Question:
<p>If the mean of the set of numbers \(x_1, x_2, x_3, \ldots, x_n\) is \(\bar{x}\), then the mean of the numbers \(x_i + 2i\), \(1 \le i \le n\) is</p>
<p>\(\bar{x} + 2n\)</p>
<p>\(\bar{x} + n + 1\)</p>
<p>\(\bar{x} + 2\)</p>
<p>\(\bar{x} - n\)</p>
Step-by-Step Solution
Key Concept: The mean of a transformed dataset equals the original mean plus the mean of the transformation terms. Since we're adding 2i to each term, find the average of the sequence 2(1), 2(2), 2(3),...,2(n).
<p><strong>Step 1:</strong> Use the linearity property of mean: Mean of (x_i + 2i) = Mean of (x_i) + Mean of (2i)</p><p><strong>Step 2:</strong> The first part is simply the original mean = bar(x)</p><p><strong>Step 3:</strong> Find the mean of the sequence 2i for i = 1 to n:<br>Mean of (2i) = [2(1) + 2(2) + 2(3) + ... + 2(n)]/n<br>= 2[1 + 2 + 3 + ... + n]/n<br>= 2 · [n(n+1)/2]/n<br>= 2 · (n+1)/2<br>= n+1</p><p><strong>Step 4:</strong> Therefore, Mean of (x_i + 2i) = bar(x) + (n+1)</p><p>∴ Answer: <strong>bar(x) + n + 1</strong></p>
Correct Answer: B