Estimating a Proportion, Given a Small Sample

In this lesson, we explain how to estimate a confidence interval for a proportion, when the sample size is small.

How to Estimate Confidence Intervals With Small Samples

In the previous lesson, we showed how to estimate a confidence interval for a proportion when the sample included at least 10 successes and 10 failures. This requirement serves two purposes:

  • It guarantees that the sample size will be at least 20 when the proportion is 0.5.
  • It ensures that the minimum acceptable sample size increases as the proportion becomes more extreme.

When the sample does not include at least 10 successes and 10 failures, the sample size will be too small to justify the estimation approach presented in the previous lesson. This lesson describes how to construct a confidence interval for a proportion when the sample size is small. The key steps are:

Estimation Requirements

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

The following examples illustrate how this works. The first example involves a binomial experiment; and the second example, a hypergeometric experiment.

Example 1: Find Confidence Interval When Sampling With Replacement

Suppose an urn contains 30 marbles. Some marbles are red, and the rest are green. Five marbles are randomly selected, with replacement, from the urn. Two of the selected marbles are red, and three are green. Construct an 80% confidence interval for the proportion of red marbles in the urn.

Solution: To solve this problem, we need to define the sampling distribution of the proportion.

  • First, we assume that the population proportion is equal to the sample proportion. Thus, since 2 of the 5 marbles were red, we assume the proportion of red marbles is equal to 0.4.

  • Second, since we sampled with replacement, the sample proportion can be considered an outcome of a binomial experiment.

  • Assuming that the population proportion is 0.4 and the sample proportion is the outcome of a binomial experiment, the sampling distribution of the proportion can be determined. It appears in the table below. (Previously, we showed how to compute binomial probabilities that form the body of the table.)
Number of red marbles in sample Sample proportion Probability Cumulative probability
0 0.0 0.07776 0.07776
1 0.2 0.2592 0.3396
2 0.4 0.3456 0.68256
3 0.6 0.2304 0.91296
4 0.8 0.0768 0.98976
5 1.0 0.01024 1.00

We see that the probability of getting 0 red marbles in the sample is 0.07776; the probability of getting 1 red marble is 0.2592; etc. Given the entries in the above table, it is not possible to create an 80% confidence interval exactly. However, we can come close. The probability the true proportion is between 0.2 and 0.6 is equal to 0.2592 + 0.3456 + 0.2304 or 0.8352. Thus, based on this sample, we are 83.52% confident that the true population proportion lies within the range 0.2 to 0.6.

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Example 2: Find Confidence Interval When Sampling Without Replacement

Let's take another look at the problem from Example 1. This time, however, we will assume that the marbles are sampled without replacement. Suppose an urn contains 30 marbles. Some marbles are red, and the rest are green. Five marbles are randomly selected, without replacement, from the urn. Two of the selected marbles are red, and three are green. Construct an 80% confidence interval for the proportion of red marbles in the urn.

Solution: To solve this problem, we need to define the sampling distribution of the proportion.

  • First, we assume that the population proportion is equal to the sample proportion. Thus, since 2 of the 5 marbles were red, we assume the proportion of red marbles is equal to 0.4.

  • Second, since we sampled without replacement, the sample proportion can be considered an outcome of a hypergeometric experiment.

  • Assuming that the population proportion is 0.4 and the sample proportion is the outcome of a hypergeometric experiment, the sampling distribution of the proportion can be determined. It appears in the table below. (Previously, we showed how to compute hypergeometric probabilities that form the body of the table.)
Number of red marbles in sample Sample proportion Probability Cumulative probability
0 0.0 0.0601 0.0601
1 0.2 0.2577 0.3178
2 0.4 0.3779 0.6957
3 0.6 0.2362 0.9319
4 0.8 0.0625 0.9944
5 1.0 0.0056 1.0000

We see that the probability of getting 0 red marbles in the sample is 0.0601; 1 red marble in the sample is 0.2577; etc. Given the entries in the above table, it is not possible to create an 80% confidence interval exactly. However, we can come close. The probability the true proportion is between 0.2 and 0.6 is equal to 0.2577 + 0.3779 + 0.2362 or 0.8718. Thus, based on this sample, we are 87.18% confident that the true population proportion lies within the range 0.2 to 0.6.

It is informative to compare the findings from Examples 1 and 2. In both problems, the interval estimate ranged from 0.2 to 0.6. However, the confidence level was greater for Example 2 (which sampled without replacement) than for Example 1 (which sampled with replacement). This illustrates the fact that precision is greater when sampling without replacement than when sampling with replacement.