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Statistics question

2019 · 10 Jan · Shift 1 · Q25
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Statistics question

2019 · 10 Jan · Shift 1 · Q25

JEE MainMathematicsStatisticsMCQ+4 / −1
The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is -
  1. A
    6 : 7
  2. B
    10 : 3
  3. C
    4 : 9
  4. D
    5 : 8
View written solutionFree

Correct answer: C

  1. Let the five observations be 1,  3,  8,  x,  y1,\;3,\;8,\;x,\;y1,3,8,x,y

  2. Use the mean

    Given mean =5=5=5 for 555 observations, so total sum is 5×5=255\times 5 = 255×5=25 Hence, 1+3+8+x+y=251+3+8+x+y=251+3+8+x+y=25 12+x+y=2512+x+y=2512+x+y=25 x+y=13x+y=13x+y=13

  3. Use the variance formula

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

    Given variance =9.20=9.20=9.20 and mean =5=5=5, so 9.20=12+32+82+x2+y25−259.20 = \frac{1^2+3^2+8^2+x^2+y^2}{5} - 259.20=512+32+82+x2+y2​−25

    Since 12+32+82=1+9+64=741^2+3^2+8^2 = 1+9+64=7412+32+82=1+9+64=74 therefore, 9.20=74+x2+y25−259.20 = \frac{74+x^2+y^2}{5} - 259.20=574+x2+y2​−25 74+x2+y25=34.20\frac{74+x^2+y^2}{5} = 34.20574+x2+y2​=34.20 74+x2+y2=17174+x^2+y^2 = 17174+x2+y2=171 x2+y2=97x^2+y^2 = 97x2+y2=97

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

    Substituting x+y=13x+y=13x+y=13 and x2+y2=97x^2+y^2=97x2+y2=97: 132=97+2xy13^2 = 97 + 2xy132=97+2xy 169=97+2xy169 = 97 + 2xy169=97+2xy 2xy=722xy = 722xy=72 xy=36xy=36xy=36

  5. Find xxx and yyy

    The numbers satisfy t2−13t+36=0t^2 - 13t + 36 = 0t2−13t+36=0 Factorizing, t2−13t+36=(t−4)(t−9)=0t^2 - 13t + 36 = (t-4)(t-9)=0t2−13t+36=(t−4)(t−9)=0 So, x=4,  y=9x=4,\; y=9x=4,y=9 (or vice versa)

  6. Required ratio x:y=4:9x:y = 4:9x:y=4:9

  7. Check options

    • A: 6:76:76:7 ❌
    • B: 10:310:310:3 ❌
    • C: 4:94:94:9 ✅
    • D: 5:85:85:8 ❌

Therefore, the correct option is C.

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