Preliminary writeup. Find the number of heads that corresponds to the cutoff for rejecting the null hypothesis of a fair coin using a two- tailed test for a significance level of 0.10, 0.05 and 0.01. [For example, to get a significance level of 0.10 in a two-tailed test, the desired z-value corresponds to an area of 100 - 2(5) = 90 percent, which the normal table says is z = 1.65. Since the null hypothesis implies an EV of 50 and an SE of 5, we should reject a result outside of the interval 50 plus or minus 1.65(5), i.e., from 41.8 to 58.3; these are the values that should go in the "bin array".]
What percentage of the tests should fail to get the correct result if the coin is actually fair when the significance level is 0.10, 0.05 and 0.1? Now a much more difficult question: Can we tell what percentage of the tests should fail to get the correct result when the coin is actually unfair? Explain.
Simulations: For each problem below, use a spreadsheet to implement 50 simulations of 1) tossing a coin 100 times and 2) applying a test of significance. For each problem, report the percentage of simulations where the test of significance failed to reach the correct decision. The letter p denotes the probability of a head on any given toss of the coin.
=IF(RAND()<.55,1,0)]. At the bottom of each column use the SUM command to count the number of heads. You can then see in how many of the 50 simulations (sets of 100 flips) the null hypothesis is rejected by using FREQUENCY or HISTOGRAM to count how many simulations gave too many or too few heads: Construct the "bin array" with the values for the lower and upper acceptable range (calculated by hand using the normal distribution). The bins should have just two numbers, there should be 3 counts produced by the FREQUENCY command.
The FREQUENCY or HISTOGRAM function then can give the number of simulations "below the lower level" (i.e., unfair, too few heads), "in the acceptable range" (fair, about the right number of heads) and "above the higher level" (unfair, too many heads). In the last three questions, the coin is unfair, so any simulation with count of heads inside the acceptable range, i.e., in which the null hypothesis is not rejected, got the wrong answer; in the first three, the coin is fair, so any simulation with count of heads outside the acceptable range, so that the null hypothesis is rejected, got the wrong answer.