**Discard Null hypothesis at $\alpha=0.05$ but retain at $\alpha=0.10$.****Discard Null hypothesis at $\alpha=0.01$ but retain at $\alpha=0.05$.****Discard Null hypothesis at $\alpha=0.10$ but retain at $\alpha=0.05$.****Discard Null hypothesis at $\alpha=0.10$, $0.05$ and $ 0.01$.****Do not Discard Null hypothesis at $\alpha=0.10$, $0.05$, or $0.01$.**

This problem aims to familiarize us with the concept of the Null Hypothesis in which we are to figure out the best feasible choice to discard or retain a **Null Hypothesis** such that the $p$-value is given. For a better understanding, you should be aware of **the Null hypothesis, alternate hypothesis, **and p **-value conclusion.**

Before starting the solution, we should understand that **Hypothesis testing** is a form of an assumption that employs data from an example to **draw conclusions** about a significant **parameter.** We can say that if the **null hypothesis** is denied, then the **research hypothesis** can be **assumed,** but If the null hypothesis is assumed, then the research hypothesis can be **denied.**

Whereas the $p$**-value** is just a mathematical value that clarifies how likely you have uncovered a particular bunch of **statements** if the null hypothesis $H_o$ were to be true.

## Expert Answer

Let’s say that the corresponded $p$-value is **lower** than the significance level $ \alpha$ which we had selected, then we **decline **the null hypothesis $H_o$, otherwise, we simply have to** retain** the null hypothesis $H_o$ if $p$-value is **greater than or equal** to $\alpha$.

In statistics, the main purpose of $p$**-value** is to create conclusions regarding** significance testings**. In which we approximate the $p$-value to the **significance level**, $\alpha$ to make deductions about our hypotheses. We can restate it as follows:

$$\lt \alpha \implies$ reject $H_o$.$

$$\ge \alpha \implies$ fail to reject $H_o$.$

So if a $p$-value is less than the **significance level** $\alpha$, we can reject the **null hypothesis** $H_o$.

Looking **one** **by one** into our given options:

**Case1**: If $\alpha = 0.05 \implies$ We retain $H_o$.

**Case2**: If $\alpha = 0.01 \implies$ We retain $H_o$.

**Case3**: If $\alpha = 0.10 \implies$ We retain $H_o$.

**Case4**: If $\alpha = 0.10, 0.05, 0.01\implies$ We decline $H_o$.

**Case5**: If $\alpha =0.10, 0.05, 0.01 \implies$ **We retain **$H_o$ at $\alpha = 0.10, 0.05, 0.01$ because $p$-value is greater than $\alpha$.

## Numerical Result

We **retain **$H_o$ at $\alpha = 0.10, 0.05, 0.01$ because $p$-value is greater than $\alpha$.

## Example

In the **null hypothesis**, we testify if the average value approves certain conditions, whereas, in the **alternate hypothesis**, we testify with the opposite of the null hypothesis.

The conclusion thus relies on the $p$-value:

Since the $p$-value is **less ****than** the significance level $\alpha$ if $\alpha=0.05 $ , then we reject the **null hypothesis** $H_o$ but at the same time reatin it at $\alpha = 0.01 $. A large $p$-value does not give **evidence** for the **rejection** of the null hypothesis.

So the correct **assumption** would be $\alpha=0.05 \implies$ we reject $H_o$.