Let two independent random samples of sizes \(n_1\) and \(n_2\) respectively have been drawn from the same normal population. Let \(\overline{X_1}\) and \(\overline{X_2}\) be the means and let \(s_1\) and \(s_2\) be their standard deviations. In order to test whether the two sample means \(\overline{X_1}\) and \(\overline{X_2}\) differ significantly or not, the t-test statistic is given by
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(t=\displaystyle\frac{\overline{X_1}-\overline{X_2}}{S\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}},\quad S=\sqrt{\displaystyle\frac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}}\)
Both samples come from the same normal population, so the two samples are combined into one pooled estimate S of the population standard deviation:
\( \displaystyle S = \sqrt{\frac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}} \)
The variances of the two sample means, \(S^2/n_1\) and \(S^2/n_2\), add, so the standard error of the difference of the means is
\( \displaystyle S\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \)
The t statistic is the difference of the sample means divided by this standard error, with \(n_1 + n_2 - 2\) degrees of freedom:
\( \displaystyle t = \frac{\overline{X_1}-\overline{X_2}}{S\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \)
Why the other options are incorrect
Option (1) \(t=\displaystyle\frac{\overline{X_1}-\overline{X_2}}{\frac{S}{\sqrt{\frac{1}{n_1}-\frac{1}{n_2}}}},\quad S=\sqrt{\displaystyle\frac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}}\) — Incorrect. The variances of the two means add, so the root holds \(\frac{1}{n_1} + \frac{1}{n_2}\); also S must multiply the root, not be divided by it.
Option (2) \(t=\displaystyle\frac{\overline{X_1}-\overline{X_2}}{\frac{S}{\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}},\quad S=\sqrt{\displaystyle\frac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}}\) — Incorrect. The standard error is S multiplied by \(\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\), not S divided by it.
Option (4) \(t=\displaystyle\frac{\overline{X_1}+\overline{X_2}}{S\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}},\quad S=\sqrt{\displaystyle\frac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}}\) — Incorrect. The test is about the difference of the means, so the numerator is \(\overline{X_1}-\overline{X_2}\), not their sum.
Note: This topic was not in the CUET 2026 syllabus.