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EXACT and APPROXIMATE (normal) BINOMIAL PROBABILITIES 1. Find P(Y 2 | N = 10, p = .5) a. Find the exact binomial probability. b. Use the normal curve to approximate the probability without making the correction for continuity. Compute the amount by which the approximate probability differs from the exact probability [ the error due to using an approximation ]. c. Repeat b. but with the correction for continuity. 2. Find P(Y 7 | N =10, p = .9) a. as in 1a above b. as in 1b above c. as in 1c above 3. Compare the error in 1b with that in 1c and compare the error in 2b with that in 2c. What do you conclude about the correction for continuity? 4. Compare the error in 1c with that in 2c. What do you conclude about the relationship between p and the error of approximation? BINOMIAL POWER ASSIGNMENT You are testing the null hypothesis that the proportion of newly registered voters in North Carolina that are Republican is .50 or less. If you set an upper limit of .10 for the probability of a Type I error (you wish to have no more than a 10% chance of rejecting the null hypothesis, given that it is true), and if the true proportion is .60, what are your chances of rejecting the null hypothesis if you: a. employ an exact binomial test with a random sample of 10 newly registered voters. b. use a normal approximation (making a correction for continuity) with 100 subjects. SHOW YOUR WORK. COMPUTE ANSWERS (PROBABILITIES) TO THREE SIGNIFICANT DIGITS. Bino-HW-Democracy.doc The Statistics of Democracy One of my statistics students posed an interesting question. Suppose that individuals are making a decision about whether or not to take some particular action. Each individual has a 51% probability of making the correct decision (correct in terms of the good of the group). The decision will be made by a group vote. We gather N randomly selected individuals and they vote, without having deliberated. What is the probability that the majority will make the correct decision? This is a binomial probability. SPSS or SAS can be used to find a cumulative binomial probability. We want to know the probability that a majority will make the correct decision. Since these programs compute cumulative (lower-tailed) probabilities, we find the probability that a minority will make the wrong decision (logically, this probability must be the same as the probability that the majority will make the correct decision). To keep the problem simple, we shall assume that we have an odd number of persons voting. If N = 3, the binomial probability can be obtained in SAS using the command p3 = PROBBNML(.49, 3, 1), and in SPSS by using the command COMPUTE P = CDF.BINOM(1, 3, .49). The probability comes out to be .515. That is, a group of 3 is a little more likely to make the correct decision than is an individual. What if we increase the size of the group? Here are SAS commands with different values of N: p25 = PROBBNML(.49, 25, 12); p51 = PROBBNML(.49, 51, 25); p101 = PROBBNML(.49, 101, 50); p1001 = PROBBNML(.49, 1001, 500); and here are the results Group Size Probability of Correct Decision 3 25 51 101 1001 .515 .540 .557 .580 .737 As you can see, the larger the number of voters, the more likely a correct decision. You may want to play around with the values of N and p to see how they affect these binomial probabilities. So far, democracy seems great. Consider, however, that in actual practice the voters do deliberate before voting -- that is, they discuss the issue at hand before taking a vote. Suppose that we are voting whether or not to approve a permit to allow a new housing development in our wetlands. Further suppose that a decision not to approve this construction would be best in the long term for our group and that more than 50% of the individuals are initially opposed to the project. Consider, however, what will happen if a few of the individuals are strongly in favor of the project (those in the construction trade, for example). If they have superior lobbying resources (intellectual resources and/or other resources), they may be much more effective in their persuasive communications, enabling them to change the minds of enough individuals to produce a majority vote in favor of the plan. bino-hw1.doc