Calculating Probabilities in the Normal Distribution: Example Problems

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來自 University System of Georgia 的課程

Six Sigma Tools for Analyze

268 個評分

University System of Georgia

Six Sigma Tools for Analyze

268 個評分

課程 3（共 4 門，Specialization Six Sigma Yellow Belt）

This course will cover the Measure phase and portions of the Analyze phase of the Six Sigma DMAIC (Define, Measure, Analyze, Improve, and Control) process. You will learn about lean tools for process analysis, failure mode and effects analysis (FMEA), measurement system analysis (MSA) and gauge repeatability and reproducibility (GR&R), and you will be introduced to basic statistics. This course will outline useful measure and analysis phase tools and will give you an overview of statistics as they are related to the Six Sigma process. The statistics module will provide you with an overview of the concepts and you will be given multiple example problems to see how to apply these concepts. Every module will include readings, discussions, lecture videos, and quizzes to help make sure you understand the material and concepts that are studied. Our applied curriculum is built around the latest handbook The Certified Six Sigma Handbook (2nd edition) and students will develop /learn the fundamentals of Six Sigma. Registration includes online access to course content, projects, and resources but does not include the companion text The Certified Six Sigma Handbook (2nd edition). The companion text is not required to complete the assignments. However, the text is a recognized handbook used by professionals in the field. Also, it is a highly recommended text for those wishing to move forward in Six Sigma and eventually gain certification from professional agencies such as American Society for Quality (ASQ).

從本節課中

Data Analysis

In this module you will be diving into the statistical side of Six Sigma. You will begin with learning about the basic distribution types which include normal and binomial. You will then proceed to variation and will learn the difference between common and special cause variation.

Characteristics of Probability Distributions2:05

Calculating Probabilities Using the Normal Distribution4:56

Calculating Probabilities in the Normal Distribution: Example Problems3:50

The Binomial Distribution2:44

Binomial Distribution Calculations3:31

Binomial Distribution Calculations: More Examples3:12

與講師見面

Christina Scherrer, PhD
Professor of Industrial Engineering
Department of Systems and Industrial Engineering
David Cook, PhD
Assistant Professor of Mechanical Engineering Technology
Department of Mechanical Engineering Technology
Gregory Wiles, PhD
Interim Chair and Assistant Professor
Department of Systems and Industrial Engineering
Bill Bailey, PhD
Assistant Professor of Industrial Engineering
Department of Systems and Industrial Engineering

Calculating probabilities in the normal distribution, example problems.

Let's take a look at our first example.

Suppose the weights of pound cakes produced by your bakery are normally

distributed with a mean of 16.08 ounces and a standard deviation of 0.05 ounces.

What percent of the pound cakes will way less than 16.00 ounces?

Please pause the video and attempt to solve the problem.

When you have your answer resume the video and I will go over the solution.

We begin by drawing a normal curve, center at the mean of 16.08 ounces.

We want the probability of being less than 16.00 on this curve, so

mark 16.00 somewhere to the left of 16.08.

It doesn't matter that you get in the perfect spot, and then shade that area in.

Notice that you will have shaded less than half the curve, so

you should expect your final answer to be less than 0.5.

We next calculate our z score, this is our x of 16.00

minus our mu of 16.08 all divided by sigma of 0.05.

We obtain -1.6, We then go to our table and

look up z of -1.60 by going down to -1.6.0 and over to 0.00.

This is 0.0548, this means that the area to the left of

-1.6 on the standard normal curve is 0.0548.

And that means the probability of being less than 16.0 ounces in our example

is also 0.0548.

This problem can also be solved in Excel.

When using Excel, remember that you don't need to calculate your z.

You can just enter the x, mean, standard deviation, into the formula and

then we will always use the word true as the last argument in this syntax.

Notice here that we obtain the same solution.

Let's do one more example.

Suppose now we want to know the probability of being greater than 16.121

ounces in the same scenario.

How does that change the problem?

Go ahead and pause the video and see if you can work out this problem.

We again draw a normal curve centered at 16.08, but now we mark 16.121

on the right of 16.08, since 16.121 is greater than 16.08.

And we shade the area to the right of 16.121,

since we're interested in the probability of being greater than 16.121.

Notice that your shaded area is again less than one-half of the curve, so

you expect your final answer to be less than one-half.

We again calculate our z, and this time obtain z=0.82.

We again look up our z value in the table, this time going down to 0.8, and

over to 0.02, since 2 is in the hundredths place of our z.

The value is 0.7939, but remember this is the area to the left of 0.82,

and this time we want the area to the right.

So we must subtract our value from 1.

Our correct answer is 1-0.7939

which is 0.2061 or 20.61%.

Remember from when we do the curve, that we expected an answer less than one-half.

Taking that extra step can keep us from making mistakes, and

that if we had tried to stop at 0.07939.

We would hopefully realize our answer didn't make sense since it was too large.

This can save you from a lot of mistakes.

And again, this problem could have been solved in Excel instead.

Just as when solving by hand, we need to subtract our answer

from Excel from 1 to get the area to the right of the x value.

We obtain the same answer as when solving by hand.

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