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Confidence Intervals

Confidence Intervals

An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a sample proportion. However, unlike the point estimate which is a single value, the confidence interval contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
A confidence...

Interpretation of Confidence Intervals

Interpretation of Confidence Intervals

A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
Confidence intervals have confidence coefficients that are crucial for their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99, which can be written as percentages–90%, 95%, and 99%, respectively.
Suppose a person calculates a confidence interval with a confidence coefficient of 0.95. In that case, they can...

Estimating Population Mean with Known Standard Deviation

Estimating Population Mean with Known Standard Deviation

To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate - error bound, point estimate + error bound)
The...

Confidence Interval for Estimating Population Mean

Confidence Interval for Estimating Population Mean

A point estimate of the population mean is obtained from a single sample. Such a point estimate does not represent a population well because it needs to account for variability in the population. Single point estimate can also be biased despite the sample being selected randomly. Thus, a point estimate is often unreliable. A confidence interval is needed to reduce this unreliability.
A confidence interval for the mean is a range of values that provides an estimate of the population mean. As the...

Estimating Population Mean with Unknown Standard Deviation

Estimating Population Mean with Unknown Standard Deviation

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the Guinness...

Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor 't,' or...

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Related Experiment Video

Updated: Jun 21, 2026

Establishing a Competing Risk Regression Nomogram Model for Survival Data

Establishing a Competing Risk Regression Nomogram Model for Survival Data

Published on: October 23, 2020

Improved approximate confidence intervals for the mean of a log-normal random variable.

Douglas J Taylor¹, Lawrence L Kupper, Keith E Muller

¹Family Health International, 2224 E. NC Hwy 54, Durham, North Carolina 27713, USA. dtaylor@bios.unc.edu

Statistics in Medicine

|August 21, 2002

Summary

Approximate confidence intervals for log-normal means often fail. New methods provide accurate coverage, even for small sample sizes, improving statistical analysis reliability.

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Area of Science:

Statistics
Computational Statistics

Background:

Log-normal distribution is common in various fields.
Exact confidence intervals for the mean are computationally intensive.
Approximate intervals are frequently used but may lack accuracy.

Purpose of the Study:

To evaluate the coverage error of existing approximate confidence intervals for log-normal means.
To develop and present new, more accurate procedures for constructing these intervals.
To ensure reliable statistical inference for log-normal data.

Main Methods:

Evaluation of two simple approximation methods.
Assessment of a sophisticated approximation using numerical integration or bootstrap sampling.
Development of a new procedure involving standard distribution function integration.
Introduction of a related method avoiding integration.

Main Results:

Existing simple approximations exhibit significant coverage errors, especially for small to moderate sample sizes (n ≤ 100).
Sophisticated approximations show improvement but still have unacceptable errors for n ≤ 25.
The newly developed procedure achieves satisfactory coverage errors for n as small as 5.
The related method without integration outperforms simple approximations for n ≤ 100, maintaining coverage error below alpha.

Conclusions:

Existing approximate confidence intervals for log-normal means are unreliable across various sample sizes.
New computational procedures offer improved accuracy and reliability.
The developed methods are practical and suitable for small sample sizes, enhancing statistical analysis.