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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:
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Testing a Claim about Mean: Unknown Population SD01:21

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A complete procedure of testing a hypothesis about a population mean when the population standard deviation is unknown is explained here.
Estimating a population mean requires the samples to be approximately normally distributed. The data should be collected from the randomly selected samples having no sampling bias. There is no specific requirement for sample size. But if the sample size is less than 30, and we don't know the population standard deviation, a different approach is used;...
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Distributions to Estimate Population Parameter01:26

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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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Testing a Claim about Mean: Known Population SD01:11

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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.
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Estimating Population Mean with Unknown Standard Deviation01:22

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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

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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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Log-ratio type estimation for the finite population mean under simple random sampling without replacement with

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Two new logarithmic ratio-type estimators improve finite-population mean estimation using auxiliary variables. These methods offer significant efficiency gains, especially for skewed data, enhancing statistical accuracy in surveys.

Keywords:
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Area of Science:

  • Statistics
  • Survey Methodology

Background:

  • Accurate estimation of finite-population means is crucial in statistical surveys.
  • Traditional methods can be sensitive to outliers and nonlinear relationships.

Purpose of the Study:

  • To introduce two novel logarithmic ratio-type estimators for finite-population mean estimation.
  • To enhance variance stabilization and outlier handling using auxiliary variables.

Main Methods:

  • Development of logarithmic ratio-type estimators under simple random sampling without replacement (SRSWOR).
  • Derivation of closed-form expressions for bias and mean squared error (MSE).
  • Analytic determination of optimal tuning constants via MSE minimization.

Main Results:

  • Proposed estimators consistently reduce MSE compared to classical and competing methods.
  • Achieved substantial percent-relative-efficiency (PRE) gains (empirical ≈ 283%, simulation up to ≈ 670%).
  • Demonstrated robust performance with skewed and heavy-tailed populations.

Conclusions:

  • The novel estimators provide significant improvements in efficiency and robustness.
  • Logarithmic transformation effectively handles nonlinear relationships and outliers.
  • The methods are theoretically sound and practically applicable for survey practitioners.