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Mathematics

Assertion (A) : The mean of 15 observations was found to be 21. Later it was detected that one value 15 was wrongly copied as 18, while calculating the mean. The correct mean is 20.

Reason (R) : The mean of n observations x1, x2, x3, ….., xn is xˉ\bar{x}. If each observation is increased by p, then the new mean is increased by p, i.e., the new mean is xˉ\bar{x} + p.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Statistics

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Answer

Given,

Wrong mean = 21

Total observations = 15

So,

Total wrong sum = 21 × 15 = 315

One value 15 was wrongly copied as 18

∴ 18 - 15 = 3

New total sum = 315 - 3 = 312

New mean = 31215\dfrac{312}{15}

= 20.8.

But given mean = 20.

∴ Assertion (A) is false.

If given observation is increased by p, then

⇒New Mean=(x1+p)+(x2+p)+⋯+(xn+p)n⇒New Mean=(x1+x2+⋯+xn)+npn⇒New Mean=xˉ+p\Rightarrow \text{New Mean} = \dfrac{(x1 + p) + (x2 + p) + \dots + (xn + p)}{n} \\[1em] \Rightarrow \text{New Mean} = \dfrac{(x1 + x2 + \dots + xn) + np}{n} \\[1em] \Rightarrow \text{New Mean} = \bar{x} + p \\[1em]

∴ Reason (R) is true.

Assertion (A) is false, Reason (R) is true.

Hence option 2 is the correct option.

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