Problem:

We are given:

• Null hypothesis $H_0: \mu = 25$
• Alternative hypothesis $H_a: \mu \neq 25$ (a two-tailed test)
• $p = 0.027$
• Level of significance $\alpha = 0.02$

We need to determine whether to reject or fail to reject the null hypothesis $H_0$ at $\alpha = 0.02$.

Solution:

1. Compare $p$-value with $\alpha$:

   • If $p < \alpha$, reject $H_0$.
   • If $p \geq \alpha$, fail to reject $H_0$.

   Here:

   • $p = 0.027$
   • $\alpha = 0.02$

   Since $p = 0.027 > 0.02$, we fail to reject $H_0$.

2. Conclusion: At the 0.02 level of significance, there is insufficient evidence to reject the null hypothesis. We fail to reject $H_0$, meaning we do not have strong enough evidence to conclude that $\mu \neq 25$.

Do you want further details or have any questions? 😊

Related Questions:

1. What is the difference between a one-tailed and two-tailed test in hypothesis testing?
2. How do you calculate the $p$-value in a hypothesis test?
3. What does the level of significance ($\alpha$) represent in hypothesis testing?
4. What steps are involved in hypothesis testing?
5. How does increasing the significance level affect the results of hypothesis testing?

Tip:

Always remember to compare the $p$-value directly to $\alpha$ in hypothesis testing—it simplifies the decision-making process!