Question

If dataset $A=\{1,2,3,\ldots,19\}$ and dataset $B=\{ax+b;\,x\in A\}$. If mean of $B$ is $30$ and variance of $B$ is $750$, then sum of possible values of $b$ is

Hint:

Variance is unaffected by addition of constants but is multiplied by the square of the scaling factor.

Answer 1

Correct Answer: 60

Step 1: Determine mean and variance of dataset A

Given,

A = {1, 2, 3, … , 19}

Mean of an arithmetic sequence:

x̄ = (first term + last term) / 2

x̄ = (1 + 19) / 2 = 10

Variance of the first n natural numbers is given by:

σ² = (n² − 1) / 12

σ² = (19² − 1) / 12 = 30

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Step 2: Use linear transformation for dataset B

Dataset B is obtained using the transformation:

B = ax + b

Mean under linear transformation becomes:

Mean(B) = a · Mean(A) + b

Given Mean(B) = 30,

10a + b = 30  ……(1)

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Step 3: Apply variance transformation rule

Variance under linear transformation:

σ_B² = a² σ_A²

Given σ_B² = 750,

750 = a² × 30

a² = 25

a = 5 or a = −5

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Step 4: Find corresponding values of b

Using equation (1):

If a = 5:

b = 30 − 50 = −20

If a = −5:

b = 30 + 50 = 80

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Step 5: Required sum

Sum of all possible values of b:

−20 + 80 = 60

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Final Answer:

The required sum is
60