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 P₁ = the proportion of Republican voters in the first state, P₂ = the proportion of Republican voters in the second state, p₁ = the proportion of Republican voters in the sample from the first state, and p₂ = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n₁) = 100, and the number of voters sampled from the second state (n₂) = 100.

The solution involves four steps.

• Make sure the sample size is big enough to model differences with a normal population. Because n₁P₁ = 100 * 0.52 = 52, n₁(1 - P₁) = 100 * 0.48 = 48, n₂P₂ = 100 * 0.47 = 47, and n₂(1 - P₂) = 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(p₁ - p₂) = P₁ - P₂ = 0.52 - 0.47 = 0.05.
  
  
• Find the standard deviation of the difference.
  σ_d = sqrt{ [ P₁(1 - P₁) / n₁ ] + [ P₂(1 - P₂) / n₂ ] }
  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 p₁ is less than p₂. This is equivalent to finding the probability that p₁ - p₂ is less than zero. To find this probability, we need to transform the random variable (p₁ - p₂) into a z-score. 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 z-score 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.