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Question #40999

Difficulty: 8/10 | 🟢 Free

The Question

A student calculates the mean and variance of 10 real observations $x_1, x_2, \dots, x_{10}$ , where $x_k = \text{Re}(z_k) + \text{Im}(z_k)$ and $z_1, z_2, \dots, z_{10}$ are the distinct roots of the equation $z^{10} - 1 = 0$ . Later, it was discovered that the student misread the observation corresponding to the root $z = 1$ as $11$ instead of its correct value. What is the incorrect variance calculated by the student?

Options

A.
10
B.
11
C.
12
D.
13

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Correction of Incorrect Mean/Variance|Roots of Unity Properties|Vieta’s Formulas in Complex Domain

Why This Question Is Tricky

Common Mistakes

The question tests the integration of statistical variance correction with the algebraic properties of the $n$ -th roots of unity.

The primary trap lies in the calculation of the initial mean and variance.

Students often attempt to manually sum the trigonometric forms $\cos(2k\pi/10) + \sin(2k\pi/10)$ and their squares, which leads to calculation fatigue and high probability of errors.

The elegant bypass forces the use of Vieta's formulas in the complex domain, recognizing that $\sum z_k = 0$ and $\sum z_k^2 = 0$ for $z^{10}-1=0$ , instantly yielding the correct sum and sum of squares.

The problem then seamlessly transitions into a classic incorrect observation correction, rewarding algebraic insight over brute-force trigonometry.

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