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Statistics question

2018 · 15 Apr · Shift 1 · Q37
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Statistics question

2018 · 15 Apr · Shift 1 · Q37

JEE MainMathematicsStatisticsMCQ+4 / −1
The mean of set of 30 observations is 75. If each observation is multiplied by a non-zero number λ\lambdaλ and then each of them is decreased by 25, their mean remains the same. Then λ\lambdaλ is equal to :
  1. A
    13{1 \over 3}31​
  2. B
    23{2 \over 3}32​
  3. C
    43{4 \over 3}34​
  4. D
    103{10 \over 3}310​
View written solutionFree

Correct answer: C

  1. Let the original mean of the 30 observations be xˉ=75.\bar{x} = 75.xˉ=75.

  2. Each observation is transformed as follows:

    • first multiplied by λ\lambdaλ
    • then decreased by 252525

    So if an observation is xxx, the new observation is x′=λx−25.x' = \lambda x - 25.x′=λx−25.

  3. The mean under a linear transformation also changes similarly. Hence the new mean is xˉ′=λxˉ−25=λ(75)−25.\bar{x}' = \lambda \bar{x} - 25 = \lambda(75) - 25.xˉ′=λxˉ−25=λ(75)−25.

  4. It is given that the mean remains the same, so xˉ′=75.\bar{x}' = 75.xˉ′=75. Therefore, 75λ−25=75.75\lambda - 25 = 75.75λ−25=75.

  5. Solve for λ\lambdaλ: 75λ=10075\lambda = 10075λ=100 λ=10075=43.\lambda = \frac{100}{75} = \frac{4}{3}.λ=75100​=34​.

  6. Thus the correct option is 43.\boxed{\frac{4}{3}}.34​​.

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