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Statistics question

2008 · Shift 0 · Q30
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Statistics question

2008 · Shift 0 · Q30

JEE MainMathematicsStatisticsMCQ+4 / −1
The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?
  1. A
    a = 0, b = 7
  2. B
    a = 5, b = 2
  3. C
    a = 1, b = 6
  4. D
    a = 3, b = 4
View written solutionFree

Correct answer: D

  1. Use the mean condition

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

Now check which options satisfy a+b=7a+b=7a+b=7:

  • A: 0+7=70+7=70+7=7 ✔️
  • B: 5+2=75+2=75+2=7 ✔️
  • C: 1+6=71+6=71+6=7 ✔️
  • D: 3+4=73+4=73+4=7 ✔️

So all options satisfy the mean. We now use the variance.

  1. Use the variance formula

Mean xˉ=6\bar{x}=6xˉ=6.

Variance of the five observations is

σ2=15∑(xi−xˉ)2\sigma^2=\frac{1}{5}\sum (x_i-\bar{x})^2σ2=51​∑(xi​−xˉ)2

Given variance is 6.806.806.80,

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,

15[(a−6)2+(b−6)2+4+1+16]=6.8\frac{1}{5}\left[(a-6)^2+(b-6)^2+4+1+16\right]=6.851​[(a−6)2+(b−6)2+4+1+16]=6.8 15[(a−6)2+(b−6)2+21]=6.8\frac{1}{5}\left[(a-6)^2+(b-6)^2+21\right]=6.851​[(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
  1. Check each option

Option A: a=0,b=7a=0, b=7a=0,b=7

(a−6)2+(b−6)2=(0−6)2+(7−6)2=36+1=37(a-6)^2+(b-6)^2=(0-6)^2+(7-6)^2=36+1=37(a−6)2+(b−6)2=(0−6)2+(7−6)2=36+1=37

Not equal to 131313. ❌

Option B: a=5,b=2a=5, b=2a=5,b=2

(a−6)2+(b−6)2=(5−6)2+(2−6)2=1+16=17(a-6)^2+(b-6)^2=(5-6)^2+(2-6)^2=1+16=17(a−6)2+(b−6)2=(5−6)2+(2−6)2=1+16=17

Not equal to 131313. ❌

Option C: a=1,b=6a=1, b=6a=1,b=6

(a−6)2+(b−6)2=(1−6)2+(6−6)2=25+0=25(a-6)^2+(b-6)^2=(1-6)^2+(6-6)^2=25+0=25(a−6)2+(b−6)2=(1−6)2+(6−6)2=25+0=25

Not equal to 131313. ❌

Option D: a=3,b=4a=3, b=4a=3,b=4

(a−6)2+(b−6)2=(3−6)2+(4−6)2=9+4=13(a-6)^2+(b-6)^2=(3-6)^2+(4-6)^2=9+4=13(a−6)2+(b−6)2=(3−6)2+(4−6)2=9+4=13

This matches. ✔️

  1. Conclusion

The possible values are

a=3,b=4a=3,\quad b=4a=3,b=4

So the correct option is D.

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