Question:
<p>If the arithmetic mean of the numbers \(x_1, x_2, x_3, \ldots, x_n\) is \(\bar{x}\), then the arithmetic mean of the numbers \(ax_1 + b,\ ax_2 + b,\ ax_3 + b,\ \ldots,\ ax_n + b\), where \(a, b\) are two constants, would be</p>
<p>\(\bar{x}\)</p>
<p>\(n a\bar{x} + nb\)</p>
<p>\(a\bar{x}\)</p>
<p>\(a\bar{x} + b\)</p>
Step-by-Step Solution
Key Concept: The mean of a linearly transformed dataset equals the same linear transformation applied to the original mean: if original mean is x̄, then mean of (axᵢ + b) is a·x̄ + b. This follows from the linearity property of summation and the definition of arithmetic mean.
<p><strong>Step 1:</strong> Write the definition of arithmetic mean for the transformed data:</p><p>New mean = (1/n)∑(axᵢ + b) for i=1 to n</p><p><strong>Step 2:</strong> Use linearity of summation to separate terms:</p><p>= (1/n)[∑axᵢ + ∑b]</p><p>= (1/n)[a∑xᵢ + nb]</p><p><strong>Step 3:</strong> Factor out constants and use the original mean x̄ = (1/n)∑xᵢ:</p><p>= a·(1/n)∑xᵢ + (1/n)·nb</p><p>= a·x̄ + b</p><p><strong>Verification:</strong> If x₁=1, x₂=2, x₃=3, then x̄=2. For a=2, b=5: new values are 7,9,11 with mean (7+9+11)/3 = 9 = 2(2)+5 ✓</p><p>∴ Answer: <strong>ax̄ + b</strong> (Option D)</p>
Correct Answer: D