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Statistics question

2020 · 4 Sep · Shift 1 · Q29
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Statistics question

2020 · 4 Sep · Shift 1 · Q29

JEE MainMathematicsStatisticsMCQ+4 / −1
The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :
  1. A
    5
  2. B
    3
  3. C
    7
  4. D
    9
View written solutionFree

Correct answer: C

  1. Use the mean to find the sum of all 8 observations

Given mean =10=10=10 and number of observations n=8n=8n=8,

Total sum=8×10=80.\text{Total sum}=8\times 10=80.Total sum=8×10=80.

The given 6 observations are 5,7,10,12,14,155,7,10,12,14,155,7,10,12,14,15. Their sum is

5+7+10+12+14+15=63.5+7+10+12+14+15=63.5+7+10+12+14+15=63.

Let the remaining two observations be xxx and yyy. Then

x+y=80−63=17.x+y=80-63=17.x+y=80−63=17.
  1. Use the variance to find x2+y2x^2+y^2x2+y2

Variance is given by

σ2=∑xi2n−(∑xin)2.\sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2.σ2=n∑xi2​​−(n∑xi​​)2.

Given variance =13.5=13.5=13.5 and mean =10=10=10,

13.5=∑xi28−102.13.5=\frac{\sum x_i^2}{8}-10^2.13.5=8∑xi2​​−102.

So,

13.5=∑xi28−10013.5=\frac{\sum x_i^2}{8}-10013.5=8∑xi2​​−100 ∑xi28=113.5\frac{\sum x_i^2}{8}=113.58∑xi2​​=113.5 ∑xi2=8×113.5=908.\sum x_i^2=8\times 113.5=908.∑xi2​=8×113.5=908.

Now calculate the sum of squares of the known 6 observations:

52+72+102+122+142+1525^2+7^2+10^2+12^2+14^2+15^252+72+102+122+142+152 =25+49+100+144+196+225=739.=25+49+100+144+196+225=739.=25+49+100+144+196+225=739.

Hence,

x2+y2=908−739=169.x^2+y^2=908-739=169.x2+y2=908−739=169.
  1. Find xyxyxy using (x+y)2=x2+y2+2xy(x+y)^2=x^2+y^2+2xy(x+y)2=x2+y2+2xy

We have

(x+y)2=x2+y2+2xy.(x+y)^2=x^2+y^2+2xy.(x+y)2=x2+y2+2xy.

Substitute x+y=17x+y=17x+y=17 and x2+y2=169x^2+y^2=169x2+y2=169:

172=169+2xy17^2=169+2xy172=169+2xy 289=169+2xy289=169+2xy289=169+2xy 2xy=1202xy=1202xy=120 xy=60.xy=60.xy=60.
  1. Find the absolute difference ∣x−y∣|x-y|∣x−y∣

Using

(x−y)2=(x+y)2−4xy,(x-y)^2=(x+y)^2-4xy,(x−y)2=(x+y)2−4xy,

we get

(x−y)2=172−4⋅60=289−240=49.(x-y)^2=17^2-4\cdot 60=289-240=49.(x−y)2=172−4⋅60=289−240=49.

Thus,

∣x−y∣=7.|x-y|=7.∣x−y∣=7.
  1. Check the options

The absolute difference is 777, which corresponds to:

  • A: 555
  • B: 333
  • C: 777 ✅
  • D: 999

So the correct option is C.

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