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Related Concept Videos

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 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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Testing a Claim about Mean: Unknown Population SD01:21

Testing a Claim about Mean: Unknown Population SD

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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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Regression Toward the Mean01:52

Regression Toward the Mean

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Regression toward the mean (“RTM”) is a phenomenon in which extremely high or low values—for example, and individual’s blood pressure at a particular moment—appear closer to a group’s average upon remeasuring. Although this statistical peculiarity is the result of random error and chance, it has been problematic across various medical, scientific, financial and psychological applications. In particular, RTM, if not taken into account, can interfere when...
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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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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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Second-Order Inference for the Mean of a Variable Missing at Random.

Iván Díaz, Marco Carone, Mark J van der Laan

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    |May 27, 2016
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    Summary
    This summary is machine-generated.

    We developed a new statistical method for handling missing data, improving accuracy and reliability in analysis. This second-order estimator offers better performance than existing approaches for estimating means with incomplete datasets.

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    Area of Science:

    • Statistics
    • Biostatistics
    • Data Science

    Background:

    • Missing data is a common challenge in statistical analysis.
    • Existing methods for handling missing data, such as doubly robust methods, have limitations.
    • Accurate estimation of means with missing data is crucial for reliable scientific conclusions.

    Purpose of the Study:

    • To develop a novel second-order estimator for the mean of a variable with missing data.
    • To improve upon existing first-order doubly robust methods by incorporating a second-order expansion.
    • To provide a more robust and accurate estimation strategy under the missing at random assumption.

    Main Methods:

    • Utilized the targeted minimum loss-based estimation (TMLE) framework.
    • Developed an approximate second-order expansion of the parameter functional.
    • Introduced a novel first-order estimator inspired by a second-order expansion, requiring only one-dimensional smoothing.

    Main Results:

    • The second-order TMLE consistently achieved coverage probabilities closer to the nominal 0.95 compared to first-order methods.
    • In simulations, the second-order TMLE reached a coverage probability of 0.86 where the first-order TMLE reached zero.
    • The novel first-order estimator improved coverage probability from 0 to 0.90 in simulations.

    Conclusions:

    • The proposed second-order TMLE offers improved accuracy and robustness for estimating means with missing data.
    • The novel first-order estimator provides a computationally simpler alternative with enhanced finite sample performance.
    • These methods offer valuable tools for researchers dealing with incomplete datasets, with practical applications demonstrated in a clinical dataset.