Sample Proportion

Definition

The sample proportion is a statistical measure that represents the proportion or percentage of a characteristic of interest within a sample drawn from a population. It is a crucial concept in understanding population inferences, confidence intervals, and hypothesis testing.

5 Must Know Facts For Your Next Test

1. The sample proportion is denoted as $\hat{p}$ and is calculated as the number of successes in the sample divided by the total sample size.
2. The sample proportion is used to estimate the unknown population proportion, which is denoted as $p$.
3. The sampling distribution of the sample proportion is approximately normal when the sample size is large, as per the Central Limit Theorem.
4. The standard error of the sample proportion is $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.
5. The sample proportion is a key component in constructing confidence intervals and conducting hypothesis tests for population proportions.

Review Questions

• Explain how the sample proportion is used to estimate the unknown population proportion.
  • The sample proportion $\hat{p}$ is used to estimate the unknown population proportion $p$. This is done by taking a random sample from the population and calculating the proportion of successes in the sample. The sample proportion $\hat{p}$ is then used as the best point estimate of the true population proportion $p$. As the sample size increases, the sample proportion $\hat{p}$ becomes a more reliable estimate of the population proportion $p$.
• Describe the role of the sample proportion in constructing confidence intervals for population proportions.
  • The sample proportion $\hat{p}$ is a crucial component in constructing confidence intervals for population proportions. When the sample size is large, the sampling distribution of the sample proportion is approximately normal, and the standard error of the sample proportion can be used to construct a confidence interval for the unknown population proportion $p$. The confidence interval is of the form $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $z^*$ is the appropriate z-score for the desired confidence level.
• Analyze the role of the sample proportion in hypothesis testing for a single population proportion.
  • The sample proportion $\hat{p}$ is the key statistic used in hypothesis testing for a single population proportion. The null hypothesis typically takes the form $H_0: p = p_0$, where $p_0$ is the hypothesized population proportion. The test statistic is then calculated as $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, which follows a standard normal distribution under the null hypothesis. The sample proportion $\hat{p}$ is used to compute the test statistic and determine whether to reject or fail to reject the null hypothesis based on the p-value or critical value.

Related terms

Population Proportion: The population proportion is the true proportion or percentage of a characteristic of interest within the entire population, which is often unknown and must be estimated using sample data.

Sampling Distribution: The sampling distribution is the probability distribution of a statistic, such as the sample proportion, across all possible samples of a given size drawn from the population.

Standard Error: The standard error is the standard deviation of the sampling distribution of a statistic, such as the sample proportion, and is used to quantify the uncertainty in estimating the population parameter from a sample.