– 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, 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:
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.