is, therefore, the probability that the observed lies as far from 0.5 in either direction as the observed . Figure 22.3 shows this probability as area under the Normal curve. It is P = 0.37.
The conclusion. A truly balanced coin would give a result this far or farther from 0.5 in 37% of all repetitions of Buffon’s trial. His result gives no reason to think that his coin was not balanced.