Here we look at a quick example of conducting a hypothesis test. In this video, we'll walk you through a step-by-step process for testing hypotheses with a known normal distribution. Using a real-world example, you'll learn how to set up null and alternative hypotheses, calculate the test statistic, and use the rejection region approach as well as the p-value approach to decide whether or not to reject the null hypothesis.

What You Will Learn:

Understanding the basics of hypothesis testing and its significance in statistics.

How to formulate the null hypothesis and alternative hypothesis

Calculating the test statistic and interpreting results.

Using the rejection region and significance level to make decisions about the null hypothesis.

Applying the concepts of p-values and rejection regions to draw meaningful conclusions about data.

This video is ideal for students aiming to build a strong foundation in hypothesis testing,

particularly when the standard deviationis known, and the sample follows a normal distribution.

Perfect for statistics learners, math enthusiasts, and teachers looking for an engaging, practical demonstration of hypothesis testing.

(https://www.youtube.com/watch?v=NSgzTnrpakk)
(https://www.youtube.com/watch?v=PZPJTgwQbww)
(https://www.youtube.com/watch?v=9kNt-7KE_wA)

About Pearson Education:

This video is part of an educational series for Pearson Education, a leading provider of high-quality learning materials. Through clear explanations and practical examples, we aim to make statistics concepts like hypothesis testing accessible and engaging.

For more great statistics videos, please check out (https://www.pearson.com/en-us/search.html/Statistics)


#HypothesisTesting #Statistics #MathTutorial #PearsonEducation #NullHypothesis #AlternativeHypothesis #StatisticsLessons #MathHelp #SignificanceLevel #ZScore #RejectionRegion #PValue #StatisticsTutorial #mathconcepts



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Constructing Confidence Intervals

In this video I look at a quick intro to hypothesis testing. I try to give some motivation as well as highlight some important formulas for future reference.

No description provided.

In this video, we break down the concept of confidence intervals—a fundamental topic in statistics that helps you estimate a range for a population parameter. Using real-world examples, we explore how to calculate and interpret confidence intervals step by step, using sample data to provide insights into a population mean.

What You Will Learn:

What confidence intervals are and why they matter in statistical analysis.

How to use sample data to construct a 90% confidence interval.

Understanding how changing confidence levels (like 95% and 99%) affect the interval.

Step-by-step guide to using the formula for estimating population means.

How to compare confidence intervals and interpret results for decision-making.

This video is perfect for students looking to gain a deeper understanding of confidence intervals, including how they are used to provide a range estimate for parameters when direct measurement of an entire population isn't feasible.

Whether you're a statistics student, a teacher, or a math enthusiast, you'll find practical examples and an easy-to-follow explanation of how to construct and interpret confidence intervals effectively.
If you want to learn how confidence intervals can be used to make inferences about populations or apply them in various real-world scenarios (like political polling or quality control), this video will guide you through the key concepts, formulas, and applications.

(https://www.youtube.com/watch?v=YB7P2ZlmY7c)

(https://www.youtube.com/watch?v=8KYsyMdMPyc)

(https://www.youtube.com/watch?v=uF3YxC6SL70)

About Pearson Education:

This video is part of a project for Pearson Education, a leader in creating educational content to help students succeed in their studies. We aim to provide clear and engaging lessons that simplify complex math and statistics topics, helping you excel in your coursework.

For more great statistics videos, please check out (https://www.pearson.com/en-us/search.html/Statistics)


#ConfidenceIntervals #Statistics #MathTutorial #PearsonEducation #ConfidenceIntervalFormula #SampleData #StatisticsLessons #MathHelp #PopulationMean #InferentialStatistics #ConfidenceIntervalCalculation #MathConcepts #StatisticsTutorial


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Cumulative Probability Distribution Functions (CDFs)
Learn how to work with cumulative probability distribution functions (CDFs) for both discrete and continuous random variables. In this video, we define CDFs, show how they accumulate probabilities, and compare their formulas:

Discrete:

F(x)=∑P(X=x_i)

Continuous:

F(x) = ∫ f(t)dt on (−∞, x)
​

We also dive into the exponential distribution as an example, covering its probability density function (PDF) and CDF. By the end, you'll know how to apply CDFs to solve real-world problems, such as finding probabilities and proportions for events.

What You Will Learn:
What cumulative distribution functions are and why they matter.
The key differences between discrete and continuous CDFs.
Practical applications, including working with the exponential distribution.
If you're studying probability and statistics, this is an essential concept to master!

Support my work on Patreon: https://www.patreon.com/patrickjmt?ty=c

#CumulativeProbability #CDF #Probability #Statistics #RandomVariables #ExponentialDistribution #ProbabilityDensityFunction #PatrickJMT #MathHelp #MathTutorial #LearnProbability #MathExplained #EducationalMath #ProbabilityBasics #DiscreteVariables #ContinuousVariables

Visual Proof for the Area of a Circle

Unlock a powerful and intuitive way to understand the formula for the area of a circle! In this video, we’ll break down the steps to visually demonstrate why the area of a circle is given by A=(pi)*r^2. By dissecting a circle and reassembling its parts, we reveal the reasoning behind this classic formula in a clear, visual manner.

