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Constructing a Confidence Interval for a Population Proportion
A confidence interval provides an estimated range of values which is likely to include an unknown population parameter, in this case, the population proportion ( $ p $ ). Instead of just a single point estimate (our sample proportion $ \hat{p} $ ), a confidence interval gives us a range of plausible values for $ p $ and quantifies our confidence that this range captures the true parameter.
Purpose and Point Estimate
Our goal is to estimate an unknown population proportion $ p $ (e.g., the true proportion of all U.S. adults who support a certain policy). We use a sample proportion, denoted by $ \hat{p} $ , as our point estimate for $ p $ . The formula for the sample proportion is: $$ \hat{p} = \frac{\text{number of successes in the sample}}{\text{sample size}} = \frac{x}{n} $$
General Formula
A confidence interval generally follows the form: $$ \text{Point Estimate} \pm \text{Margin of Error} $$ For a population proportion, this translates to: $$ \hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ Where:
- $ \hat{p} $ is the sample proportion.
- $ z^* $ is the Critical Values for the desired confidence level.
- $ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $ is the standard error of the sample proportion.
Conditions for Inference
Before constructing a confidence interval for a population proportion, several conditions must be met to ensure the validity of the procedure. These are often remembered as “RANDOM, NORMAL, INDEPENDENT”:
| Condition | Description |
|---|---|
| Random | The data must come from a Random Sampling and a Collection or a Randomized Comparative Experiment. This ensures the sample is representative of the population. |
| Normal | The sampling distribution of $ \hat{p} $ must be approximately normal. This is checked by verifying that the expected number of successes ( $ np $ ) and failures ( $ n(1-p) $ ) are both at least 10. Since $ p $ is unknown, we use $ \hat{p} $ : $ n\hat{p} \ge 10 $ and $ n(1-\hat{p}) \ge 10 $ . This condition allows us to use $ z^* $ critical values. |
| Independent | Individual observations must be independent. This is ensured if sampling without replacement, the population size is at least 10 times the sample size ( $ N \ge 10n $ ). This is the 10% Condition. |
If these conditions are met, we can use the Normal approximation for the Sampling Distributions for Sample Proportions.
Steps for Construction (PANIC/PHANTOMS)
A common mnemonic to remember the steps for constructing a confidence interval is PANIC (Parameter, Assumptions, Name, Interval, Conclude) or PHANTOMS (Parameter, Hypotheses, Assumptions, Name, Test Statistic, Obtain p-value, Make a decision, State conclusion). Here, we’ll focus on the construction:
- Parameter (P): Define the population proportion of interest, $ p $ .
- Assumptions (A): Check the three conditions (Random, Normal, Independent). If any are violated, the interval may not be reliable.
- Name (N): State the specific inference procedure you are using: a one-sample z-interval for a population proportion.
- Interval (I):
- Calculate the point estimate $ \hat{p} $ .
- Find the appropriate Critical Values|critical value $ z^* $ for your chosen confidence level (e.g., for 95% confidence, $ z^* \approx 1.96 $ ).
- Calculate the standard error: $ SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $ .
- Construct the interval using the formula: $ \hat{p} \pm z^* \cdot SE_{\hat{p}} $ .
- Conclude (C): Interpret the confidence interval in the context of the problem.
Interpretation of a Confidence Interval
The interpretation of a confidence interval is crucial and must be phrased carefully. For example, a 95% confidence interval for the proportion of voters who support a candidate might be (0.48, 0.52).
A correct interpretation would be: “We are 95% confident that the interval from 0.48 to 0.52 captures the true proportion of all voters who support the candidate.”
Common Misinterpretations to Avoid:
- Do NOT say: “There is a 95% probability that the true proportion is in this interval.” (Once the interval is calculated, the true proportion is either in it or it isn’t; probability applies to the method, not the specific interval).
- Do NOT say: “95% of all samples would have a sample proportion in this interval.” (The interval estimates $ p $ , not $ \hat{p} $ ).
- Do NOT say: “95% of the data falls within this interval.” (This is about the population distribution, not the parameter $ p $ ).
The confidence level refers to the success rate of the method: If we were to take many, many samples and construct a confidence interval from each, about 95% of those intervals would capture the true population proportion.
This note page sets the foundation for understanding how to quantify uncertainty when estimating a population proportion. For applying this to specific claims, refer to Justifying a Claim Based on a Confidence Interval for a Population Proportion.