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Statistics question

2022 · 26 Jun · Shift 1 · Q31
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Statistics question

2022 · 26 Jun · Shift 1 · Q31

JEE MainMathematicsStatisticsMCQ+4 / −1
The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :
  1. A
    60
  2. B
    55
  3. C
    50
  4. D
    45
View written solutionFree

Correct answer: A

  1. Use the mean to find a+ba+ba+b

The five numbers are a,b,8,5,10a,b,8,5,10a,b,8,5,10 and their mean is 666.

So,

a+b+8+5+105=6\frac{a+b+8+5+10}{5}=65a+b+8+5+10​=6 a+b+23=30a+b+23=30a+b+23=30 a+b=7a+b=7a+b=7
  1. Use the variance condition

For these five numbers, mean =6=6=6 and variance =6.8=6.8=6.8.

Using

Variance=15∑(xi−6)2\text{Variance} = \frac{1}{5}\sum (x_i-6)^2Variance=51​∑(xi​−6)2

we get

15[(a−6)2+(b−6)2+(8−6)2+(5−6)2+(10−6)2]=6.8\frac{1}{5}\left[(a-6)^2+(b-6)^2+(8-6)^2+(5-6)^2+(10-6)^2\right]=6.851​[(a−6)2+(b−6)2+(8−6)2+(5−6)2+(10−6)2]=6.8

Now,

(8−6)2=4,(5−6)2=1,(10−6)2=16(8-6)^2=4, \quad (5-6)^2=1, \quad (10-6)^2=16(8−6)2=4,(5−6)2=1,(10−6)2=16

so

(a−6)2+(b−6)2+215=6.8\frac{(a-6)^2+(b-6)^2+21}{5}=6.85(a−6)2+(b−6)2+21​=6.8 (a−6)2+(b−6)2+21=34(a-6)^2+(b-6)^2+21=34(a−6)2+(b−6)2+21=34 (a−6)2+(b−6)2=13(a-6)^2+(b-6)^2=13(a−6)2+(b−6)2=13

Expand:

a2−12a+36+b2−12b+36=13a^2-12a+36+b^2-12b+36=13a2−12a+36+b2−12b+36=13 a2+b2−12(a+b)+72=13a^2+b^2-12(a+b)+72=13a2+b2−12(a+b)+72=13

Using a+b=7a+b=7a+b=7,

a2+b2−84+72=13a^2+b^2-84+72=13a2+b2−84+72=13 a2+b2=25a^2+b^2=25a2+b2=25

Now use

(a+b)2=a2+b2+2ab(a+b)^2=a^2+b^2+2ab(a+b)2=a2+b2+2ab 49=25+2ab49=25+2ab49=25+2ab 2ab=24Rightarrowab=122ab=24 Rightarrow ab=122ab=24Rightarrowab=12

So a,ba,ba,b are roots of

t2−7t+12=0t^2-7t+12=0t2−7t+12=0

which gives

(t−3)(t−4)=0(t-3)(t-4)=0(t−3)(t−4)=0

Hence,

{a,b}={3,4}\{a,b\}=\{3,4\}{a,b}={3,4}
  1. Now the numbers are
3,4,8,5,103,4,8,5,103,4,8,5,10

with mean 666.

  1. Find mean deviation about the mean

Mean deviation about mean:

M=15(∣3−6∣+∣4−6∣+∣8−6∣+∣5−6∣+∣10−6∣)M=\frac{1}{5}\left(|3-6|+|4-6|+|8-6|+|5-6|+|10-6|\right)M=51​(∣3−6∣+∣4−6∣+∣8−6∣+∣5−6∣+∣10−6∣) M=15(3+2+2+1+4)M=\frac{1}{5}(3+2+2+1+4)M=51​(3+2+2+1+4) M=125M=\frac{12}{5}M=512​

Therefore,

25M=25⋅125=6025M=25\cdot \frac{12}{5}=6025M=25⋅512​=60
  1. Compare with the stored answer

Derived answer is 606060, which matches Option A.

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