<p>In a set of \(2n\) distinct observations, each of the observation below the median of all the observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations</p>
Step-by-Step Solution
Key Concept: When observations below the median are shifted by +5 and observations above/at the median are shifted by -3, the change in mean equals the weighted average of these shifts based on their frequencies.
<p><strong>Step 1:</strong> Identify the structure. With 2n distinct observations, when arranged in order, the median lies between the n-th and (n+1)-th observation.</p><p><strong>Step 2:</strong> Count affected observations. Exactly n observations are below the median (increased by 5) and exactly n observations are above the median (decreased by 3).</p><p><strong>Step 3:</strong> Calculate total change in sum. Change in sum = n(+5) + n(-3) = 5n - 3n = 2n</p><p><strong>Step 4:</strong> Calculate change in mean. New mean = Original mean + (Change in sum)/(Total observations) = Original mean + 2n/(2n) = Original mean + 1</p><p><strong>∴ Answer: The mean increases by 1, or new mean = old mean + 1 (Option A)</strong></p>
Correct Answer: A