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Related Concept Videos

Estimating Population Mean with Known Standard Deviation01:16

Estimating Population Mean with Known Standard Deviation

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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 +...
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Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

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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.
William S. Gosset (1876–1937) of the...
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Estimating Population Standard Deviation01:26

Estimating Population Standard Deviation

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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...
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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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Testing a Claim about Mean: Known Population SD01:11

Testing a Claim about Mean: Known Population SD

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A complete procedure of testing the hypothesis about a population mean is explained here.
Estimating a population mean requires the samples to be distributed normally. The data should be collected from the randomly selected samples having no sampling bias. The sample size needed to be higher than 30, and most importantly, the population standard deviation should be already known.
In most realistic situations, the population standard deviation is often unknown, but in rare circumstances, when it...
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Empirical Method to Interpret Standard Deviation01:09

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The empirical rule, also known as the three-sigma rule, allows a statistician to interpret the standard deviation in a normally distributed dataset. The rule states that 68% of the data lies within one standard deviation from the mean, 95% lies within two standard deviations from the mean, and 99.7% lies within three standard deviations from the mean. Additionally, this rule is also called the 68-95-99.7 rule.
This rule is used widely in statistics to calculate the proportion of data values...
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Robust estimation of mean-variance relation.

Mushan Li1, Yanyuan Ma1

  • 1Department of Statistics, Pennsylvania State University, University Park, Pennsylvania, USA.

Statistics in Medicine
|November 23, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a new statistical method to accurately assess the mean-variance relationship in biomedical data, improving analysis by accounting for data uncertainties and varying experimental conditions.

Keywords:
mean-variance relationmeasurement errorrobustnesssemiparametrics

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Area of Science:

  • Biostatistics
  • Biomedical Data Analysis
  • Statistical Modeling

Background:

  • Accurate mean-variance relation assessment is crucial for biomedical research analysis.
  • True mean and variance are often unavailable in biomedical datasets, necessitating the use of sample statistics.
  • Variations in experimental conditions can lead to differing mean-variance relationships within the same dataset.

Purpose of the Study:

  • To develop a robust semiparametric estimator for the mean-variance relationship in biomedical data.
  • To address challenges posed by unavailable true parameters and heterogeneous experimental conditions.
  • To improve the accuracy of statistical analyses in biomedical research.

Main Methods:

  • A semiparametric estimation approach is proposed.
  • Uncertainty in sample mean is treated as a measurement error.
  • Uncertainty in sample variance is modeled as model error.
  • A mixture model is employed to handle varying mean-variance relations.

Main Results:

  • Asymptotic normality of the proposed semiparametric estimator is theoretically established.
  • Simulation studies confirm the finite sample properties of the method.
  • A data application demonstrates the method's effectiveness compared to existing approaches.

Conclusions:

  • The proposed semiparametric method provides sensible results for mean-variance relation assessment in biomedical data.
  • The method effectively accounts for uncertainties in sample means and variances.
  • It successfully addresses differing mean-variance relationships arising from diverse experimental conditions.