Confidence Interval
Interval estimate for population mean
About Confidence Interval
The Confidence Interval represents interval estimate for population mean. This statistics formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Statistics and Mathematical statistics. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Confidence Interval include Quality control, Medical research, Market research, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}Formula Information
Difficulty Level
Prerequisites
Discovered
20th century
Discoverer
Jerzy Neyman
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Important Notes
Confidence intervals provide a range of plausible values for the population parameter with a specified confidence level.
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Frequently Asked Questions
What is a confidence interval?
A confidence interval is a range of values that likely contains the true population parameter (like the mean) with a specified level of confidence. For example, a 95% confidence interval means if you repeated the sampling process many times, 95% of the intervals would contain the true population mean.
What does 95% confidence mean?
95% confidence means that if you constructed many confidence intervals from different samples, about 95% of them would contain the true population parameter. It does NOT mean there's a 95% chance the specific interval contains the parameter - the parameter is either in it or not (it's fixed).
How do I interpret a confidence interval?
If you have a 95% CI of [10, 20] for a mean, you can say: 'We are 95% confident that the true population mean lies between 10 and 20.' The interval gives you a range of plausible values. Wider intervals indicate more uncertainty; narrower intervals indicate more precision.
What's the difference between z-interval and t-interval?
Use z-interval when you know the population standard deviation σ (rare in practice). Use t-interval when you only know the sample standard deviation s (common). The t-distribution accounts for the extra uncertainty from estimating σ. For large samples (n ≥ 30), z and t are very similar.
How does sample size affect the confidence interval?
Larger sample sizes produce narrower (more precise) confidence intervals. The margin of error is proportional to 1/√n, so doubling the sample size makes the interval √2 ≈ 1.41 times narrower. To halve the margin of error, you need 4 times the sample size.
What affects the width of a confidence interval?
The width depends on: 1) Confidence level (higher = wider), 2) Sample size (larger = narrower), 3) Variability (higher σ or s = wider). To get a narrower interval, increase sample size or decrease confidence level. The trade-off: higher confidence or larger samples cost more.
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- Category
- Statistics
- Difficulty
- Intermediate
- Discovered
- 20th century
- Discoverer
- Jerzy Neyman
- Formula ID
- confidence-interval