Explanation

When a paired comparison test (paired t-test) supports rejecting the null hypothesis, it means we have sufficient evidence to conclude that the mean difference between the paired observations is statistically significant (not zero).

Let's analyze each option:

A. difference in means is not statistically significant.

• Incorrect - Rejecting the null hypothesis means the opposite: the difference in means IS statistically significant.

B. independence of the samples is statistically significant.

• Incorrect - Paired samples are by definition dependent (they come from the same subjects or matched pairs). The test doesn't assess independence; it assumes dependence.

C. standard error of the mean differences is low relative to the sample mean difference.

• Correct - In hypothesis testing, we reject the null when the test statistic (t-value) is large. The t-statistic for a paired t-test is calculated as:

  $t = \frac{\bar{d}}{s_d/\sqrt{n}}$

  Where:

  • $\bar{d}$ = sample mean difference
  • $s_d$ = standard deviation of the differences
  • $n$ = number of pairs
  • $s_d/\sqrt{n}$ = standard error of the mean differences

  A large t-value (leading to rejection) occurs when:

  1. The sample mean difference ($\bar{d}$) is large (numerator)
  2. The standard error ($s_d/\sqrt{n}$) is small relative to the mean difference (denominator)

  Therefore, when we reject the null, it means the standard error is low relative to the sample mean difference, making the t-statistic large enough to be statistically significant.

Key Concept: The t-statistic represents how many standard errors the observed mean difference is from zero. A large absolute t-value indicates the observed difference is unlikely to have occurred by chance alone.