<p>The mean of a set of numbers is \(\bar{X}\). If each number is divided by 3, then the new mean is</p>
Step-by-Step Solution
Key Concept: The mean is a linear operator: if each data point is transformed by dividing by 3, the mean transforms in the same way. Therefore, new mean = (old mean)/3 = X̄/3.
<p><strong>Step 1:</strong> Recall the definition of mean. If the original set has numbers x₁, x₂, ..., xₙ, then:</p><p>Mean = X̄ = (x₁ + x₂ + ... + xₙ)/n</p><p><strong>Step 2:</strong> After dividing each number by 3, the new set becomes: x₁/3, x₂/3, ..., xₙ/3</p><p><strong>Step 3:</strong> Calculate the new mean:</p><p>New Mean = (x₁/3 + x₂/3 + ... + xₙ/3)/n</p><p>= (1/3)(x₁ + x₂ + ... + xₙ)/n</p><p>= (1/3) × X̄</p><p>= <strong>X̄/3</strong></p><p><strong>Key Property:</strong> For any linear transformation aX + b applied to each data point, the new mean = a(X̄) + b</p><p>∴ Answer: <strong>X̄/3</strong></p>
Correct Answer: D