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Estimating Population Mean with Unknown Standard Deviation01:22

Estimating Population Mean with Unknown Standard Deviation

8.3K
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...
8.3K
Estimating Population Mean with Known Standard Deviation01:16

Estimating Population Mean with Known Standard Deviation

8.9K
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 +...
8.9K
Estimating Population Standard Deviation01:26

Estimating Population Standard Deviation

3.1K
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...
3.1K
Confidence Interval for Estimating Population Mean01:25

Confidence Interval for Estimating Population Mean

8.0K
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...
8.0K
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

14.5K
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
14.5K
Prediction Intervals01:03

Prediction Intervals

2.3K
The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y. 
2.3K

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Related Experiment Video

Updated: Aug 24, 2025

Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine
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Distributed Estimation of Support Vector Machines for Matrix Data.

Wangli Xu, Jiamin Liu, Heng Lian

    IEEE Transactions on Neural Networks and Learning Systems
    |October 21, 2022
    PubMed
    Summary
    This summary is machine-generated.

    This study analyzes nuclear-norm regularized linear support vector machines (SVMs), establishing estimator convergence rates in high dimensions. A communication-efficient distributed estimator is also proposed, achieving similar performance for machine learning discrimination tasks.

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    Area of Science:

    • Machine Learning
    • Statistical Learning Theory
    • High-Dimensional Data Analysis

    Background:

    • Discrimination problems are central to machine learning.
    • Growing interest in extending vector-based methods to matrix forms.
    • Support Vector Machines (SVMs) are a key tool for classification.

    Purpose of the Study:

    • Investigate statistical properties of nuclear-norm regularized linear SVMs.
    • Establish convergence rates for these estimators in high-dimensional settings.
    • Propose a communication-efficient distributed estimator.

    Main Methods:

    • Analysis of nuclear-norm-based regularized linear SVMs.
    • Theoretical investigation of estimator convergence rates.
    • Development of a distributed estimation algorithm.

    Main Results:

    • Established the convergence rate of the nuclear-norm regularized linear SVM estimator in the high-dimensional setting.
    • Proposed a communication-efficient distributed estimator achieving the same convergence rate.
    • Demonstrated estimator performance through simulations and empirical data analysis.

    Conclusions:

    • The proposed methods offer efficient solutions for discrimination problems in high-dimensional and distributed settings.
    • Nuclear-norm regularization provides a robust approach for linear SVMs.
    • The study contributes to the advancement of scalable machine learning algorithms.