Hypothesis Testing Basics

This question examines the fundamental principles of hypothesis testing, focusing on the role and justification of the null hypothesis.

Assertion (A) Analysis

Assertion (A) states that hypothesis testing can proceed using the null hypothesis. This is a core principle in statistical inference. We formulate a null hypothesis ($H_0$) representing a default or status quo assumption, which we then attempt to disprove with evidence from sample data.

Therefore, Assertion (A) is true.

Reason (R) Analysis

Reason (R) explains that if the null hypothesis is true, probabilities can be assigned to different possible sample results. This refers to the process of determining the likelihood of obtaining specific sample outcomes (or more extreme ones) under the assumption that $H_0$ is correct. This likelihood is crucial for calculating test statistics and p-values.

For instance, the p-value is the probability of observing data as extreme as, or more extreme than, the actual sample result, assuming the null hypothesis is true.

Therefore, Reason (R) is true.

Relationship Between Assertion and Reason

The ability described in Reason (R) – assigning probabilities to sample results under the assumption that the null hypothesis ($H_0$) is true – is precisely what enables hypothesis testing as stated in Assertion (A). Without this ability, we could not determine how likely our observed sample data is under the null hypothesis, and thus could not decide whether to reject it.

Reason (R) provides the statistical foundation and justification for conducting hypothesis tests based on the null hypothesis.

Therefore, Reason (R) is the correct explanation of Assertion (A).

Conclusion

Both statements are factually correct regarding hypothesis testing, and the reason given directly supports and explains the assertion.

The correct option is that both (A) and (R) are true, and (R) is the correct explanation of (A).