One of the biggest challenges Six Sigma students encounter is Hypothesis Testing. There are so many types! And there are so many different equations? How on earth are you supposed to be able to make sense of those problems – especially if you haven’t done word problems in years or, like some people believe, ‘you’re just no good at math’? Not to Worry. I’ve got you covered. Here I’ve put together the Hypothesis Test Study Guide.

My goal is to provide a cohesive overview of the types of hypothesis tests you’ll be responsible for on your Six Sigma exam, which hypothesis test to use under which circumstances and to introduce an outline how to actually do them.

## First, A Word on Hypothesis Tests

To me, these concepts fall under the phrase ‘Simple, but not easy.’

They’re simple in the manner that they are easily repeatable algorithms – or a series of instructions – that you can follow time and time again to get the result you want.

The not easy part is that you have to learn all of the different types and decide when to use each one.

## If you can bake cookies, you can do hypothesis tests following this study guide.

Make fun of me all you want. My mental model for learning hypothesis tests for Six Sigma certification was baking.

Instead of thinking of hypothesis testing, consider baking. What if they were asking you questions about when to bake cookies instead of cake. Or bake bread instead of pie? And then imagine they were asking you questions about how to bake that pie, cookie, cake, or bread loaf.

There’s a time and a place for each of those baked goods. And there’s a time and a place for each hypothesis test.

All you have to do is decide what kind of baked good you want to cook and then pick the right recipe for the given situation!

Let’s dive in by learning about the different types covered in this Hypothesis Test Study Guide.

Below I give you a guided tour of the different types of hypothesis tests – aka baked goods. I introduce the variations of each type and when you’d want to ‘bake’ that item. I then link to other articles on this website for the ‘recipe’ and examples.

## Hypothesis Testing Basics

Since this is a long article I’ve added additional resources that can help provide a great foundation for the Hypothesis Tests Study Guide.

If you’re absolutely new to Hypothesis tests, or just want a refresher, please see my Hypothesis Testing Overview here and my Basic Hypothesis Testing Process overview here.

Another resource that is helpful to have on hand is the Hypothesis Testing Terminology page.

Ok – let’s dive into how you can pick which hypothesis test to use and when.

### What You Need to Know about the Basics of Hypothesis Testing for Your Six Sigma Certification Exam

#### Six Sigma Green Belts

The ASQ Six Sigma Green Belt BOK requires:

Hypothesis testing Basics

Define and distinguish between statistical and practical significance and apply tests for significance level, power, type I and type II errors. Determine appropriate sample size for various test. (Apply).

The IASSC Six Sigma Green Belt BOK requires:

3.3 Hypothesis Testing

3.3.1 General Concepts & Goals of Hypothesis Testing

3.3.2 Significance; Practical vs. Statistical

3.3.3 Risk; Alpha & Beta

3.3.4 Types of Hypothesis Test

The Villanova Six Sigma Green Belt BOK requires:

#### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK requires:

1. Terminology

Define and interpret the significance level, power, type I and type II errors of statistical tests. (Evaluate)

2. Statistical vs. practical significance

Define, compare and interpret statistical and practical significance. (Evaluate)

3. Sample size

Calculate sample size for common hypothesis tests (e.g., equality of means, equality of proportions, etc.). (Apply)

4. Point and interval estimates

Define and distinguish between confidence and prediction intervals. Define and interpret the efficiency and bias of estimators. Calculate tolerance and confidence intervals. (Evaluate)

The IASSC Six Sigma Black Belt BOK requires:

3.3 Hypothesis Testing

3.3.1 General Concepts & Goals of Hypothesis Testing

3.3.2 Significance; Practical vs. Statistical

3.3.3 Risk; Alpha & Beta

3.3.4 Types of Hypothesis Test

The Villanova Six Sigma Black Belt BOK requires:

Hypothesis testing

i. Fundamental concepts of hypothesis testing

1. Statistical vs. practical significance

Define, compare and contrast statistical and practical significance.

2. Significance level, power, type I and type II errors

Apply and interpret the significance level, power, type I and type II errors of statistical tests.

3. Sample size

Understand how to calculate sample size for any given hypothesis test.

ii. Point and interval estimation

Define and interpret the efficiency and bias of estimators; compute, interpret and draw conclusions from statistics such as standard error, tolerance intervals, and confidence intervals; understand the distinction between confidence intervals and prediction intervals.

