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Related Concept Videos

Confidence Intervals01:21

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.
A...
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Uncertainty: Confidence Intervals00:54

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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...
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Interpretation of Confidence Intervals01:19

Interpretation of Confidence Intervals

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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...
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Determining the Empirical Formula07:05

Determining the Empirical Formula

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Source: Laboratory of Dr. Neal Abrams - SUNY College of Environmental Science and Forestry
Determining the chemical formula of a compound is at the heart of what chemists do in the laboratory every day. Many tools are available to aid in this determination, but one of the simplest (and most accurate) is the determination of the empirical formula. Why is this useful? Because of the law of conservation of mass, any reaction can be followed gravimetrically, or by change in mass. The empirical...
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Experimental Determination of Chemical Formula02:37

Experimental Determination of Chemical Formula

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The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most concise way of representing this elemental makeup. When a compound’s formula is unknown, measuring the mass of its constituent elements is often the first step in determining the formula experimentally.
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Confidence Interval for Estimating Population Mean01:25

Confidence Interval for Estimating Population Mean

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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...
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Related Experiment Video

Updated: Jan 19, 2026

Confidence Intervals
01:21

Confidence Intervals

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Confidence interval-based sample size determination formulas and some mathematical properties for hierarchical data.

Satoshi Usami1

  • 1Department of Education, University of Tokyo, Japan.

The British Journal of Mathematical and Statistical Psychology
|September 8, 2019
PubMed
Summary
This summary is machine-generated.

Researchers can now determine sample sizes for hierarchical data using new formulas. This ensures desired confidence interval widths in multilevel research, minimizing the risk of statistically underpowered experiments.

Keywords:
confidence intervalexperimental designhierarchical datasample size

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Area of Science:

  • Behavioral and Psychological Research
  • Statistical Modeling
  • Quantitative Psychology

Background:

  • Hierarchical data, common in behavioral and psychological research, involves nested units (e.g., students within classes).
  • Advances in statistical methods over 25 years have improved sample size calculations for such data.
  • Existing methods for sample size determination in hierarchical models can be complex.

Purpose of the Study:

  • To provide practical, closed-form and iterative formulas for sample size determination in hierarchical data.
  • To ensure desired confidence interval widths for experimental effect estimates in multilevel studies.
  • To offer guidance on minimizing the risk of underpowered experiments in hierarchical research designs.

Main Methods:

  • Development of closed-form and iterative formulas for sample size calculation.
  • Application to a four-level hierarchical linear model, accommodating slope variances and covariates.
  • Analysis of mathematical properties related to standard errors of experimental effect estimates.

Main Results:

  • Formulas are provided for both balanced and unbalanced hierarchical designs.
  • The impact of various variance components (e.g., random intercept/slope) on standard errors is analyzed.
  • Minimum required sample sizes at the highest level are identified to enhance statistical power.

Conclusions:

  • The provided formulas offer a robust method for sample size determination in hierarchical data.
  • Understanding the impact of variance components aids in efficient experimental design.
  • These methods help researchers conduct statistically sound experiments with hierarchical data.