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Question #40990

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{x} = (x_1, x_2, \dots, x_{10}) \in \mathbb{R}^{10}$ be a vector representing a dataset of 10 observations. Let $\vec{u} = (-1, -1, \dots, -1) \in \mathbb{R}^{10}$ . The vector $\vec{x}$ is projected onto the line spanned by $\vec{u}$ to obtain the vector $\vec{p}$ . It is given that $\|\vec{x} - \vec{p}\|^2 = 90$ and the dot product $\vec{x} \cdot \vec{u} = 50$ . If a new dataset is defined as $y_i = 3x_i - 2$ for $i = 1, 2, \dots, 10$ , find the value of $\frac{1}{10} \sum_{i=1}^{10} y_i^2$ .

Options

A.
250
B.
370
C.
81
D.
304

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

Variance Invariance under Translation|Linear Independence vs Dependence|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of variance as the squared Euclidean distance from the data vector to the subspace spanned by the constant vector $\vec{1}$ , normalized by $n$ .

The 'Lethal Collision' occurs when students misinterpret the dot product projection sign.

By using $\vec{u} = -\vec{1}$ and giving $\vec{x} \cdot \vec{u} = 50$ , students who blindly use the magnitude or ignore the negative components will incorrectly calculate the mean as $\mu_x = 5$ instead of $-5$ .

This sign error propagates through the linear transformation $y_i = 3x_i - 2$ , leading to an incorrect mean $\mu_y = 13$ and a final second moment of 250 instead of 370.

It rigorously separates rote memorization from true vector-space comprehension in statistics.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Variance Invariance under Translation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free