Confidence Intervals for Proportions Study Guide

Inference is the process of using sample data to make conclusions about a population.
The relationship between the population and the sample is defined as follows: - Population Truth: Represented by the parameter $p$ (the true percentage). - Sampling: Data is collected from the population using random sampling methods. - Sample Statistic: The result of the sample is represented by $\hat{p}$ . - Sample Size: Represented by $n$ .
There are two primary methods to infer the truth about a population: - Hypothesis Test: Testing a claim about a population parameter (H_0: p = \text{#}) to find a P-value. - Confidence Interval: If the null hypothesis ( $H_0$ ) is rejected, we may try to estimate the true proportion ( $p$ ) using a confidence interval.

Definition: A confidence interval tries to infer the true population proportion (parameter) by creating a numeric interval and assigning a percent qualifier known as confidence.
Applicability: This method is used when specific conditions (randomness, independence, and sample size) are met to find the confidence interval for the population proportion, $p$ .
General Formula: The structure of the confidence interval is modeled as: $\text{CI} = \hat{p} \pm z^* \times SE(\hat{p})$
Components of the Formula: - Center of the Interval (Statistic): The sample proportion $\hat{p}$ . - Critical Value ( $z^*$ ): A value picked based on the desired level of confidence. - Standard Error ( $SE(\hat{p})$ ): This is the estimated standard deviation of the proportion calculated from a sample.

Standard Error ( $SE(\hat{p})$ ): Because we do not know the true population parameter, we use the sample variation to estimate the standard deviation. The formula is: $SE(\hat{p}) = \sqrt{\frac{\hat{p}\hat{q}}{n}}$ - Note: $\hat{q}$ is the complement of the sample proportion ( $1 - \hat{p}$ ).
Margin of Error (ME): This represents the range above and below the sample statistic. It accounts for sample variation (it is an expected variation, not a mistake). The formula is: $\text{ME} = z^* \times \sqrt{\frac{\hat{p}\hat{q}}{n}}$
Visual Representation of the Interval: $\hat{p} - \text{ME} \leftarrow \hat{p} \rightarrow \hat{p} + \text{ME}$

The critical value ( $z^<em>$ ) is determined by the specific level of confidence chosen for the interval. The most commonly used values are: - 90% Confidence: $z^</em> = 1.645$ - 95% Confidence: $z^* = 1.96$ - 98% Confidence: $z^* = 2.326$ - 99% Confidence: $z^* = 2.576$