Evaluating Assertion (A): Hypothesis Acceptance Threshold

Assertion (A) states that a hypothesis is accepted if the p-value is less than 0.01.

• In hypothesis testing, a p-value represents the probability of observing the data, or more extreme data, assuming the null hypothesis ($H_0$) is true.
• A small p-value indicates that the observed data is unlikely under $H_0$, providing evidence against $H_0$.
• A pre-determined significance level ($\alpha$) is used. If the p-value falls below $\alpha$, the null hypothesis ($H_0$) is rejected.
• A p-value < 0.01 signifies a very low probability of the data occurring by chance if $H_0$ were true. This is a strict criterion ($\alpha = 0.01$).
• Meeting this criterion (p < 0.01) provides strong statistical evidence to reject $H_0$ or, conversely, to accept the alternative hypothesis ($H_1$). Therefore, Assertion (A) is considered true in this context.

Understanding Reason (R): P-Value and Chance Factor

Reason (R) states that estimating the p-value accounts for correcting the chance factor.

• The definition of a p-value inherently incorporates the concept of random chance. It is specifically calculated as the probability of the observed results arising solely due to random variation, assuming the null hypothesis is correct.
• Therefore, the process of estimating the p-value directly quantifies the influence of the chance factor on the observed data.
• Reason (R) is a correct statement regarding the nature and function of p-values in statistical inference.

Analyzing the Relationship Between Assertion and Reason

We need to determine if Reason (R) correctly explains Assertion (A).

• Statement (R) accurately describes that p-values measure the probability related to the chance factor under the null hypothesis.
• Statement (A) proposes a specific decision rule based on a threshold (p < 0.01) for accepting a hypothesis.
• While the p-value's role in quantifying chance (R) is fundamental to hypothesis testing, it does not explain the *reason* for choosing the specific threshold of 0.01 mentioned in (A). The choice of the significance level ($\alpha$) is an independent decision made by the researcher based on the desired balance between Type I and Type II errors.
• Thus, (R) is true but does not provide the correct explanation for why the threshold in (A) is significant for hypothesis acceptance.

Conclusion

Both Assertion (A) and Reason (R) are true statements.

However, Reason (R) explains the general principle of what a p-value represents (quantifying chance), while Assertion (A) states a specific decision rule using a particular threshold. The principle described in (R) does not explain the specific choice of threshold in (A).

Therefore, the correct answer is that both statements are true, but (R) is not the correct explanation of (A).