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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.
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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.
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Confidence Intervals

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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.
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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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Sample size and the width of the confidence interval for mean difference.

Xiaofeng Steven Liu¹

¹National Education Association, Washington, DC, USA. xliu@mailbox.sc.edu

The British Journal of Mathematical and Statistical Psychology

|January 23, 2008

Summary

Sample size calculations for confidence intervals must account for the width's random nature. This study reconciles unconditional and conditional probabilities, improving sample size accuracy for mean difference comparisons.

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Area of Science:

Statistics
Biostatistics
Statistical Inference

Background:

The width of a confidence interval for mean difference is often treated as fixed.
Ignoring the stochastic nature of confidence interval width can lead to inadequate sample sizes.
Existing methods may not accurately reflect the probability of achieving a desired interval width.

Purpose of the Study:

To highlight the importance of the confidence interval width as a random variable.
To reconcile unconditional and conditional probabilities for achieving a desired interval width.
To propose a method for determining unequal sample sizes in multiple mean comparisons.

Main Methods:

Viewed confidence interval width for mean difference as a random variable.
Derived the lower bound of conditional probability to reconcile unconditional and conditional probabilities.
Utilized the harmonic mean for calculating unequal sample sizes.

Main Results:

Overlooking the random nature of interval width leads to underestimated sample sizes.
The derived lower bound clarifies the relationship between unconditional and conditional probabilities.
The harmonic mean provides a practical approach for unequal sample size determination.

Conclusions:

Accurate sample size calculation requires acknowledging the random variability of confidence interval width.
The proposed reconciliation of probabilities enhances the reliability of sample size planning.
The harmonic mean method facilitates efficient sample size allocation in complex comparisons.