## The Three Types of Hypothesis Tests

You can divide the types of hypothesis tests you need into 3 main types:

- Tests for means
- Tests for variances
- Tests for proportions

If this sounds intimidating, just think cookies, cake, and pie! ;’)

Let’s step through each type and how to select each one.

**Note**: Black belt candidates be aware that this article will focus on the hypothesis test you use on normal distributions. You will be responsible for those AND hypothesis tests you use under non-normal / non-parametric distributions. Not to worry, I’ve made a study guide for those tricky situations, too.

## Hypothesis Tests for Means Study Guide

### What You Need to Know about the Hypothesis Testing for Means for Your Six Sigma Certification Exam

#### Six Sigma Green Belts

The ASQ Six Sigma Green Belt BOK requires:

Tests for means, variances, and proportions

Define, compare, and contrast statistical and practical significance. (Apply)

Paired-comparison tests

Define and describe paired-comparison parametric hypothesis tests. (Understand)Single-factor analysis of variance (ANOVA)

Define terms related to one-way ANOVAs and interpret their results and data plots. (Apply)

The IASSC Six Sigma Green Belt BOK requires:

3.4.1 1 & 2 sample t-tests

The Villanova Six Sigma Green Belt BOK requires:

#### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK requires:

Tests for means, variances and proportions

Use and interpret the results of hypothesis tests for means, variances and proportions. (Evaluate)

The IASSC Six Sigma Black Belt BOK requires:

One Way ANOVA

a. Including Tests of Equal Variance, Normality Testing and Sample Size calculation, performing tests and interpreting results.

The Villanova Six Sigma Black Belt BOK requires:

Test for means, variances, and proportions

Apply hypothesis tests for means, variances and proportions, and interpret the results.Paired-comparison tests

Define, determine applicability, and apply paired-comparison parametric hypothesis tests and interpret the results.

### What is a Hypothesis Test for Means?

Remember Mean, Median, and Mode? Those are fancy statistical words for basic math principles. Mean is a synonym for average. So you might infer that a Hypothesis test for means involves testing averages – and you’d be right!

### When do you use Hypothesis Tests for Means?

You use a hypothesis test for means when you want to see if two sample populations have a characteristic that is on average the same.

A good example of this is when you need to know if you’re actually comparing like samples. For instance, if you needed to see if two samples were on average roughly equivalent before proceeding to use them in some manner.

Or you might want to see if the sample you just took is actually reflective of the whole population. For instance, if your sample is just the honors students, that sample is unlikely representative of the whole school.

### What is Needed to Run Hypothesis Tests for Means?

Another way to think about these kind of questions is to have a checklist and eliminate hypothesis tests when they don’t meet a criteria.

In order to perform a Hypothesis test for means, you need 2 things:

### How to do a Hypothesis Tests for Means

#### Step 1: State a Hypothesis

State both your Null Hypothesis & Alternative hypothesis.

Sometimes you have to interpret the question yourself. Other times it will be given for you. Sometimes you can use the details in step 2 to figure out what your hypotheses should be.

Remember, your **null and alternative hypothesis must be mutually exclusive!**

#### Step 2: Pick a type of means test!

There are a few different types of test for means. Be sure to pick the right one for th right job.

##### Z Test (is the sample mean the same as the population mean?)

You use a Z test to **compare if your sample mean is the same as the population mean**. Use to validate you have a good sample

Like many of the hypothesis tests we are covering here, you must have normal data.

Use a Z test when you **DON’T **know the mean and Standard Deviation of the population.

See specific examples of the Z test here.

##### T Tests

There are several kinds of T tests. You’ll have to decide on each one. The names of each are fairly descriptive. Below I’ve added additional common indicators of when to use each one.

##### One Sample T Test

Use the One Sample T Test when you want to know if the **mean of your sample different than another one** – either from your imagination or a specification. Or use it when you need to **test the mean of a single group against a known mean.**

A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value…. Each makes a statement about how the true population mean μ is related to some hypothesized value M .

##### Two sample T Test

You guessed it! This one is for **two different samples**.

A two-sample t-test is used to **test the difference between two population means.** A common application is to determine whether the means are equal.

Each makes a statement about the difference d0 between the mean of one population μ1 and the mean of another population μ2.

A neat example of a 2 sample T test is this one about Groundhog’s day from MiniTab.

