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Related Concept Videos

Estimating Population Mean with Known Standard Deviation01:16

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...
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...
Estimating Population Standard Deviation01:26

Estimating Population Standard Deviation

When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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...
Confidence Interval for Estimating Population Mean01:25

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...
Distributions to Estimate Population Parameter01:26

Distributions to Estimate Population Parameter

The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...

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Basics of Multivariate Analysis in Neuroimaging Data
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Published on: July 24, 2010

Estimating the Correlation in Bivariate Normal Data with Known Variances and Small Sample Sizes().

Bailey K Fosdick1, Adrian E Raftery

  • 1Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322.

The American Statistician
|February 5, 2013
PubMed
Summary

Bayesian estimators with arc-sine priors best estimate small-sample correlations in bivariate normal data. These methods also improved hypothesis testing for zero correlation, outperforming traditional tests in small samples.

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Area of Science:

  • Statistics
  • Statistical Inference
  • Bayesian Statistics

Background:

  • Estimating correlation in bivariate normal data is crucial.
  • Small sample sizes pose challenges for accurate correlation estimation.
  • Known means and variances simplify the estimation problem.

Purpose of the Study:

  • To evaluate various estimators for bivariate normal correlation in small samples.
  • To compare Bayesian estimators against empirical and maximum likelihood methods.
  • To assess the performance of hypothesis tests for zero correlation.

Main Methods:

  • Simulation study comparing eight different correlation estimators.
  • Utilized Bayesian estimators with uniform and arc-sine priors.
  • Assessed performance in small sample cases and for varying correlation values.

Main Results:

  • Bayesian estimators with uniform and arc-sine priors outperformed other methods in small samples.
  • The arc-sine prior demonstrated superior performance for large correlation values.
  • Bayesian hypothesis tests were more effective than significance tests in small samples.

Conclusions:

  • The posterior mean with the arc-sine prior is recommended for small-sample correlation estimation when variances are known.
  • Bayesian approaches offer advantages for both estimation and hypothesis testing in specific scenarios.
  • Performance differences diminished with larger sample sizes (n=50).