# Step 1 of DOE

# Introduction/Hypothesis/Research Question

The problem formulation is the first step of the design of an experiment. Arguably, the most important step is to formulate the problem so that it is logical and legitimate enough that it justifies a research study. The goal of problem formulation or making a research question is to develop a **hypothesis**, which is defined by an educated guess or statement about something to be tested by a scientific method, including statistical methods. For example, the mean height of US male population is 69 inches. If the problem statement/hypothesis is not a well-researched one, everything followed by it will be wasted. What if we make a statement/hypothesis that the mean height of US male is 100 inches? Any research questions or statements need to be verified by the existing literature and subject matter experts. Otherwise, we will be testing the mean height of US male is 100 inches!

# Statistical Problem Formulation

## Translating the Problem into Statistical Hypothesis

As making a decision is typically a binary choice such as whether to do something or not (e.g., writing this book today or not; working in the garden or not; etc.), the hypothesis is divided into two mutually exclusive events, completing the entire set. For example, some part of the world is in daytime, while the other part is nighttime. The statistical names for the two divided hypotheses are (1) null hypothesis and (2) alternative hypothesis.

To test the research question of “the medicine works better than a placebo,” the hypotheses are written as follows.

Where, *H _{0}* = null hypothesis;

*H*= alternative hypothesis; µ

_{A}_{new medicine }= mean effect from the new medicine, µ

_{placebo }= mean effect from the placebo

### Most Common Mistake

Students typically switch the null and alternative hypothesis while translating the research questions into statistical hypothesis.

# Null Hypothesis

The word “null” means nothing, zero, no difference, equal, no effect, etc. Therefore, the **null hypothesis** means that there is no difference between the treatments of a medicine and a placebo; medicine is equal to a placebo; no effect from the medicine; etc. Therefore, the null hypothesis is unbiased, which always follows an equal sign.

**Example 1**

**Example 2**

*Note that both examples follow an equal sign, meaning that some kind of zero or null flavor in the null hypothesis.*

# Alternative Hypothesis

**Alternative Hypothesis** is anything other than what is stated in the null hypothesis. Therefore, it does not follow an equal sign and comparatively biased than the null hypothesis. The alternative hypothesis could be (1) unequal, ≠, (2) less than, < or (3) greater than, >; resulting in three different types of alternative hypothesis depending on the research questions to be answered.

Alternative Hypothesis for *Example 1*

Alternative Hypothesis for *Example 2*

Researchers want to prove the alternative and reject the null hypothesis. A significant result indicates the acceptance of the alternative hypothesis and the rejection of the null hypothesis.

### Most Common Mistake

As the alternative hypothesis includes everything other than the just the “equal” null hypothesis, it is easy to make a mistake in writing an alternative hypothesis. Subject matter expertise is required to write an alternative hypothesis. Otherwise, the research questions will be wrong.

# More Examples on Hypothesis Writing

Table 2. Examples on null and alternative hypotheses in various field of study

# Properties of Null and Alternative Hypotheses

- Null hypothesis must have some kind of zero or null flavor in it.
- Alternative must have some kind of biased and non-zero or non-null flavor in it.
- Hypothesis is about the population, not for the sample. Population is the entire world, while a sample is a representative chunk of the population.
- Greek letters are used for population parameters. Therefore, both null and alternative hypothesis shall be written using Greek letters.
- Null hypothesis always follows an equal sign, while the alternative follows a sign other than equal.
- Alternative hypothesis is the complement set of the null hypothesis. Therefore, the entire population either falls into the null or into the alternative set.
- The null hypothesis is very narrow as it contains an equal sign, while the alternative is a very wide highly defused one.