If $$ H_{0}: \mu \geq 20, H_{a}: \mu

Question

If $$ H_{0}: \mu \geq 20, H_{a}: \mu<20 $$ and $$ \alpha=0.01 $$ and $$ p $$-value $$ =0.0041 $$ then we failed to reject Null hypotheses

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**Step 1. State the hypothesis:**  
The null hypothesis is  
$$
H_{0}: \mu \geq 20,
$$  
and the alternative hypothesis is  
$$
H_{a}: \mu < 20.
$$

**Step 2. Specify the significance level:**  
The significance level is  
$$
\alpha = 0.01.
$$

**Step 3. Compare $$p$$-value and $$\alpha$$:**  
The given $$p$$-value is  
$$
0.0041.
$$  
Since  
$$
0.0041 < 0.01,
$$  
we have enough evidence to reject $$H_{0}$$.

**Step 4. Conclusion:**  
Because the $$p$$-value is less than the significance level, we **reject** the null hypothesis $$H_{0}$$. Thus, the statement "we failed to reject Null hypotheses" is incorrect.  
The correct decision is to reject $$H_{0}$$ in favor of $$H_{a}$$.
We reject the null hypothesis because the $$p$$-value (0.0041) is less than the significance level ($$ \alpha = 0.01 $$).

If and and -value then we failed
to reject Null hypotheses

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If and and -value then we failed to reject Null hypotheses