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?
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
b. use a normal approximation (making a correction for continuity) with 100
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
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
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.