$p$

Definition

$p$ is a statistical parameter that represents the proportion or probability of a characteristic of interest in a population. It is a fundamental concept in the study of proportions and is central to the understanding of the Central Limit Theorem for Proportions.

5 Must Know Facts For Your Next Test

1. The Central Limit Theorem for Proportions states that the sampling distribution of the sample proportion $\hat{p}$ approaches a normal distribution as the sample size $n$ increases, with a mean of $p$ and a standard error of $\sqrt{\frac{p(1-p)}{n}}$.
2. The parameter $p$ represents the true proportion or probability of a characteristic in the population, which is unknown and must be estimated from sample data.
3. The sample proportion $\hat{p}$ is an unbiased estimator of the population proportion $p$, meaning that the expected value of $\hat{p}$ is equal to $p$.
4. The Central Limit Theorem for Proportions allows us to make inferences about the population proportion $p$ based on the sample proportion $\hat{p}$ and the standard error of the proportion.
5. The normal approximation to the binomial distribution, which is the basis for the Central Limit Theorem for Proportions, is valid when the sample size $n$ is large and the population proportion $p$ is not too close to 0 or 1.

Review Questions

• Explain the relationship between the population proportion $p$ and the sample proportion $\hat{p}$.
  • The population proportion $p$ represents the true proportion or probability of a characteristic in the entire population, which is unknown and must be estimated. The sample proportion $\hat{p}$ is the proportion of the characteristic observed in a sample drawn from the population. The sample proportion $\hat{p}$ is an unbiased estimator of the population proportion $p$, meaning that the expected value of $\hat{p}$ is equal to $p$. As the sample size increases, the sample proportion $\hat{p}$ will converge to the true population proportion $p$.
• Describe how the Central Limit Theorem for Proportions is used to make inferences about the population proportion $p$.
  • The Central Limit Theorem for Proportions states that as the sample size $n$ increases, the sampling distribution of the sample proportion $\hat{p}$ approaches a normal distribution with a mean of $p$ and a standard error of $\sqrt{\frac{p(1-p)}{n}}$. This allows us to use the normal distribution to make inferences about the population proportion $p$ based on the sample proportion $\hat{p}$ and the standard error of the proportion. For example, we can construct confidence intervals or perform hypothesis tests about the value of $p$ using the normal approximation.
• Analyze the factors that influence the validity of the normal approximation to the binomial distribution in the context of the Central Limit Theorem for Proportions.
  • The normal approximation to the binomial distribution, which is the basis for the Central Limit Theorem for Proportions, is valid when the sample size $n$ is large and the population proportion $p$ is not too close to 0 or 1. When the sample size is large, the sampling distribution of the sample proportion $\hat{p}$ will be approximately normal, regardless of the underlying distribution of the population. However, if the population proportion $p$ is very close to 0 or 1, the binomial distribution will be highly skewed, and the normal approximation may not be accurate. In these cases, the sample size needed for the normal approximation to be valid may be larger.

Related terms

Sample Proportion: The proportion of a characteristic observed in a sample, denoted as $\hat{p}$.

Standard Error of the Proportion: The standard deviation of the sampling distribution of the sample proportion, given by $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

Binomial Distribution: The probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure).