Statistics
Variance and Mean
Grade 11
Question:
<p>Let \(x_1, x_2, \ldots, x_n\) be \(n\) observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^2\) be the variance.<br><strong>Statement 1:</strong> Variance of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\sigma^2\).<br><strong>Statement 2:</strong> Arithmetic mean of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\bar{x}\).</p>
<p>Statement 1 is false, statement 2 is true.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.</p>
<p>Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.</p>
<p>Statement 1 is true, statement 2 is false.</p>
Step-by-Step Solution
Key Concept: When all observations are multiplied by a constant k, the mean is multiplied by k but variance is multiplied by k². Statement 1 uses the variance transformation rule correctly, while Statement 2 incorrectly applies a squaring to the mean transformation.
<p><strong>Step 1: Analyze Statement 1 (Variance)</strong></p><p>If original data: x₁, x₂, ..., xₙ with variance σ²</p><p>New data: 2x₁, 2x₂, ..., 2xₙ</p><p>Variance property: Var(kX) = k² · Var(X)</p><p>New variance = 2² · σ² = 4σ²</p><p><strong>Statement 1 is TRUE ✓</strong></p><p><strong>Step 2: Analyze Statement 2 (Mean)</strong></p><p>Original mean: x̄ = (x₁ + x₂ + ... + xₙ)/n</p><p>New mean = (2x₁ + 2x₂ + ... + 2xₙ)/n = 2(x₁ + x₂ + ... + xₙ)/n = 2x̄</p><p>Mean property: E(kX) = k · E(X) (linear, NOT squared)</p><p><strong>Statement 2 claims the new mean is 4x̄, but it's actually 2x̄</strong></p><p><strong>Statement 2 is FALSE ✗</strong></p><p>∴ Answer: D (Statement 1 is true, Statement 2 is false)</p>
Correct Answer: D