Good questions to ask yourself are:

- How is the two sample t test different from a Paired T?
- When would you use two sample t test and not a Paired T?

##### Paired Sample T Test

The paired sample t-test, sometimes called the dependent sample t-test, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. In a paired sample t-test, each subject or entity is measured twice, resulting in pairs of observations.

Use the Paired Sample T test when you need to know if **the means of 2 sets the same.** Or if you need to compare means from the same sample group taken at different times.

Good questions to ask yourself:

- Why not just take the average value of each set and compare them against each other?

##### One-Way ANOVA

The one-way analysis of variance (ANOVA) is used to determine whether there are **any statistically significant differences between the means of three or more independent (unrelated) groups.**

The null hypothesis for ANOVA is that the mean (average value of the dependent variable) is the same for all groups. The alternative or research hypothesis is that the average is not the same for all groups. The ANOVA test procedure produces an F-statistic, which is used to calculate the p-value

###### Multiple ANOVA

Note that Black Belts will need to have a cursory knowledge of MANOVA – Multiple Analysis of Variation. You can learn more here.

## Hypothesis Tests for Variances Study Guide

### What You Need to Know about the Basics of Hypothesis Testing for Variances for Your Six Sigma Certification Exam

#### Six Sigma Green Belts

The ASQ Six Sigma Green Belt BOK requires:

Tests for means, variances, and proportions

Define, compare, and contrast statistical and practical significance. (Apply)

Chi square

Define and interpret chi square and use it to determine statistical significance. (Analyze)

The IASSC Six Sigma Green Belt BOK requires:

1 sample variance

The Villanova Six Sigma Green Belt BOK requires:

#### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK requires:

Tests for means, variances and proportions

Use and interpret the results of hypothesis tests for means, variances and proportions. (Evaluate)Goodness-of-fit (chi square) tests

Define, select and interpret the results of these tests. (Evaluate)

The IASSC Six Sigma Black Belt BOK requires:

3.5.8 Chi-Squared (Contingency Tables)

a. Including Tests of Equal Variance, Normality Testing and Sample Size calculation, performing tests and interpreting results.

The Villanova Six Sigma Black Belt BOK requires:

Test for means, variances, and proportions

Apply hypothesis tests for means, variances and proportions, and interpret the results.Goodness-of-fit tests

Define, determine applicability, and apply chi-square tests and interpret the results.

### What is a Hypothesis Test for Variances?

The first thing we need to understand is ‘what in the world is variance?’ The root of the word is variability. In the statistical sense, we’re thinking about how large a range of options are in a sample or population.

To get a graphical sense of this, think about a standard bell curve graph. The longer it is – meaning the further it stretches across the y axis – the more variability it has. “Spread” is another word that refers to variance – as in how far is the range of possible values in a given group “spread” across.

Another way to think about Variance questions is to automatically replace the word Variance with Standard Deviation in your head.

To learn more about variance see this article.

### When do you use Hypothesis Tests for Variances?

You use these kind of tests when you want to know if the variability of the possible values are equal amongst two groups.

Ex. Does the new process have greater/lesser/equal variability than the old process?

### What is Needed to Run Hypothesis Tests for Variances?

We generally need:

- Two groups to test against each other.
- The populations that the two groups were selected from be normally distributed.

### How to do a Hypothesis Tests for Variance

There are generally two types of Hypothesis Tests for Variance: Chi-Square-tests and F-tests for variance or standard deviation both require that the original population be normally distributed.

#### Step 1: Pick a Variance Test

This is straight forward. When you are trying to see if the variance of 2 populations are equal each other, pick the F test.

If you are comparing the variance of a threshold of some kind, pick a Chi-Square tesm

##### Notes on the F test

Pick when you’re trying to determine **“Are the variances of 2 populations equal?”**

The null hypothesis of an F Test is that the two populations are equal variance (to each other).

##### Notes on the Chi-Square test for variance

**Use the Chi-Square when you want to determin if the variance equal to a specific value.**

The chi-square test can be used to answer the following questions:

- Is the variance equal to some pre-determined threshold value?
- Is the variance greater than some pre-determined threshold value?
- Is the variance less than some pre-determined threshold value?

See more about Chi-Squared Hypothesis tests for Variance here.

