6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion

A confidence interval for a population proportion estimates a likely range of values for the true population proportion, usually written $p$. To justify a claim with an interval, check whether the claimed value is inside the interval.

The interpretation should name the confidence level, the true population proportion, and the context. In repeated random sampling with the same sample size, about C% of such intervals will capture the true proportion.

Since each confidence interval is based on a random sample of data, the interval will either contain the population proportion or it will not. In repeated random sampling with the same sample size, about C% of such intervals will capture the true proportion. For example, if the confidence level is 95%, then in many repeated samples, about 95% of the intervals computed would capture the true population proportion.

Concluding a Confidence Interval

Let's return to our confidence interval that was given before:

We are estimating a 95% confidence interval of what proportion of high school math students pass their class. We were given a sample of 85 students where ~75% of them passed. We calculated a confidence interval for the true population proportion based off of our sample. The interval is given in the calculator output below:

We are given the 95% confidence interval (0.66125, 0.84463) as an estimate of the population proportion of high school students who are passing their math class.

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Huh? What the Heck Does This Mean?

In terms of what this means, it means we are 95% confident that the true population proportion of high school students who pass their math class is between 0.66125 and 0.84463. Because we are estimating a proportion, the true population proportion is between 0 and 1. However, using the normal-approximation method, a calculated confidence interval can occasionally extend slightly below 0 or above 1. This does not invalidate the interval; interpret it in context (and, if desired, acknowledge that the parameter itself must be between 0 and 1).

Interpretation Templates

Interpreting a confidence interval for a one-sample proportion should include a reference to the sample taken and details about the population it represents.

• "Based on a random sample of [n] [context], we are C% confident that the true proportion of [population + context] is between [L] and [U]."
• "We are C% (90%, 95%, 99%, etc.) confident that the confidence interval for a population proportion captures the population proportion of (CONTEXT)."

Example: For interpreting a 99% confidence interval of (0.268, 0.292), based on the proportion of a nationally representative sample of twelfth-grade students who answered a particular multiple choice question correctly: “We are 99 percent confident that the interval from 0.268 to 0.292 contains the population proportion of all United States twelfth-grade students who would answer this question correctly”

• "In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the population proportion of (CONTEXT)."

The Big Four

When interpreting a confidence interval for a population proportion, there are four things necessary to receive full credit: confidence level, context, reference to true population proportion, and reference to the sample and the population it represents.

(1) Confidence Level

Our confidence level is generally given in the problem. This is the 95%, 90%, 98%, etc. This impacts the z* for our confidence interval and is necessary in including in our interpretation of the interval.

(2) Context

As with anything in AP Stats, context is needed for full credit. Anytime we write out an answer, we need to include it in context of the problem being asked. It is no different when interpreting a confidence interval. We need to ask ourselves, "What is this interval estimating?" and include that in our response.

(3) True Population Parameter

We also need to be sure that our answer implies that we are estimating a population proportion, not just a sample proportion. After all, there's no reason to estimate something for our sample because we have the EXACT sample proportion as it was given to us. We are using that sample to estimate the bigger picture with our population.

(4) Reference to the Sample

Include a brief reference to the sample on which the interval is based and the population it represents. For example: "Based on a random sample of 85 U.S. high school math students..." Then describe the population/context the sample represents.

Testing a Claim Using Intervals

When we are given a population proportion that maybe we don't necessarily believe, we can use a confidence interval based off of a random sample to assess that claim. The main way we are going to check the statistical claim is by seeing if the claimed population proportion is within our confidence interval. If it is in our confidence interval, then it is possible that the claim is true. If the claimed value is not in our interval, the interval provides evidence against the claim (at the given confidence level), making the claim not plausible based on this sample.

Example

In our example above dealing with students passing their math class, let's say that we recently read an article that said only 55% of all US students are passing their math class. Therefore, we took a random sample of 85 US math students and we were given the interval above: (0.66125, 0.84463).

Since 0.55 is not in our interval, we have reason to doubt the article that we read; the interval suggests the true proportion differs from 0.55. We should definitely investigate it further.

Important Note!

When constructing a confidence interval, check Large Counts with p-hat: n p-hat ≥ 10 and n(1 − p-hat) ≥ 10. If you instead conduct a one-proportion z test for a claim, check Large Counts with the hypothesized p0: n p0 ≥ 10 and n(1 − p0) ≥ 10. In this section we are using a CI to assess plausibility of the claim, so conditions should be based on p-hat.

Sample Size, Confidence Interval Width, Confidence Level, and Margin of Error

Confidence Interval Width & Sample Size -- INVERSE relationship: When all other things remain the same, the width of the confidence interval for a population proportion tends to decrease as the sample size increases. For a population proportion, the width of the interval is proportional to 1/sqrt(n). This is because as the sample size increases, the standard deviation of the sampling distribution decreases, which results in a narrower confidence interval.

Confidence Interval Width & Confidence Level -- DIRECT relationship: For a given sample, the width of the confidence interval for a population proportion increases as the confidence level increases. This is because a higher confidence level results in a wider confidence interval, as the interval is constructed using a larger critical value.

Confidence Interval Width & Margin of Error -- 2 x MOE = Width: A confidence interval is p-hat ± (margin of error). The width of the interval is exactly 2 × (margin of error).