$\hat{p}_1 - \hat{p}_2$

Definition

$\hat{p}_1 - \hat{p}_2$ is a key term in the context of comparing two independent population proportions. It represents the difference between the sample proportions of two populations, which is used to make inferences about the difference between the true population proportions.

5 Must Know Facts For Your Next Test

1. The difference between sample proportions, $\hat{p}_1 - \hat{p}_2$, is used to estimate the difference between the true population proportions, $p_1 - p_2$.
2. When the sample sizes are large enough, the sampling distribution of $\hat{p}_1 - \hat{p}_2$ follows a normal distribution with a mean of $p_1 - p_2$ and a standard deviation of $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$.
3. Hypothesis tests for the difference between two population proportions are often used to determine if there is a statistically significant difference between the proportions of two populations.
4. The test statistic for comparing two population proportions is given by $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$.
5. Confidence intervals for the difference between two population proportions can be constructed using the formula $\hat{p}_1 - \hat{p}_2 \pm z_{\alpha/2}\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

Review Questions

• Explain the purpose of comparing the difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, in the context of 10.3 Comparing Two Independent Population Proportions.
  • The difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, is used to make inferences about the difference between the true population proportions, $p_1 - p_2$. This comparison is the focus of 10.3 Comparing Two Independent Population Proportions, where the goal is to determine if there is a statistically significant difference between the proportions of two populations based on sample data. By analyzing the difference between the sample proportions, researchers can draw conclusions about the relationship between the underlying population parameters and test hypotheses about the equality of the population proportions.
• Describe the sampling distribution of the difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, and explain how it is used to construct confidence intervals and perform hypothesis tests.
  • When the sample sizes are large enough, the sampling distribution of $\hat{p}_1 - \hat{p}_2$ follows a normal distribution with a mean of $p_1 - p_2$ and a standard deviation of $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. This sampling distribution is the basis for constructing confidence intervals and performing hypothesis tests for the difference between two population proportions. Confidence intervals are constructed using the formula $\hat{p}_1 - \hat{p}_2 \pm z_{\alpha/2}\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$, while the test statistic for hypothesis testing is given by $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$.
• Analyze the factors that influence the standard error of the difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, and explain how these factors affect the precision of the comparison between the population proportions.
  • The standard error of the difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, is given by $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. This standard error is influenced by several factors, including the true population proportions ($p_1$ and $p_2$) and the sample sizes ($n_1$ and $n_2$). As the sample sizes increase, the standard error decreases, leading to more precise estimates of the difference between the population proportions. Similarly, as the true population proportions move away from 0.5, the standard error also decreases, resulting in more accurate and reliable comparisons between the population parameters. Understanding these factors is crucial in designing effective studies and interpreting the results of hypothesis tests and confidence intervals for the difference between two population proportions.

Related terms

Sample Proportion: The proportion of a sample that exhibits a certain characteristic, calculated as the number of successes divided by the total sample size.

Population Proportion: The true proportion of a population that exhibits a certain characteristic, which is typically unknown and must be estimated from sample data.

Hypothesis Testing: A statistical procedure used to determine whether a claim about a population parameter is supported by the sample data.