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Question #41000

Difficulty: 8/10 | 🟢 Free

The Question

A student uses the step-deviation method $u_i = \frac{x_i - a}{h}$ with assumed mean $a = 10$ and scale $h = \frac{1}{\sqrt{3} + \sqrt{2}}$ to find the variance of $40$ observations $x_1, x_2, \dots, x_{40}$ . The student calculates $\sum_{i=1}^{40} u_i = 40(\sqrt{3} + \sqrt{2} + 1)$ and $\sum_{i=1}^{40} u_i^2 = 2760 + 400\sqrt{6}$ . Later, it was discovered that an observation $x_k = 10 + 42(\sqrt{3} - \sqrt{2})$ was incorrectly recorded. If the correct observation is $x_k = 10 + 2(\sqrt{3} - \sqrt{2})$ , what is the correct variance of the original data $x$ ?

Options

A.
4
B.
$20 + 8\sqrt{6}$
C.
$4(\sqrt{3} + \sqrt{2})$
D.
$196 + 80\sqrt{6}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Correction of Incorrect Mean/Variance|Step-Deviation Method|Algebraic Simplification

Why This Question Is Tricky

Common Mistakes

This question tests the Step-Deviation method for variance correction, specifically targeting the 'Algebraic Simplification' trap.

The scale factor $h$ is given as a disguised denominator $\frac{1}{\sqrt{3} + \sqrt{2}}$ , forcing students to rationalize it to $\sqrt{3} - \sqrt{2}$ to correctly map the incorrect/correct $x$ values to $u$ values.

A common critical error occurs at the final step: students either forget the relation $\text{Var}(x) = h^2 \text{Var}(u)$ , mistakenly divide by $h^2$ because of its initial fractional form, or fail to recognize the conjugate pair multiplication $(5 - 2\sqrt{6})(5 + 2\sqrt{6})$ that elegantly collapses the complex surds into a clean integer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Correction of Incorrect Mean/Variance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free