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Related Concept Videos

Student t Distribution01:31

Student t Distribution

The population standard deviation is rarely known in many day-to-day examples of statistics. When the sample sizes are large, it is easy to estimate the population standard deviation using a confidence interval, which provides results close enough to the original value. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
The Student t distribution was developed by William S. Goset (1876–1937) of the...
Estimating Population Mean with Unknown Standard Deviation01:22

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...
Central Limit Theorem01:14

Central Limit Theorem

The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
The sample size, n, that...
Unusual Results01:16

Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.
According to the range rule of thumb, any value above or below two standard deviations, 2σ  from the mean, μ  is considered unusual.
Maximum unusual value = μ + 2σ
Minimum unusual value...
Applications of Normal Distribution01:22

Applications of Normal Distribution

The normal distribution is a useful statistical tool. One of its practical applications is determining the door height after considering the normal distribution of heights of persons, such that many can pass through it easily without striking their heads. The normal distribution can also determine the probability of a person having a height less than a specific height.
The heights of 15 to 18-year-old males from Chile from 1984 to 1985 followed a normal distribution. The mean height is 172.36...
Hazard Ratio01:12

Hazard Ratio

The hazard ratio (HR) is a widely used measure in clinical trials to compare the risk of events, such as death or disease recurrence, between two groups over time. It reflects the ratio of hazard rates—the instantaneous risk of the event occurring—between a treatment group and a control group. This measure provides valuable insights into the relative effectiveness of a treatment by assessing how the risk of an event differs between the two groups.
For example, in a clinical trial evaluating a...

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Related Experiment Video

Updated: Jun 17, 2026

Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method
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Optimization of Processing of Tiebangchui with Highland Barley Wine Based on the Box-Behnken Design Combined with the Entropy Method

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Tolerance limits for a ratio of normal random variables.

Lanju Zhang1, Thomas Mathew, Harry Yang

  • 1Department of Biostatistics, MedImmune, Inc., Gaithersburg, Maryland 20878, USA. zhangla@MedImmune.com

Journal of Biopharmaceutical Statistics
|January 16, 2010
PubMed
Summary

This study develops methods for upper tolerance limits for ratios of normal random variables, applicable to both independent and bivariate normal distributions. These statistical techniques are crucial for applications like reverse transcriptase assays.

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Area of Science:

  • Statistics
  • Probability Theory
  • Biostatistics

Background:

  • Deriving upper tolerance limits for ratios of random variables is essential in various scientific fields.
  • Existing methods may not adequately address scenarios involving bivariate or independent normal distributions.

Purpose of the Study:

  • To develop and present methods for deriving upper tolerance limits for the ratio of two normally distributed random variables.
  • To address cases where variables are bivariate normal or independent normal.
  • To illustrate the application and performance of the proposed methods.

Main Methods:

  • Utilizing the relationship between upper tolerance limits and lower confidence limits for cumulative distribution functions (cdf).
  • Employing generalized confidence intervals to derive the necessary cdf confidence limits.
  • Applying specific representations of the cdf for the ratio in bivariate normal cases.
  • Developing a simplified derivation for cases with a small coefficient of variation.

Main Results:

  • The study successfully derives upper tolerance limits for the ratio of two normal random variables under different distribution assumptions.
  • A simplified approach is presented for specific conditions, enhancing applicability.
  • Numerical results demonstrate the performance of the proposed tolerance limit methods.

Conclusions:

  • The proposed methods provide a robust framework for establishing upper tolerance limits for ratios of normal random variables.
  • The techniques are validated through a reverse transcriptase assay example, showing practical utility.
  • The findings contribute to statistical inference in applications involving ratios of normally distributed data.