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

Definition

$\hat{p}_1 - \hat{p}_2$ is the difference between two sample proportions, which is used to compare the proportions of two independent populations. It is a key concept in the context of 10.3 Comparing Two Independent Population Proportions, as it forms the basis for statistical inference and hypothesis testing when comparing the characteristics of two populations.

5 Must Know Facts For Your Next Test

1. The difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, is used to estimate the difference between the corresponding population proportions, $p_1 - p_2$.
2. The sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal when the sample sizes are large enough, which allows for the use of a z-test or a t-test to compare the two population proportions.
3. The standard error of $\hat{p}_1 - \hat{p}_2$ is given by $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$, where $n_1$ and $n_2$ are the sample sizes of the two populations.
4. The test statistic for comparing two population proportions is $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$, which follows a standard normal distribution under the null hypothesis.
5. The confidence interval for the difference between two population proportions, $p_1 - p_2$, is given by $\hat{p}_1 - \hat{p}_2 \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

Review Questions

• Explain the purpose of calculating the difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, in the context of comparing two independent population proportions.
  • The difference between two sample proportions, $\hat{p}_1 - \hat{p}_2$, is used to estimate the difference between the corresponding population proportions, $p_1 - p_2$. This difference is the key statistic used in hypothesis testing and confidence interval estimation when comparing the characteristics of two independent populations. By calculating $\hat{p}_1 - \hat{p}_2$, researchers can determine if there is a statistically significant difference between the proportions of the two populations and make inferences about the underlying population parameters.
• Describe the sampling distribution of $\hat{p}_1 - \hat{p}_2$ and explain how it is used to conduct statistical inference.
  • The sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal when the sample sizes are large enough. This allows for the use of a z-test or a t-test to compare the two population proportions. The standard error of $\hat{p}_1 - \hat{p}_2$ is used to standardize the difference, resulting in a test statistic that follows a standard normal distribution under the null hypothesis. This test statistic can then be used to determine the p-value and make decisions about the null hypothesis, such as whether to reject it and conclude that there is a significant difference between the two population proportions.
• Explain how to construct a confidence interval for the difference between two population proportions, $p_1 - p_2$, using the sample proportions $\hat{p}_1$ and $\hat{p}_2$.
  • The confidence interval for the difference between two population proportions, $p_1 - p_2$, is given by $\hat{p}_1 - \hat{p}_2 \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$. This interval provides a range of plausible values for the true difference between the population proportions, based on the observed sample data. The margin of error, given by $z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$, represents the level of uncertainty in the estimate and is influenced by the sample sizes and the observed sample proportions. Constructing this confidence interval allows researchers to make inferences about the population parameters and assess the practical significance of the difference between the two proportions.

Related terms

Sample Proportion: The sample proportion, denoted as $\hat{p}$, is the proportion of successes in a sample. It is calculated as the number of successes divided by the total number of observations in the sample.

Null Hypothesis: The null hypothesis, denoted as $H_0$, is a statement that there is no difference between the population proportions being compared. It is the hypothesis that is tested in a hypothesis test.

Alternative Hypothesis: The alternative hypothesis, denoted as $H_1$, is a statement that there is a difference between the population proportions being compared. It is the hypothesis that is accepted if the null hypothesis is rejected.