Formulating Hypothesis

Let’s see how to formulate hypothesis

Suppose we have a Population

with Probability Model = f(x,θ)

θ is a parameter of the population but its true value is unknown.

Therefore, we can formulate null hypothesis as

H0: θ = θ0

which means, the existing null hypothesis has stated that the parameter θ  could have a value of θ0

This θ0 value is a hypothesized value of θ whose true value is unknown to us.

Now, we are challenged with a new information about the population under study. This challenge with the new information requires us to formulate another hypothesis which is called alternative hypothesis denoted as Ha or H1.

Therefore, we will formulate alternative hypothesis as

H1: θ ≠ θ0

(True value of  θ could not be equal to θ0)

H1: θ < θ0

(True value of θ could be less than θ0)

H1: θ > θ0

(True value of θ could be greater than θ0)

There can only be three different ways as written above regarding the value of θ that could be in the alternative hypothesis in comparison to the value θ0 stated in the null hypothesis.

DNA Pot (c) 2009 - Current

H0: θ = θ0

H1: θ ≠ θ0

H0: θ = θ0

H1: θ < θ0

H0: θ = θ0

H1: θ > θ0

Alternative Hypothesis

two-tail alternative

one-tail alternative

As you can see, the null hypothesis assigns a specific value θ0 of unknown θ. This type of hypothesis which assigns a specific value to an unknown parameter of a population is called a simple hypothesis.

Now, when we look at the three possible alternative hypothesis, there is no specific value of θ being assigned. This type of hypothesis in which there is no specific value of an unknown parameter of a population is called a composite hypothesis.

Therefore, the null hypothesis is a simple hypothesis and the alternative hypotheses are composite hypotheses.

Suppose, new value 𝚹 is the value of parameter θ, then if 𝚹 lies in this region or interval θ < θ0 , then, the alternative hypothesis H1: θ < θ is true, so we reject the null hypothesis H0: θ = θ0 .

Suppose, new value 𝚹 is the value of parameter θ, then if 𝚹 lies in this region or interval θ > θ0 , then, the alternative hypothesis H1: θ > θ0   is true , so we reject the null hypothesis H0: θ = θ0 .

Suppose, new value 𝚹 is the value of parameter θ, then if 𝚹 lies in this region or interval θ ≠ θ0 , then, the alternative hypothesis H1: θ ≠ θis true , so we reject the null hypothesis H0: θ = θ0 .

H0: θ = θ0

H1: θ < θ0

H0: θ = θ0

H1: θ > θ0

one-tail alternative

Left-tail alternative

Right-tail alternative

The new value for parameter θ is on the left of the existing value θ0

The new value for parameter θ is on the right of the existing value θ0

if new θ in the rejection region of θ < θ

H0: θ = θ0

H1: θ ≠ θ0

two-tail alternative

The new value for parameter θ is on the left and on the right of the existing value θ0

if new θ is in the rejection region of θ< θ

rejection  region for H0: θ = θ0 is located on both sides, thus it is called two-tail alternative

if new θ in the rejection region of θ < θ

if new θ is in the rejection region of θ< θ

rejection  region for H0: θ = θ0 is located on one side either left or right, thus it is called one-tail alternative