To determine whether the null hypothesis $H_0$ should be rejected, compare the P-value to the significance level $\alpha$. The null hypothesis is rejected if the $P$-value is less than $\alpha$. Let’s evaluate each case:

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a) $P \text{-value} = 0.258, \alpha = 0.05, \text{one-tailed test}$

• Decision: $P\text{-value} = 0.258 > \alpha = 0.05$
• Conclusion: Do not reject $H_0$.

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b) $P \text{-value} = 0.0684, \alpha = 0.10, \text{two-tailed test}$

• Decision: $P\text{-value} = 0.0684 < \alpha = 0.10$
• Conclusion: Reject $H_0$.

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c) $P \text{-value} = 0.0153, \alpha = 0.01, \text{one-tailed test}$

• Decision: $P\text{-value} = 0.0153 > \alpha = 0.01$
• Conclusion: Do not reject $H_0$.

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d) $P \text{-value} = 0.0232, \alpha = 0.05, \text{two-tailed test}$

• Decision: $P\text{-value} = 0.0232 < \alpha = 0.05$
• Conclusion: Reject $H_0$.

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e) $P \text{-value} = 0.002, \alpha = 0.01, \text{one-tailed test}$

• Decision: $P\text{-value} = 0.002 < \alpha = 0.01$
• Conclusion: Reject $H_0$.

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Summary of Decisions:

Case	$P\text{-value}$	$\alpha$	Test Type	Decision	Conclusion
a	0.258	0.05	One-tailed	Do not reject $H_0$	Insufficient evidence
b	0.0684	0.10	Two-tailed	Reject $H_0$	Sufficient evidence
c	0.0153	0.01	One-tailed	Do not reject $H_0$	Insufficient evidence
d	0.0232	0.05	Two-tailed	Reject $H_0$	Sufficient evidence
e	0.002	0.01	One-tailed	Reject $H_0$	Sufficient evidence

Would you like a detailed explanation for any of these cases or concepts?

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Related Questions:

1. What does the $P$-value represent in hypothesis testing?
2. How do you decide between a one-tailed and a two-tailed test?
3. Why is it important to choose the appropriate significance level $\alpha$?
4. What are the implications of Type I and Type II errors in hypothesis testing?
5. How can the $P$-value approach be compared to the critical value method?

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Tip:

Always confirm whether a test is one-tailed or two-tailed before comparing the $P$-value with $\alpha$, as this directly affects the decision.