## Hypothesis Tests for Proportions Study Guide

### What You Need to Know about the Basics of Hypothesis Testing for Proportions for Your Six Sigma Certification Exam

#### Six Sigma Green Belts

The ASQ Six Sigma Green Belt BOK requires:

Tests for means, variances, and proportions

Define, compare, and contrast statistical and practical significance. (Apply)

The IASSC Six Sigma Green Belt BOK requires:

One Way ANOVA

a. Including Tests of Equal Variance, Normality Testing and Sample Size calculation, performing tests and interpreting results.

The Villanova Six Sigma Green Belt BOK requires:

#### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK requires:

Tests for means, variances and proportions

Use and interpret the results of hypothesis tests for means, variances and proportions. (Evaluate)Analysis of variance (ANOVA)

Select, calculate and interpret the results of ANOVAs. (Evaluate)

The IASSC Six Sigma Black Belt BOK requires:

3.5.7 One and Two Sample Proportion

One Way ANOVA

a. Including Tests of Equal Variance, Normality Testing and Sample Size calculation, performing tests and interpreting results.

The Villanova Six Sigma Black Belt BOK requires:

Test for means, variances, and proportions

Apply hypothesis tests for means, variances and proportions, and interpret the results.Analysis of Variance (ANOVA)

Define, determine applicability, and apply ANOVAs and interpret the results.

### What is a Hypothesis Test for Proportions?

A hypothesis test for proportion is a way to see if there is a difference between two groups – or between a group and a given standard.

### When do you use Hypothesis Tests for Proportions?

You would use a Hypothesis test for proportion when you’re asked about the difference between two variables.

### What is Needed to Run a Hypothesis Tests for Proportions?

- Simple, random sampling
- Just (2) possible outcomes.
- Label 1 outcome a success, the other a failure.
- Have at least 10 successes and 10 failures
- Population is 20 times as large as the sample size.

### How to do a Hypothesis Tests for Proportions?

#### Step 1: Write the Hypothesis and decide which type of tailed test you need.

We’ve covered hypothesis writing earlier. It’s very easy when writing Proportion tests.

Just remember that the null hypothesis ALWAYS includes some kind of equals. (Ex =, <=, or >=)

An alternative **hypothesis** is a statement that there is a relationship between two variables or there is a difference between two groups or there is a difference from a previous or existing standard.

Let’s move on to the kind of tailed test.

##### One tail

If you want reject if something is too small or too big, you’re doing a one-tail test.

But which tail?

##### Left tail vs Right tail

Left tail: alternative hypothesis expresses ‘less than’

Right tail: alternative hypothesis expresses ‘greater than’

##### Two tail

Ex. If the alternative hypothesis has l=

#### Step 2: Compute Standard Deviation (of the sample!)

σ = sqrt[ P *(1 – P)/n]

where P is the hypothesized value of population proportion in the null hypothesis, and n is the sample size.

#### Step 3: Compute the test statistic

The test statistic is a z-score (z) defined by the following equation.

z = (p – P)/σ

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and σ is the standard deviation of the sampling distribution.

#### Step 4: Find the P value

The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a z-score, use that process to assess the probability associated with the z-Score.

#### Step 5: Interpret your results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

## Beyond Normal: Non-Parametric Hypothesis Tests

Finally, Black belt candidates are widely expected to be conversant in non-parametric hypothesis testing across all 3 types; Means, Variance, and Proportion.

Here are the specific requirements for each:

### Six Sigma Green Belts

IASSC Six Sigma Green Belt BOK requires:

3.5 Hypothesis Testing with Non-Normal Data

3.5.1 Mann-Whitney

3.5.2 Kruskal-Wallis

3.5.3 Mood’s Median

3.5.4 Friedman

3.5.5 1 Sample Sign

3.5.6 1 Sample Wilcoxon

### Six Sigma Black Belts

The ASQ Six Sigma Black Belt BOK requires:

Non-parametric tests

Select, develop and use various non-parametric tests, including Mood’s Median, Levene’s test, Kruskal-Wallis, Mann-Whitney, etc. (Evaluate)

The IASSC Six Sigma Black Belt BOK requires:

Hypothesis Testing with Non-Normal Data

3.5.1 Mann-Whitney

3.5.2 Kruskal-Wallis

3.5.3 Mood’s Median

3.5.4 Friedman

3.5.5 1 Sample Sign

3.5.6 1 Sample Wilcoxon

The Villanova Six Sigma Black Belt BOK requires: