Problem 1
In one state, 52% of the voters are Republicans, and 48% are Democrats.
In a second state, 47% of the voters are Republicans, and 53% are
Democrats. Suppose a simple random sample of 100 voters are
surveyed from each state.
What is the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state?
(A) 0.04
(B) 0.05
(C) 0.24
(D) 0.71
(E) 0.76
Solution
The correct answer is C. For this analysis, let P_{1} =
the proportion of Republican voters in the first state,
P_{2} = the proportion of Republican voters in the second state,
p_{1} = the proportion of Republican voters in the
sample from the first state, and
p_{2} = the proportion of Republican voters in the
sample from the second state. The number of voters sampled from
the first state (n_{1}) = 100, and the number of voters
sampled from the second state (n_{2}) = 100.
The solution involves four steps.

Make sure the sample size is big enough to model differences
with a normal population. Because
n_{1}P_{1} = 100 * 0.52 = 52,
n_{1}(1  P_{1}) = 100 * 0.48 = 48,
n_{2}P_{2} = 100 * 0.47 = 47, and
n_{2}(1  P_{2}) = 100 * 0.53 = 53
are each greater than 10,
the sample size is large enough.

Find the mean of the difference in sample proportions:
E(p_{1}  p_{2}) = P_{1}  P_{2}
= 0.52  0.47 = 0.05.

Find the standard deviation of the difference.
σ_{d} =
sqrt{ [ P_{1}(1  P_{1}) / n_{1} ] +
[ P_{2}(1  P_{2}) / n_{2} ] }
σ_{d} =
sqrt{ [ (0.52)(0.48) / 100 ] +
[ (0.47)(0.53) / 100 ] }
σ_{d} =
sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

Find the probability. This problem requires us to find the
probability that p_{1} is less than p_{2}.
This is equivalent to finding the probability that
p_{1}  p_{2} is less than zero. To find this
probability, we need to transform the random variable
(p_{1}  p_{2}) into a
zscore.
That transformation appears below.
z_{
p1  p2
} =
(x  μ_{
p1  p2
}) /
σ_{d} = = (0  0.05)/0.0706 = 0.7082
Using Stat Trek's
Normal Distribution Calculator,
we find that the probability of a zscore being 0.7082 or less
is 0.24.
Therefore, the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state is 0.24.