When the two independent small samples… - CUET UG 2025 Mathematics PYQ

CUET UG 2025 — Mathematics Statistics & Applications

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When the two independent small samples of sizes $n_1$ and $n_2$ with means $\bar{x}_1$ and $\bar{x}_2$ respectively are drawn from populations with identical population variances, the test-statistic is computed as

Official previous-year question

Held on 13 May 2025 · Verified 13 Jul 2026.

Options

A
$t = \frac{\bar{x}_1 - \bar{x}_2}{S_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$ , where $S_p$ is pooled standard deviation.
B
$t = \frac{\bar{x}_1 - \bar{x}_2}{S_p\sqrt{\frac{1}{n_1} - \frac{1}{n_2}}}$ , where $S_p$ is pooled standard deviation.
C
$t = \frac{\bar{x}_1 + \bar{x}_2}{S_p\sqrt{\frac{1}{n_1} - \frac{1}{n_2}}}$ , where $S_p$ is pooled standard deviation.
D
$t = \frac{\bar{x}_1 + \bar{x}_2}{\sqrt{\frac{1}{n_1} - \frac{1}{n_2}}}$ , where $S_p$ is pooled standard deviation.

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Solution

The position vectors represent two points in 3D space:

Point A: $2\hat{i} - 2\hat{j} + 3\hat{k}$ has coordinates $(2, -2, 3)$

Point B: $5\hat{i} + 2\hat{j} + 3\hat{k}$ has coordinates $(5, 2, 3)$

The difference vector from Point A to Point B:

$\vec{AB} = (5\hat{i} + 2\hat{j} + 3\hat{k}) - (2\hat{i} - 2\hat{j} + 3\hat{k})$

For each component:

$\hat{i}$ component: $5 - 2 = 3$

$\hat{j}$ component: $2 - (-2) = 4$

$\hat{k}$ component: $3 - 3 = 0$

Therefore:

$\vec{AB} = 3\hat{i} + 4\hat{j} + 0\hat{k}$

The length is the magnitude of the difference vector:

$\text{Length} = \sqrt{(3)^2 + (4)^2 + (0)^2}$

$= \sqrt{9 + 16 + 0}$

$= \sqrt{25}$

$= 5$

The length of the line segment is $5$ units.

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