P̂2

Definition

The term $p̂2$ represents the sample proportion squared, which is a key concept in the context of comparing two independent population proportions. It is a statistical measure used to estimate the variance of the sampling distribution of the difference between two sample proportions.

5 Must Know Facts For Your Next Test

1. $p̂2$ is used to estimate the variance of the sampling distribution of the difference between two sample proportions, which is a crucial step in comparing two independent population proportions.
2. The formula for the variance of the sampling distribution of the difference between two sample proportions is $ ext{sigma}_{p1-p2}^2 = p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2$, where $p̂1$ and $p̂2$ are the sample proportions, and $n1$ and $n2$ are the sample sizes.
3. The standard error of the difference between two sample proportions is calculated as $ ext{sqrt}{p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2}$, which is the square root of the variance.
4. The test statistic used to compare two independent population proportions is the $z$-statistic, which is calculated as the difference between the two sample proportions divided by the standard error of the difference.
5. The $p$-value associated with the $z$-statistic is used to determine the statistical significance of the difference between the two population proportions.

Review Questions

• Explain the role of $p̂2$ in the context of comparing two independent population proportions.
  • $p̂2$ is a key component in the formula for the variance of the sampling distribution of the difference between two sample proportions. This variance is used to calculate the standard error of the difference, which is a crucial step in the $z$-test for comparing two independent population proportions. The $z$-test statistic is calculated by dividing the difference between the two sample proportions by the standard error, and the resulting $p$-value is used to determine the statistical significance of the difference between the two population proportions.
• Describe how the formula for the variance of the sampling distribution of the difference between two sample proportions is derived using $p̂2$.
  • The formula for the variance of the sampling distribution of the difference between two sample proportions is $ ext{sigma}_{p1-p2}^2 = p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2$. This formula is derived by recognizing that the variance of the difference between two random variables is the sum of their individual variances. The variance of a sample proportion $ ext{hat}{p}$ is $p(1-p)/n$, where $p$ is the true population proportion. Substituting $p̂1$ and $p̂2$ for the sample proportions, and $n1$ and $n2$ for the respective sample sizes, we arrive at the formula that includes $p̂2$ as a key component.
• Explain how the $z$-statistic used to compare two independent population proportions is calculated using $p̂2$, and discuss the interpretation of the resulting $p$-value.
  • The $z$-statistic used to compare two independent population proportions is calculated as $(p̂1 - p̂2) / ext{sqrt}{p̂1(1-p̂1)/n1 + p̂2(1-p̂2)/n2}$, where $p̂1$ and $p̂2$ are the sample proportions, and $n1$ and $n2$ are the respective sample sizes. This formula incorporates $p̂2$ as part of the standard error of the difference between the two sample proportions. The resulting $z$-statistic is then used to calculate the $p$-value, which represents the probability of observing a difference between the two population proportions at least as extreme as the observed difference, assuming the null hypothesis (that the population proportions are equal) is true. The $p$-value is used to determine the statistical significance of the difference and guide decision-making regarding the research hypothesis.

Related terms

Sample Proportion:

The proportion of a sample that exhibits a certain characteristic or attribute, denoted as $ ext{hat}{p}$.

Sampling Distribution:

The distribution of a statistic, such as the sample proportion, obtained from all possible samples of the same size drawn from a population.

Variance:

A measure of the spread or dispersion of a dataset, calculated as the average squared deviation from the mean.