Question

The degree of freedom for the statistic \(t\) for paired \(t\)-test based on \(n\) pairs of observations is :

• A. \(2n - 1\)
• B. \(n - 1\)
• C. \(2(n - 1)\)
• D. \(n - 2\)

Hint:

Paired t-test reduces n pairs to n differences; treat it as a one-sample t-test on those n differences.

Answer 1

The Correct Option is B

Step 1: In a paired \(t\)-test, we have \(n\) pairs of observations \((x_i, y_i)\), \(i = 1, 2, \ldots, n\). Instead of treating the two samples separately, we form the \(n\) differences \(d_i = x_i - y_i\).

Step 2: The problem now reduces to a one-sample \(t\)-test on the single set of \(n\) values \(d_1, d_2, \ldots, d_n\), testing whether the population mean difference is zero. The test statistic is \[t = \frac{\bar{d}}{s_d/\sqrt{n}}\] where \(\bar d\) is the mean of the differences and \(s_d^2 = \frac{1}{n-1}\sum (d_i - \bar d)^2\) is their sample variance.

Step 3: The sample variance \(s_d^2\) is computed from \(n\) values but uses one degree of freedom to estimate \(\bar d\) first, leaving \(n - 1\) degrees of freedom, exactly as in an ordinary one-sample \(t\)-test on \(n\) observations.

Final Answer: \[\boxed{n - 1}\], option (B).