<p>The mean of a set of 30 observations is 75. If each observation is multiplied by a non-zero number \(\lambda\) and then each of them is decreased by 25, their mean remains the same. Then \(\lambda\) is equal to</p>
Step-by-Step Solution
Key Concept: When observations are transformed linearly (multiplied by λ and decreased by 25), the mean transforms by the same rule. Set up the equation: λ(75) - 25 = 75 to find λ.
<p><strong>Step 1:</strong> Original mean of 30 observations is 75.</p><p><strong>Step 2:</strong> Each observation is transformed as: x → λx - 25</p><p><strong>Step 3:</strong> Apply the transformation rule to the mean. If original mean is m, then new mean is λm - 25.</p><p><strong>Step 4:</strong> According to the problem, the new mean equals the original mean:</p><p>λ(75) - 25 = 75</p><p><strong>Step 5:</strong> Solve for λ:</p><p>75λ - 25 = 75</p><p>75λ = 100</p><p>λ = 100/75 = 4/3</p><p>∴ Answer: λ = 4/3 (Option C)</p>
Correct Answer: C