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.

Confidence Interval: Proportion (Small Sample)

In the previous lesson, we showed how to estimate a confidence interval for a proportion when a simple random sample includes at least 10 successes and 10 failures.

When the sample does not include at least 10 successes and 10 failures, the sample size will often 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 has fewer than 10 successes and/or fewer than 10 failures. 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. When the true population proportion is 0.4, the probability the probability that a sample proportion falls between 0.2 and 0.6 is equal to 0.2592 + 0.3456 + 0.2304 or 0.8352. Thus, based on this sample, we can say that an 83.52% confidence interval is described by the range from 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; the probability of getting 1 red marble 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. When the true population proportion is 0.4, the probability the probability that a sample proportion falls between 0.2 and 0.6 is equal to 0.2577 + 0.3779 + 0.2362 or 0.8718. Thus, based on this sample, we can say that an 87.18% confidence interval is described by the range from 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.