**– Reject $H_o$ at $\alpha = 0.05$ but not at $\alpha = 0.10$**

**– Reject $H_o$ at $\alpha = 0.01$ but not at $\alpha = 0.05$**

**– Reject $H_o$ at $\alpha = 0.10$ but not at $\alpha = 0.05$**

**– Reject $H_o$ at $\alpha $ equal to $0.10$, $0.05$ and $ 0.01$**

**– Do not reject $H_o$ at $ \alpha$ equal to $0.10$, $0.05$, or $0.01$**

This problem aims to find the best possible choice to reject or not to reject a **Null Hypothesis** given the $p$-value of a conducted test. To better understand the problem, you should be familiar with **significance testing**, $p$**-value conclusion **and **hypothesis testing.**

**Hypothesis testing** is a state of the statistical assumption that utilizes data from a model to lure deductions about a populated parameter or a populated** probability distribution**. Willingly, an uncertain assumption is carried out about the parameter or the distribution.

A $p$**-value** is a numerical value that explains how presumably you are to have discovered a precise bunch of observations if the null hypothesis $H_o$ were to be true. The $p$-value is utilized in **hypothesis testing** which helps determine whether to reject or accept the null hypothesis.

## Expert Answer

The main purpose of $p$**-values** is to construct conclusions in **significance testings**. More precisely, we approximate the $p$-value to the **significance level**, $ \alpha$ in order to make deductions about our hypotheses.

If the approximated $p$-value is **lower** than the significance level $ \alpha$ we selected, then we can **reject** the null hypothesis $H_o$. But if the $p$-value comes out to be **greater** **than** **or equal** **to** the $ \alpha$, then we surely **fail** to reject the null hypothesis $H_o$. We can summarize 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** $\alpha$, then we can reject the **null hypothesis** $H_o$.

Looking one by one into our given options:

**Case1**: If $\alpha = 0.05 \implies$ We fail to reject $H_o$.

**Case2**: If $\alpha = 0.01 \implies$ We fail to reject $H_o$.

**Case3**: If $ \alpha = 0.10 \implies$ **We reject** $H_o$ at $\alpha = 0.10$ but not at $\alpha = 0.05$ because $p$-value becomes less than $\alpha$.

## Numerical Result

We **reject** $H_o$ at $ \alpha = 0.10$ but not at $ \alpha = 0.05$ because $p$-value becomes less than $ \alpha$.

## Example

Given the pieces of **evidence**, which one proves to be the strongest against the null hypothesis?

**– A low test statistic data.**

**– Utilizing a smallish level of significance.**

**– A large $p$-value data.**

**– A small $p$-value data.**

In the **null hypothesis**, we experiment if the mean admires certain conditions, and in the **alternate hypothesis**, we experiment with the contrary of the null hypothesis.

The conclusion relies on the $p$-value:

If the $p$-value is **less** **than** the significance level $\alpha$, then we can reject the **null hypothesis** $H_o$. A large $p$-value does not give evidence for the rejection of the null hypothesis.

So the correct answer is **small** $p$-**value data.**