P̂ ± z* √(p̂(1-p̂)/n)

Definition

The term $p̂ ± z* √(p̂(1-p̂)/n)$ represents the formula used to calculate a confidence interval for a population proportion. It combines the point estimate of the population proportion ($p̂$), the z-score corresponding to the desired confidence level ($z$), and the standard error of the proportion ($\sqrt{p̂(1-p̂)/n}$) to determine the range within which the true population proportion is likely to fall.

5 Must Know Facts For Your Next Test

1. The point estimate of the population proportion, $p̂$, is calculated by dividing the number of successes in the sample by the total sample size.
2. The z-score, $z$, represents the number of standard deviations from the mean that corresponds to the desired confidence level, such as 1.96 for a 95% confidence level.
3. The standard error, $\sqrt{p̂(1-p̂)/n}$, accounts for the variability in the sample proportion and the size of the sample.
4. The confidence interval provides a range of values that is likely to contain the true population proportion, with the desired level of confidence.
5. The confidence interval is used to make inferences about the population proportion and assess the precision of the sample estimate.

Review Questions

• Explain the purpose of the confidence interval formula $p̂ ± z* √(p̂(1-p̂)/n)$ in the context of population proportions.
  • The confidence interval formula $p̂ ± z* √(p̂(1-p̂)/n)$ is used to estimate the range of values within which the true population proportion is likely to fall, given a sample statistic and a desired confidence level. The formula combines the point estimate of the population proportion ($p̂$), the z-score corresponding to the confidence level ($z$), and the standard error of the proportion ($\sqrt{p̂(1-p̂)/n}$) to provide a range of values that captures the uncertainty in the sample estimate. This allows researchers to make inferences about the population proportion and assess the precision of their findings.
• Describe how the components of the confidence interval formula $p̂ ± z* √(p̂(1-p̂)/n)$ are used to determine the reliability of a sample estimate of a population proportion.
  • The components of the confidence interval formula $p̂ ± z* √(p̂(1-p̂)/n)$ work together to determine the reliability of a sample estimate of a population proportion. The point estimate $p̂$ represents the sample proportion, which serves as the best guess for the true population proportion. The z-score $z$ reflects the desired level of confidence, with a larger z-score indicating a higher level of confidence. The standard error $\sqrt{p̂(1-p̂)/n}$ accounts for the variability in the sample proportion and the size of the sample, with a smaller standard error indicating a more precise estimate. By combining these elements, the confidence interval provides a range of values that is likely to contain the true population proportion, allowing researchers to assess the reliability and precision of their sample-based conclusions.
• Analyze how the sample size $n$ and the confidence level affect the width of the confidence interval calculated using the formula $p̂ ± z* √(p̂(1-p̂)/n)$.
  • The sample size $n$ and the confidence level, as represented by the z-score $z$, both have a significant impact on the width of the confidence interval calculated using the formula $p̂ ± z* √(p̂(1-p̂)/n)$. As the sample size $n$ increases, the standard error $\sqrt{p̂(1-p̂)/n}$ decreases, resulting in a narrower confidence interval. This means that larger samples provide more precise estimates of the population proportion. Conversely, as the desired confidence level increases (e.g., from 90% to 95%), the corresponding z-score $z$ also increases, leading to a wider confidence interval. This trade-off between precision and confidence level must be considered when designing a study and interpreting the results. Researchers must balance the need for a high level of confidence with the desire for a narrow, more informative confidence interval.

Related terms

Population Proportion:

The proportion or percentage of a population that possesses a certain characteristic or attribute.

Confidence Interval:

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

Standard Error:

The standard deviation of the sampling distribution of a statistic, which represents the amount of variability in the statistic from sample to sample.