Pooled Estimate

Definition

The pooled estimate is a statistical measure used to compare two independent population proportions. It represents a combined or average estimate of the proportions from the two populations, taking into account the variability within each population.

5 Must Know Facts For Your Next Test

1. The pooled estimate is used to determine whether there is a significant difference between the proportions of two independent populations.
2. The pooled estimate is calculated as a weighted average of the sample proportions from the two populations, with the weights based on the sample sizes.
3. The pooled estimate is a crucial component in the test statistic used to compare two independent population proportions.
4. The pooled estimate is used to construct a confidence interval for the difference between the two population proportions.
5. Assumptions for using the pooled estimate include independence of the two samples, normality of the sampling distributions, and equal variances between the populations.

Review Questions

• Explain the purpose of the pooled estimate in the context of comparing two independent population proportions.
  • The pooled estimate is used to compare the proportions of a characteristic or attribute between two independent populations. It provides a combined or average estimate of the proportions from the two populations, taking into account the variability within each population. The pooled estimate is a key component in the hypothesis testing and confidence interval procedures used to determine if there is a significant difference between the two population proportions.
• Describe the process of calculating the pooled estimate and discuss the assumptions required for its use.
  • To calculate the pooled estimate, the sample proportions from the two independent populations are combined using a weighted average, where the weights are based on the sample sizes. The formula for the pooled estimate is: $\hat{p}_{pooled} = \frac{n_1 \hat{p}_1 + n_2 \hat{p}_2}{n_1 + n_2}$, where $n_1$ and $n_2$ are the sample sizes, and $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions for the two populations. The key assumptions for using the pooled estimate are: independence of the two samples, normality of the sampling distributions, and equal variances between the populations.
• Explain how the pooled estimate is used in the hypothesis testing and confidence interval procedures for comparing two independent population proportions.
  • The pooled estimate is a crucial component in the test statistic used to compare two independent population proportions. Specifically, the test statistic is calculated as: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_{pooled}(1 - \hat{p}_{pooled})}{n_1} + \frac{\hat{p}_{pooled}(1 - \hat{p}_{pooled})}{n_2}}}$. This test statistic is then used to determine the p-value and make a decision about the null hypothesis. Additionally, the pooled estimate is used to construct a confidence interval for the difference between the two population proportions, which provides a range of plausible values for the true difference.

Related terms

Population Proportion:

The proportion of a characteristic or attribute present in a population, calculated as the number of individuals with the characteristic divided by the total number of individuals in the population.

Hypothesis Testing:

A statistical method used to determine whether a claim or hypothesis about a population parameter is supported by the sample data.

Confidence Interval:

A range of values that is likely to contain an unknown population parameter, calculated from a sample statistic and a desired level of confidence.