# Hypothesis testing

## 1 Null hypothesis

A default position that there is no significant relationship between two phenomena or among groups.

Often, denoted by $H_0$.

We **never** accept the null hypothesis. We only reject or fail to reject it given the level of confidence ($\alpha$-level).

## 2 Alternative hypothesis

A hypothesis that there is significant relationship between two phenomena or among groups.

Denoted by $H_a$.

## 3 What is the $\alpha$-level?

The $\alpha$-level or significance level indicates how extreme observed data must be before we can reject the null hypothesis.

## 4 A $p$-value?

The $p$-value is the probability that we observe a certain phenomenon under the null hypothesis.

## 5 Testing hypotheses

If the $p$-value is less than or equal to the $\alpha$-level, our data is unusual—more extreme than the significance level—and we reject the null hypothesis. We can say the data is statistically significant with a significance level of $\alpha$. In this case, the alternative hypothesis is **supported**, not accepted.

If the $p$-value is greater than the $\alpha$-level, the data is usual—not as extreme as the significance level—and we fail to reject the null hypothesis. We can say the data is statistically non-significant with a significance level of $\alpha$.

## 6 Exercises

## 7 Homework: Testing a new chemical product

A manufacturer plans to introduce a brand new hand sanitizer into the consumer market during the COVID-19 pandemic. They must keep the concentration of hydrogen peroxide under its maximum allowed concentration of 3%. Since this product is new, they wanted to keep their standard higher and set the $\alpha$-level to 0.10, which is double the standard significance level of 0.05. They took 1,000 samples, measured the hydrogen peroxide concentration, and obtained a $p$-value of 0.08 based on their sample test. Please state the null and alternative hypotheses in your report and draw a conclusion. Can they sell this hand sanitizer?