# Null hypothesis and alternative hypothesis

## What are Null hypothesis and alternative hypothesis?

When you're doing hypothesis testing, you'll need to construct the null hypothesis and the alternative hypothesis. But what exactly are they? At first glance, they may seem very similar to one another, but they actually are quite different.

## Null hypothesis vs alternative hypothesis

We want to explore what is a null hypothesis and what is an alternative hypothesis. Let's start with the null first.

Null Hypothesis definition: The null hypothesis shows that there's no observed effect from the experiment we carry out. The null hypothesis symbol is written as H0 and has an "=" when the hypothesis is stated.

Alternative Hypothesis definition: The alternative hypothesis shows that there's an observed effect in the experiment we carry out. It's what we're trying to prove when we do our hypothesis test. The relationship between the null and alternative hypothesis is that when the null hypothesis is rejected, we'll accept the alternative hypothesis. When the null hypothesis is not rejected, then we won't accept the alternative hypothesis. The alternative hypothesis symbol is usually either Ha or H1.

So how do we usually use the null and alternative hypothesis in math? Some common ones you'll see include:

• H0: x is equal to y. Ha: x is not equal to y
• H0: x is a maximum of y. Ha: x is greater than y
• H0: x is a minimum of y. Ha: x is less than y

## How to find Null hypothesis and alternative hypothesis

Let's put the concept we just learned into use by showing you a null hypothesis example and its alternative hypothesis.

Question:
You meet a woman on the street who says she has Extra Sensory Perception (ESP) and can predict the probability of dice rolls with 70% probability. To test this you roll the die 90 times, and see how many times she "guesses" correctly. What is the null and alternative hypothesis?

Solution:

H0: P = 0.7

H1: P $\neq$ 0.7

So firstly, we want to determine the null hypothesis. We're dealing with a proportion here. In our case, the proportion of guesses the woman should get right is a 70% proportion. We're going to use P to show this. For the null hypothesis, the question doesn't tells us that the woman is going to predict dice rolls with more than 70% probability, nor does it tell us that she'll make predictions correctly with less than 70% probability. It tells us that she can predict the dice rolls with exactly 70% probability. Therefore, our null hypothesis is P = 0.7.

Now we move on to finding the alternative hypothesis, which must be shown with either a <, >, or a ?. The alternative hypothesis is everything but the null hypothesis. In actual fact, to cover everything that's not a 0.7 probability, it's simply P $\neq$ 0.7. This tells us that the woman can make predictions with any other probability and it'll fit into the alternative hypothesis.

If you wanted to look at more examples of the null hypothesis and the alternative hypothesis being formulated and what they'll look like for different kinds of questions, here's a great resource to refer to.

### Null hypothesis and alternative hypothesis

#### Lessons

• Introduction
Hypothesis Testing is the method of testing whether or not a claim is valid

Two types of claims:
• Proportions: Data given by percentages, %
• Means: given by data measurements, $\mu$

Null Hypothesis $(H_0)$:

The result that is hoped to be proven false. It is a single parameter.
Given by: "$=$"

Alternative Hypothesis $(H_1)$:

The result that is hoped to be true. It is a wide range of parameters, where the truth of this hypothesis is tested based off the verity of the Null Hypothesis.
Given by: "< , >, $\neq$"

• 1.
Intuitively Judging Validity of Claims
You meet a woman on the street who says she has Extra Sensory Perception (ESP) and can predict the probability of dice rolls with 70% probability. To test this you roll the die 90 times, and see how many times she "guesses" correctly.
a)
If she guesses 23 rolls correctly is it likely that she has ESP?

b)
If she guesses 70 rolls correctly is it likely that she has ESP?

• 2.
Determining Claims
What were the Null Hypothesis and the Alternative Hypothesis from the previous question?

• 3.
Which claims are Null Hypotheses and which claims are Alternative Hypotheses?
a)
$\mu = 17ft$

b)
$p$ < $0.37$

c)
$\mu \neq 5cm$

d)
$p\geq 0.76$

• 4.
For each of the following claims:
i) State whether each claim refers to proportions or means.
ii) Identify which claims are $H_0$ and which claims are $H_1$.
iii) Form the following claims into mathematical statements.
a)
"More than 75% of CEO's have their MBAs."

b)
"The mean size of a cars fuel tank is less than 50 litres."

c)
"At least half of the beaches in Mexico are beautiful."

d)
"Most people like apple pie."

e)
"The average height of people in the world is 174cm."

f)
"The mean weight of a Toblerone is at most 175 grams."