P̂ ± z* √(p̂(1-p̂)/n)
from class:
Honors Statistics
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.
congrats on reading the definition of p̂ ± z* √(p̂(1-p̂)/n). now let's actually learn it.
5 Must Know Facts For Your Next Test
- The point estimate of the population proportion, $p̂$, is calculated by dividing the number of successes in the sample by the total sample size.
- 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.
- The standard error, $\sqrt{p̂(1-p̂)/n}$, accounts for the variability in the sample proportion and the size of the sample.
- The confidence interval provides a range of values that is likely to contain the true population proportion, with the desired level of confidence.
- 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.
"P̂ ± z* √(p̂(1-p̂)/n)" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.