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Sample Size, Control Distance, and Type I and Type II Errors in Control Charts

Type I and Type II errors in control charts (or in statistics) are difficult to explain concepts. When trying to explain the impact of the sample size and the control distances in the magnitude of Type I and Type II errors, it becomes exponentially more difficult to explain.

The embedded visualization shows three distributions:

• A population distribution (PDF: g(x)) with the average m = 12 and the s.d. S = 0.5. The process target is t = 12.
• A sampling distribution (PDF: f(x)) with the sample size n = 4 and the standard error = S/sqrt(n)
• A sampling distribution (PDF: h(x)) with a “derailed mean” c = 12.32

It uses the following control charts specifications.

• A control chart with the control distance of k = 1.96, ucl = t + k*s, and lcl = t – k*s

If you are familiar with the sampling distribution, the size of k will determine the size of both the Type I error and Type II error, and the size of n will determine the size of Type II error.

You can refer to the attached in-class exercise note to adapt it to your teaching/learning needs. This visualization to helps you to see how Type I error and Type II error changes as you:

• Change the sample size n
• Change the control distance k
• Change the mean of the process m

Go to the visualization: https://www.desmos.com/calculator/hccg2zkpon

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Normal Distribution Visualized (Interactive)

You can:

• See how the shape of the distribution changes with a difference choice of $\mu$ and $\sigma$.
• You can overlay another normal distribution with a set of different parameters.
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“The Area Under the Curve Concept” using Uniform Distribution

In this example

$f(x)=\frac{1}{20}$ $where 120 \leq x \leq 140$

You can change the value of $b$ to see how does it change the value of the integral, which is the area under the curve.

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This is the lecture note I have developed for fall 2020 business statistics. I will develop the next half as my course progresses.

The slide is developed using LaTeX. So, I don’t really have a PPT version. If you want to use it as PPT, you can export all the pages as images, then import those images into PPT.

Some highlights of the lecture note:

• Less number of chapters than most textbooks in the market (I did not omit any important topics, I just combined chapters so that it appears there is a smaller number of chapters to study.)
• I did eliminate some topics that I personally do not use in the descriptive statistics part. (for example, stem-and-leaf, stacked bar chart, etc.)
• Has a detailed topic index where you can conveniently jump to the point (From the students’ perspective, it is convenient for the review purpose.)
• Introducing and emphasizing the idea of the distribution from earlier chapters. (Students typically struggle with the concept of distribution because it is so theoretical. I tried to demonstrate the formation of and the practical utility of using a distribution from early sections.)
• There is a brief introduction on how to read and utilize mathematic notations, especially the summation notation. (Students struggle with this is a lot. If they do not understand notations, providing a formula sheet does not mean anything.)

Credit: I did use some images and questions from Andrew et al, Statistics for Business and Economics, 12th ed. For other sources, I tried to add the credit whenever possible. I tried to develop the visualization myself whenever possible.

Copyright: If you want to use it for your class, please just drop a line here in reply or message me over my LinkedIn. I just want to make more friends. If you want my LaTeX code, I can share with you if you agree to share your work with me afterwards, so that I can study from you.

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What could go wrong with a product design?

This is what happens when you do not respect the voice of your customers.

This illustration can also be used in the project management context to emphasize the importance of the client requirements gathering and the project scope mismanagement and what it can cause to the project deliverables.

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Essence of Linear Algebra

This channel provides a good review on Linear Algebra

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Data, Histogram, Distribution, and Probability

This set of video explains the following ideas:

• Video 1: How do you get a distribution from a bunch of numbers?
• Video 2: What to look for from a distribution?
• Video 3: How to utilize a distribution to get probability?

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From Factory to Market: Distribution Problem (Linear Programming Approach)

This exercise introduces an important topic in the operations and supply chain management – distribution problem. Essentially, it deals with a problem where you need to match the supply and demand with the least transportation cost.

To understand the impact of this kind of approach in the real world, please refer to the following talk by Jack Levis, the guy who led the optimization effort in UPS.