Sample Size, Standard Error, and Statistical Power Explained

This question examines the relationship between sample size, the standard error of the mean, and the power of a statistical test.

Assertion (A) Analysis

The assertion states: "The larger the sample size, the greater will be the power of the test."

• Statistical Power is the probability that a hypothesis test will correctly reject a false null hypothesis.
• Increasing the sample size ($n$) generally leads to more precise estimates of population parameters.
• This increased precision reduces the likelihood that random sampling fluctuations will cause a failure to detect a real effect.
• Therefore, Assertion (A) is correct.

Reason (R) Analysis

The reason states: "A large sample size results in smaller standard error of mean."

• The Standard Error of the Mean (SEM) measures the variability of sample means expected if you were to take repeated samples from the same population.
• It is calculated using the formula: $ \sigma_{SEM} = \frac{\sigma}{\sqrt{n}} $ where $\sigma$ is the population standard deviation and $n$ is the sample size.
• As the sample size ($n$) increases, the denominator ($\sqrt{n}$) increases, leading to a decrease in the SEM.
• Therefore, Reason (R) is correct.

Relationship Between Assertion and Reason

A smaller standard error (as stated in Reason R) indicates less variability among sample means. This makes it easier to detect a true difference or effect because the sample results are more likely to be representative of the population. This directly contributes to an increased probability of rejecting a false null hypothesis, which is the definition of increased statistical power (as stated in Assertion A).

Thus, the reason correctly explains why the assertion is true.

Conclusion

Both Assertion (A) and Reason (R) are correct statements, and Reason (R) provides a valid explanation for Assertion (A).