P̂1

Definition

The sample proportion of the first population, which is an estimate of the true population proportion, p1. This term is particularly relevant in the context of comparing two independent population proportions and hypothesis testing for two proportions.

5 Must Know Facts For Your Next Test

1. The sample proportion p̂1 is used to estimate the true population proportion p1 when comparing two independent populations.
2. In hypothesis testing for two proportions, p̂1 is the sample proportion from the first population, and p̂2 is the sample proportion from the second population.
3. The difference between the two sample proportions, p̂1 - p̂2, is used to test hypotheses about the difference between the true population proportions, p1 - p2.
4. The standard error of the difference between the two sample proportions, $\sqrt{\frac{p̂1(1-p̂1)}{n_1} + \frac{p̂2(1-p̂2)}{n_2}}$, is a crucial component in the test statistic and confidence interval calculations.
5. The sampling distribution of the difference between the two sample proportions, p̂1 - p̂2, follows a normal distribution when the sample sizes are large enough, allowing for the use of the z-test or t-test for hypothesis testing.

Review Questions

• Explain the role of p̂1 in the context of comparing two independent population proportions.
  • In the topic of comparing two independent population proportions (Section 10.3), p̂1 represents the sample proportion of the first population. It is used to estimate the true population proportion p1, which is then compared to the sample proportion p̂2 of the second population. The difference between the two sample proportions, p̂1 - p̂2, is the basis for testing hypotheses about the difference between the true population proportions, p1 - p2.
• Describe how p̂1 is used in the hypothesis testing for two proportions (Section 10.5).
  • When conducting hypothesis testing for two proportions, p̂1 is the sample proportion from the first population, and p̂2 is the sample proportion from the second population. The test statistic, which is used to determine if there is a significant difference between the two population proportions, is calculated using the difference between the two sample proportions, p̂1 - p̂2. The standard error of this difference, which incorporates p̂1 and p̂2, is a crucial component in the hypothesis testing process and the calculation of the confidence interval.
• Analyze the importance of the sampling distribution of p̂1 - p̂2 in the context of comparing two independent population proportions and hypothesis testing for two proportions.
  • The sampling distribution of the difference between the two sample proportions, p̂1 - p̂2, follows a normal distribution when the sample sizes are large enough. This allows for the use of the z-test or t-test for hypothesis testing, as the test statistic can be standardized and compared to the appropriate critical value or p-value. The normality of the sampling distribution is a key assumption that must be met in order to draw valid conclusions about the difference between the true population proportions, p1 - p2, based on the observed difference between the sample proportions, p̂1 - p̂2.

Related terms

Population Proportion: The true proportion of a characteristic of interest in a population, denoted as p.

Sample Proportion: The proportion of a characteristic of interest in a sample, denoted as p̂.

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