7.2: Confidence Intervals for a Population Proportion

Confidence Intervals for a Population Proportion

So far, we’ve used sample data to calculate sample proportions for example, the proportion of customers who prefer a certain brand, or the percentage of voters supporting a new policy. But what we really care about is the underlying whole-population proportion the proportion we can’t usually measure directly.

Imagine a psychology researcher is studying anxiety among college students. He surveys a random sample of 200 students at several two-year colleges. The students complete the GAD-7 where a score higher than 10 indicates "moderate or worse" levels of anxiety. In the sample of 200, 78 screen positive for moderate-or-worse anxiety.

He wants to estimate the true proportion of all two-year college students that experience moderate-or-worse anxiety, not just the proportion in his sample. But he knows the sample proportion will probably vary a little, simply due to randomness.

So what’s the solution? Instead of using a single point, he calculates a confidence interval a range that likely includes the population proportion with a known level of confidence (say, 95%).

This kind of reasoning is the heart of confidence intervals.

In this section, we’ll learn how to construct a confidence interval for population proportions. Later in the chapter we will see how to interpret it in context, and understand the factors that affect its width: sample size and confidence level.

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Example: Estimating Support for a New Bus Line

You survey a random sample of 150 residents in your city and ask: “Would you use a new express bus route if it were added to your neighborhood?” Out of 150, 93 people say yes.

Your sample proportion is:

\[ \hat{p} = \frac{93}{150} = 0.62 \]

Using this result, we want to estimate the true proportion of all city residents who would support this new transportation option. We’ll calculate a confidence interval using the normal model (when certain conditions are met), and interpret it like this:

We are 95% confident that the true proportion of city residents who would use the new bus line is between 0.54 and 0.70.

This tells us a plausible range for the true population proportion, based on our sample and our desired level of confidence.

Before we jump into the formulas, we’ll take a look at what makes confidence intervals for proportions valid, including the role of randomness, sample size, and assumptions about the sampling distribution.

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Definition: Critical Value

A critical value is a multiplier used in confidence intervals and hypothesis testing. It tells us how far we need to go from our sample statistic to capture a desired level of confidence.

The critical value depends on two things:

• The confidence level (such as 90%, 95%, or 99%)
• The shape of the distribution model — either a z-distribution (normal) or a t-distribution (when estimating a mean using sample data)

We typically write it as z* (for normal distribution) or t* (for t-distribution), and it appears in the confidence interval formula like this:

\[ \text{Confidence Interval} = \text{point estimate} \pm \text{critical value} \times \text{standard error} \]

The critical value marks off the middle area in the distribution: for example, 95% of values fall between −1.96 and +1.96 in a standard normal model.

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Now that we understand what a critical value is — the number of standard errors we need to go out in both directions to capture our desired confidence level — we’re ready to build a full confidence interval. This involves combining our sample statistic (like a mean or a proportion), our calculated standard error, and the right critical value based on how confident we want to be.

In short: the confidence interval is grounded in the idea of “estimate ± margin of error.” Let’s define exactly what that means and how we use it to make inferences about the population.

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Definition: Confidence Interval for a Population Proportion

A confidence interval for a population proportion gives a range of plausible values where the true population proportion \( p \) is likely to fall, based on a sample proportion \( \hat{p} \).

When the conditions for normal approximation are satisfied, the confidence interval is:

\[ \hat{p} - z^* \cdot \sqrt{ \frac{ \hat{p}(1 - \hat{p}) }{n} } < p < \hat{p} + z^* \cdot \sqrt{ \frac{ \hat{p}(1 - \hat{p}) }{n} } \]

• \( \hat{p} \): sample proportion
• \( n \): sample size
• \( z^* \): critical value from the standard normal distribution for the desired confidence level

This interval gives plausible values for the unknown population proportion \( p \).

Important: This method assumes the sample is random and that:

• \( n\hat{p} \geq 10 \)
• \( n(1 - \hat{p}) \geq 10 \)

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Example 1: Voter Poll

A random sample of 350 voters shows that 195 support Proposition A. Construct a 95% confidence interval for the true population proportion.

• \( \hat{p} = \frac{195}{350} \approx 0.557 \)
• Conditions: \( n\hat{p} \approx 195, n(1 - \hat{p}) \approx 155 \), both ≥ 10 ✅
• \( z^* = 1.96 \)
• Standard error: \( \sqrt{ (0.557)(0.443) / 350 } \approx 0.0265 \)
• Margin of error: \( 1.96 \cdot 0.0265 \approx 0.052 \)
• Interval: \((0.557-0.052, 0.557+0.052) = (0.505, 0.609) \)

Interpretation: We are 95% confident that between 50.5% and 60.9% of all voters support Proposition A.

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Example 2: Mobile App Usage

In a survey of 500 college students, 305 students reported using a specific study app at least once per week.

• \( \hat{p} = 305 / 500 = 0.61 \)
• Both \( n\hat{p} = 305 \) and \( n(1 - \hat{p}) = 195 \) ≥ 10 ✅
• 95% confidence: \( z^* = 1.96 \)
• SE = \( \sqrt{ (0.61)(0.39)/500 } \approx 0.0216 \)
• ME = \( 1.96 \cdot 0.0216 \approx 0.0423 \)
• Interval = \( (0.5687, 0.6523) \)

Conclusion: We are 95% confident that between 56.9% and 65.2% of all college students use the app at least once per week.

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Example 3: Defective Products

A quality control engineer inspects 160 items and finds that 7 are defective.

• \( \hat{p} = 7 / 160 = 0.04375 \)
• \( n\hat{p} = 7 \) → not large enough for normal approximation ❌

Note: In this case, we would need to use an alternative method, such as a simulation-based interval or adjusted (plus-four) method. Be cautious when sample sizes are small or proportions are extreme.

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Example 4: Political Engagement

In an online survey of 1,000 adults, 420 said they had contacted an elected official in the past year.

• \( \hat{p} = 0.42 \)
• Large sample: \( n\hat{p} = 420, n(1 - \hat{p}) = 580 \) ✅
• Standard error: \( \sqrt{ (0.42)(0.58)/1000 } \approx 0.0156 \)
• 95% CI margin of error ≈ \( 1.96 \cdot 0.0156 \approx 0.0306 \)
• Confidence interval: \( (0.3894, 0.4506) \)

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Why Interval Estimates for Proportions Matter

In many real-world studies, we are trying to understand how common something is what proportion of a population fits a category, supports a policy, uses a product, or participates in a behavior. Proportions are everywhere: in polling, safety testing, quality control, voting, healthcare, and more.

A confidence interval for a proportion provides more information than just the sample percentage. Instead of reporting a single point (like “42% of respondents…”), we can communicate the range of values that are reasonable to believe for the whole population based on the data we collected.

Confidence intervals help answer the question: “How accurate do we think this number is?”

They are also a key tool for comparing groups: If two confidence intervals for proportions do not overlap, that often indicates a meaningful difference worth investigating further. We'll return to this idea when comparing two proportions later in the course.

As with all inference, clear thinking and careful interpretation are essential: confidence does not mean certainty it means we are being reasoned and transparent about what we know and don’t know.

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We can do a similar estimation process to estimate a population mean using sample data. You'll see that in the next section.