$ ext{hat}{p}$

Definition

$ ext{hat}{p}$ is the sample proportion, which is an estimate of the true population proportion, $p$. It represents the proportion of successes in a sample drawn from a population and is used to make inferences about the population parameter $p$.

5 Must Know Facts For Your Next Test

1. $ ext{hat}{p}$ is calculated as the number of successes in the sample divided by the total number of observations in the sample.
2. The sampling distribution of $ ext{hat}{p}$ is approximately normal when the sample size is large, allowing for the construction of confidence intervals.
3. The standard error of $ ext{hat}{p}$ is $ ext{sqrt}{rac{ ext{hat}{p}(1- ext{hat}{p})}{n}}$, where $n$ is the sample size.
4. Confidence intervals for $p$ are constructed using the formula: $ ext{hat}{p} ext{pm} z_{ ext{alpha}/2} ext{sqrt}{rac{ ext{hat}{p}(1- ext{hat}{p})}{n}}$.
5. The choice of $z_{ ext{alpha}/2}$ value depends on the desired level of confidence, such as 1.96 for a 95% confidence interval.

Review Questions

• Explain the relationship between the sample proportion $ ext{hat}{p}$ and the population proportion $p$.
  • The sample proportion $ ext{hat}{p}$ is an estimate of the true population proportion $p$. It represents the proportion of successes observed in a sample drawn from the population. While $ ext{hat}{p}$ is a point estimate of $p$, the true value of $p$ is unknown. The goal is to use $ ext{hat}{p}$ to make inferences about the population parameter $p$ and construct a confidence interval that is likely to contain the true value of $p$.
• Describe the role of the standard error of $ ext{hat}{p}$ in the construction of confidence intervals.
  • The standard error of $ ext{hat}{p}$, given by $ ext{sqrt}{rac{ ext{hat}{p}(1- ext{hat}{p})}{n}}$, represents the variability of the sample proportion around the true population proportion $p$. This standard error is a key component in the formula for constructing confidence intervals for $p$, which takes the form $ ext{hat}{p} ext{pm} z_{ ext{alpha}/2} ext{sqrt}{rac{ ext{hat}{p}(1- ext{hat}{p})}{n}}$. The standard error reflects the precision of the estimate $ ext{hat}{p}$ and determines the width of the confidence interval, with smaller standard errors leading to narrower intervals.
• Explain how the choice of the $z_{ ext{alpha}/2}$ value affects the interpretation of the confidence interval for $p$.
  • The $z_{ ext{alpha}/2}$ value in the confidence interval formula $ ext{hat}{p} ext{pm} z_{ ext{alpha}/2} ext{sqrt}{rac{ ext{hat}{p}(1- ext{hat}{p})}{n}}$ determines the level of confidence associated with the interval. Commonly, a $z_{ ext{alpha}/2}$ value of 1.96 is used for a 95% confidence interval, meaning that there is a 95% probability that the true population proportion $p$ lies within the constructed interval. Increasing the confidence level, such as to 99%, would require a larger $z_{ ext{alpha}/2}$ value (e.g., 2.576), resulting in a wider confidence interval. The choice of confidence level reflects the desired balance between precision and the probability of capturing the true parameter $p$.

Related terms

Population Proportion ($p$): The true proportion of a characteristic of interest in the entire population.

Sample Proportion ($ar{x}$): The proportion of a characteristic of interest in a sample drawn from the population.

Confidence Interval:

A range of values that is likely to contain the true population parameter with a specified level of confidence.