What You Will Learn

How to decompose a circle into manageable parts
The process of rearranging these parts to form a parallelogram-like shape
A step-by-step explanation of why this shape leads to the formula A = π r ^2

Description
In this visual proof, we’ll explore the fascinating concept of finding the area of a circle using a unique approach. Instead of relying on algebra alone, this method takes a geometric approach by "breaking up" the circle into small segments and "gluing" them back together to form a shape similar to a parallelogram. This reassembly makes it easy to see why the area formula A= π * r ^2 is correct, as it corresponds directly to the dimensions of this newly formed shape. Whether you're a student, teacher, or math enthusiast, this visual proof provides an engaging and intuitive way to grasp the concept of a circle’s area. Watch along as we transform complex math into something visually memorable!

Support me on Patreon: https://www.patreon.com/patrickjmt?ty=c

#patrickjmt #areaofacircle #visualproof #geometry #mathproof

(https://youtu.be/bjt3GiM4B0A)
Diagonalization Part 2 | Solving the Example of Matrix A

In this follow-up video, we continue our journey into matrix diagonalization by working through the example introduced in the previous video. We'll apply the diagonalization process to matrix
A and fully break down each step, from calculating eigenvalues and eigenvectors to constructing matrices P and D. This video will solidify your understanding of diagonalization by seeing it applied in practice.

What You Will Learn:
Step-by-step solution to the diagonalization example from Part 1.
How to construct matrices
P and D for matrix A.
Techniques to simplify computing powers of A using diagonalization.
This example-based walkthrough will help you grasp the practical application of diagonalization, making matrix computations more intuitive and manageable.

[Patreon support link for PatrickJMT] (https://www.patreon.com/patrickjmt?ty=c)

#Diagonalization #MatrixFactorization #MatrixExample #Eigenvalues #Eigenvectors #PatrickJMT #Mathematics #LinearAlgebra #MathTutorial #MathHelp #EducationalMath #MathExplained #MathTeacher #MatrixDecomposition #MatrixSimplification #Algebra #LearnMath

(https://youtu.be/n5wcrpc0ng0)
In this introductory video, we cover the basics of matrix diagonalization, focusing on the core concept of representing a matrix A = PDP ^(−1). This powerful technique simplifies the computation of matrix powers, making it much easier to calculate A^k.
You’ll see the step-by-step process, including finding eigenvalues and eigenvectors, constructing matrices P and D, and using them to diagonalize A.
Note that while an example is introduced in this video, the solution is completed in a follow-up video.

What You Will Learn:
The concept of matrix diagonalization and why it is useful.
Step-by-step breakdown of representing a matrix in the form A=PDP ^(−1).
How to apply this technique to simplify matrix computations.
Whether you're learning linear algebra or preparing for advanced applications, understanding diagonalization is essential for simplifying complex matrix operations.

(https://www.patreon.com/patrickjmt?ty=c)

#Diagonalization #MatrixFactorization #Eigenvalues #Eigenvectors #PatrickJMT #Mathematics #MatrixTheory #LinearAlgebra #MathTutorial #MatrixDecomposition #MathHelp #EducationalMath #MathExplained #MathTeacher #MatrixPowers #LearnMath #Algebra #MatrixTransformation #MatrixSimplification

(https://youtu.be/vI6XO09JF6w)
(https://youtu.be/6tnAzRMtq3M)
Matrix LU Factorizations | Circuits and the Transfer Matrix
In this video, we'll delve into the application of matrix and LU factorizations in the context of electrical circuits, focusing on transfer matrices in ladder networks. Using matrices to model circuits helps analyze and simplify complex networks. Here, you'll see how input and output voltages and currents relate through a transfer matrix, a valuable tool in electrical engineering.

What You Will Learn:
The basics of transfer matrices in electrical circuits.
How LU factorization applies to circuit analysis.
Step-by-step example of deriving the transfer matrix for a ladder network.
This knowledge is essential for anyone interested in circuit design, linear algebra applications, or electrical engineering fundamentals. By the end of the video, you'll have a clear understanding of how to approach transfer matrices and use LU factorization to simplify circuit networks.

[Patreon support link for PatrickJMT] (https://www.patreon.com/patrickjmt?ty=c)

#MatrixFactorization #LUFactorization #TransferMatrix #LadderNetworks #ElectricalEngineering #PatrickJMT #Mathematics #CircuitAnalysis #LinearAlgebra #MathTutorial #EngineeringMath #Voltage #Current #MathHelp #MatrixTheory #MathTeacher #EducationalMath #LUDecomposition #MathExplained #LearnEngineering #Circuits #CircuitTheory