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Question #60828

Difficulty: 8/10 | 🟢 Free

The Question

An infinite dataset is partitioned into subgroups $G_i$ for $i \in \mathbb{N}$ . The weight of each subgroup $G_i$ is given by $W_i = c \left| \frac{2 \cos(i\pi)}{3} \right|^i$ , where $c > 0$ is a constant. For each subgroup $G_i$ , the arithmetic mean is $\bar{x}_i = \left(-\frac{3}{4}\right)^i$ , the variance is $\sigma_i^2 = \frac{i^3 + 2i}{i+1}$ , and the median is $M_i = \frac{1}{i}$ . What is the combined arithmetic mean of the entire dataset?

Options

A.
$-\frac{1}{6}$
B.
$\frac{5}{6}$
C.
$-\frac{3}{7}$
D.
$\frac{1}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Combined Mean of Subgroups|Infinite GP Convergence|Modulus Sign Dependency

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to correctly evaluate an infinite geometric progression where the common ratio is embedded within a modulus function that depends on the index $i$ .

The trap lies in the 'Modulus Sign Dependency': students who mechanically drop the absolute value sign will incorrectly sum the alternating sequence $c(-2/3)^i$ , leading to a false total population weight of $-2c/5$ and an incorrect combined mean of $5/6$ .

Furthermore, the inclusion of subgroup variance and median serves as extraneous data (Red Herrings) to distract from the core combined mean formula, demanding strong conceptual focus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Combined Mean of Subgroups
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free