Confidence Interval: Difference Between Proportions

This lesson describes how to construct a confidence interval for the difference between two sample proportions, p₁ - p₂.

Estimation Requirements

The approach described in this lesson is valid whenever the following conditions are met:

Both samples are simple random samples.
The samples are independent.
Each sample includes at least 10 successes and 10 failures.

The Variability of the Difference Between Proportions

To construct a confidence interval for the difference between two sample proportions, we need to know about the sampling distribution of the difference. Specifically, we need to know how to compute the standard deviation or standard error of the sampling distribution.

The standard deviation of the sampling distribution is the "average" deviation between all possible sample differences (p₁ - p₂) and the true population difference, (P₁ - P₂). The standard deviation of the difference between sample proportions σ_p₁ _- _p₂ is:
σ_p₁ _- _p₂ =
sqrt{ [P₁ * (1 - P₁) / n₁] * [(N₁ - n₁) / (N₁ - 1)] + [P₂ * (1 - P₂) / n₂] * [(N₂ - n₂) / (N₂ - 1)] }
where P₁ is the population proportion for sample 1, P₂ is the population proportion for sample 2, n₁ is the sample size from population 1, n₂ is the sample size from population 2, N₁ is the number of observations in population 1, and N₂ is the number of observations in population 2. When each sample is small (less than 5% of its population), the standard deviation can be approximated by:
σ_p₁ _- _p₂ = sqrt{ [P₁ * (1 - P₁) / n₁] + [P₂ * (1 - P₂) / n₂] }
When the population parameters (P₁ and P₂) are not known, the standard deviation of the sampling distribution cannot be calculated. Under these circumstances, use the standard error. The standard error (SE) can be calculated from the equation below.
SE_p₁ _- _p₂ =
sqrt{ [p₁ * (1 - p₁) / n₁] * [(N₁ - n₁) / (N₁ - 1)] + [p₂ * (1 - p₂) / n₂] * [(N₂ - n₂) / (N₂ - 1)] }
where p₁ is the sample proportion for sample 1, and where p₂ is the sample proportion for sample 2. When each sample is small (less than 5% of its population), the standard deviation can be approximated by:
SE_p₁ _- _p₂ = sqrt{ [p₁ * (1 - p₁) / n₁] + [p₂ * (1 - p₂) / n₂] }

Note: The Advanced Placement Statistics Examination only covers the "approximate" formulas for the standard deviation and standard error. However, students are expected to be aware of the limitations of these formulas; namely, that they should only be used when each population is at least 20 times larger than its respective sample.

How to Find the Confidence Interval for a Proportion

Previously, we described how to construct confidence intervals. For convenience, we repeat the key steps below.

Identify a sample statistic. Use the sample proportions (p₁ - p₂) to estimate the difference between population proportions (P₁ - P₂).
Select a confidence level. The confidence level describes the uncertainty of a sampling method. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.
Find the margin of error. Previously, we showed how to compute the margin of error.
Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.

In the next section, we work through a problem that shows how to use this approach to construct a confidence interval for the difference between proportions.

Test Your Understanding

Problem 1

Suppose the Cartoon Network conducts a nation-wide survey to assess viewer attitudes toward Superman. Using a simple random sample, they select 400 boys and 300 girls to participate in the study. Forty percent of the boys say that Superman is their favorite character, compared to thirty percent of the girls. What is the 90% confidence interval for the true difference in attitudes toward Superman?

(A) 0 to 20 percent more boys prefer Superman
(B) 2 to 18 percent more boys prefer Superman
(C) 4 to 16 percent more boys prefer Superman
(D) 6 to 14 percent more boys prefer Superman
(E) None of the above

Solution

The correct answer is (C). The approach that we used to solve this problem is valid when the following conditions are met.

The sampling method must be simple random sampling. This condition is satisfied; the problem statement says that we used simple random sampling.
Both samples should be independent. This condition is satisfied since neither sample was affected by responses of the other sample.
The sample should include at least 10 successes and 10 failures. Suppose we classify choosing Superman as a success, and any other response as a failure. Then, we have plenty of successes and failures in both samples.
The sampling distribution should be approximately normally distributed. Because each sample size is large, we know from the central limit theorem that the sampling distribution of the difference between sample proportions will be normal or nearly normal; so this condition is satisfied.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval.

Identify a sample statistic. Since we are trying to estimate the difference between population proportions, we choose the difference between sample proportions as the sample statistic. Thus, the sample statistic is p_boy - p_girl = 0.40 - 0.30 = 0.10.
Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 90% confidence level.
Find the margin of error. Elsewhere on this site, we show how to compute the margin of error when the sampling distribution is approximately normal. The key steps are shown below.
- Find standard deviation or standard error. Since we do not know the population proportions, we cannot compute the standard deviation; instead, we compute the standard error. And since each population is more than 20 times larger than its sample, we can use the following formula to compute the standard error (SE) of the difference between proportions:
  SE = sqrt{ [p₁ * (1 - p₁) / n₁] + [p₂ * (1 - p₂) / n₂] }
  SE = sqrt{ [0.40 * 0.60 / 400] + [0.30 * 0.70 / 300] }
  SE = sqrt[ (0.24 / 400) + (0.21 / 300) ] = sqrt(0.0006 + 0.0007) = sqrt(0.0013) = 0.036
- Find critical value. The critical value is a factor used to compute the margin of error. Because the sampling distribution is approximately normal and the sample sizes are large, we can express the critical value as a z score by following these steps.
  - Compute alpha (α): α = 1 - (confidence level / 100) = 1 - (90/100) = 0.10
  - Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.10/2 = 0.95
  - The critical value is the z score having a cumulative probability equal to 0.95. From the Normal Distribution Calculator, we find that the critical value is 1.645.
- Compute margin of error (ME): ME = critical value * standard error = 1.645 * 0.036 = 0.06
Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.

Therefore, the 90% confidence interval is 0.04 to 0.16. That is, we are 90% confident that the true difference between population proportion is in the range defined by 0.10 + 0.06. Since both ends of the confidence interval are positive, we can conclude that more boys than girls choose Superman as their favorite cartoon character.