Statistics
Variance
Grade None
Question:
<p>Suppose a population \(A\) has 100 observations 101, 102, ..., 200 and another population \(B\) has 100 observations 151, 152, ..., 250. If \(V_A\) and \(V_B\) represent the variances of the two populations, respectively, then \(V_A/V_B\) is equal to</p>
<p>1</p>
<p>9/4</p>
<p>4/9</p>
<p>2/3</p>
Step-by-Step Solution
Key Concept: Variance is invariant under translation but scales quadratically with linear transformations. Both populations are arithmetic progressions with the same common difference (1), so they have identical variance regardless of their position on the number line.
<p><strong>Step 1:</strong> Identify the structure of each population.</p><p>Population A: 101, 102, ..., 200 (arithmetic progression with first term a=101, last term l=200, n=100 terms, common difference d=1)</p><p>Population B: 151, 152, ..., 250 (arithmetic progression with first term a=151, last term l=250, n=100 terms, common difference d=1)</p><p><strong>Step 2:</strong> Use the variance formula for arithmetic progression.</p><p>For an arithmetic progression with n terms and common difference d:</p><p>V = (n² - 1)d²/12</p><p><strong>Step 3:</strong> Calculate V_A.</p><p>V_A = (100² - 1)(1)²/12 = (10000 - 1)/12 = 9999/12</p><p><strong>Step 4:</strong> Calculate V_B.</p><p>V_B = (100² - 1)(1)²/12 = (10000 - 1)/12 = 9999/12</p><p><strong>Step 5:</strong> Find the ratio.</p><p>V_A/V_B = (9999/12)/(9999/12) = 1</p><p>∴ Answer: A (which equals 1)</p>
Correct Answer: A