 # Suppose you conduct a test and your p-value is equal to 0.93. What can you conclude? 1. Discard Null hypothesis at $\alpha=0.05$ but retain at $\alpha=0.10$.
2. Discard Null hypothesis at $\alpha=0.01$ but retain at $\alpha=0.05$.
3. Discard Null hypothesis at $\alpha=0.10$ but retain at $\alpha=0.05$.
4. Discard Null hypothesis at $\alpha=0.10$, $0.05$ and $0.01$.
5. 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.

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, to make deductions about our hypotheses. We can restate it as follows:

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

Assume you run a test and your $p$-value comes to $0.016$. What can you create from this assumption?

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 $H_o$.