key term - $\hat{p}$

Definition

$\hat{p}$ is the sample proportion, which is an estimate of the true population proportion $p$. It is calculated by dividing the number of successes in a sample by the total sample size. $\hat{p}$ is a crucial parameter in the context of confidence interval estimation, particularly when working with categorical data such as place of birth.

5 Must Know Facts For Your Next Test

1. $\hat{p}$ is an unbiased estimator of the population proportion $p$, meaning that the expected value of $\hat{p}$ is equal to $p$.
2. The sampling distribution of $\hat{p}$ follows a normal distribution when the sample size is large enough (typically $n \geq 30$), according to the Central Limit Theorem.
3. The standard error of $\hat{p}$ is given by $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the true population proportion and $n$ is the sample size.
4. When constructing a confidence interval for the population proportion $p$, the sample proportion $\hat{p}$ is used as the point estimate, and the margin of error is calculated using the standard error of $\hat{p}$.
5. The confidence interval for $p$ is of the form $\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $z_{\alpha/2}$ is the appropriate critical value from the standard normal distribution.

Review Questions

• Explain the relationship between the sample proportion $\hat{p}$ and the population proportion $p$.
  • The sample proportion $\hat{p}$ is an estimate of the true population proportion $p$. $\hat{p}$ is calculated by dividing the number of successes in the sample by the total sample size. $\hat{p}$ is an unbiased estimator of $p$, meaning that the expected value of $\hat{p}$ is equal to $p$. This means that as the sample size increases, the sample proportion $\hat{p}$ will converge to the true population proportion $p$.
• Describe the role of the standard error of $\hat{p}$ in constructing a confidence interval for the population proportion $p$.
  • The standard error of $\hat{p}$ is a measure of the variability of the sample proportion as an estimate of the population proportion. It is calculated as $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $\hat{p}$ is the sample proportion and $n$ is the sample size. The standard error of $\hat{p}$ is used to determine the margin of error in the confidence interval for $p$, which is of the form $\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. The larger the standard error, the wider the confidence interval will be, reflecting greater uncertainty in the estimate of $p$.
• Analyze the impact of the sample size $n$ on the precision of the estimate $\hat{p}$ and the width of the confidence interval for the population proportion $p$.
  • The sample size $n$ plays a crucial role in the precision of the estimate $\hat{p}$ and the width of the confidence interval for the population proportion $p$. As the sample size increases, the standard error of $\hat{p}$ decreases, according to the formula $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. This means that the margin of error in the confidence interval, which is proportional to the standard error, also decreases. Consequently, larger sample sizes result in more precise estimates of $p$ and narrower confidence intervals, allowing for more accurate inferences about the true population proportion. Therefore, the sample size is a key factor in determining the reliability and precision of the confidence interval for the population proportion.

Related terms

Population Proportion ($p$): The true proportion of a characteristic in the entire population, which is an unknown parameter that needs to be estimated.

Sample Size ($n$): The number of observations or individuals in a sample drawn from the population.

Standard Error of $\hat{p}$: The standard deviation of the sampling distribution of $\hat{p}$, which is used to quantify the variability of the sample proportion as an estimate of the population proportion.