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Statistics question

2016 · Shift 0 · Q23
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Statistics question

2016 · Shift 0 · Q23

JEE MainMathematicsStatisticsMCQ+4 / −1
If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?
  1. A
    3 aaa 2 - 26 aaa + 55 = 0
  2. B
    3 aaa 2 - 32 aaa + 84 = 0
  3. C
    3 aaa 2 - 34 aaa + 91 = 0
  4. D
    3 aaa 2 - 23 aaa + 44 = 0
View written solutionFree

Correct answer: B

  1. Given data

The numbers are 2,3,a,112, 3, a, 112,3,a,11 and their standard deviation is 3.53.53.5.

For n=4n=4n=4 observations, the standard deviation is

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

So here,

σ2=22+32+a2+1124−(2+3+a+114)2.\sigma^2 = \frac{2^2+3^2+a^2+11^2}{4} - \left(\frac{2+3+a+11}{4}\right)^2.σ2=422+32+a2+112​−(42+3+a+11​)2.

Since σ=3.5=72\sigma = 3.5 = \frac{7}{2}σ=3.5=27​,

σ2=(72)2=494.\sigma^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4}.σ2=(27​)2=449​.
  1. Compute the required sums

Sum of observations:

2+3+a+11=a+16.2+3+a+11 = a+16.2+3+a+11=a+16.

Sum of squares:

22+32+a2+112=4+9+a2+121=a2+134.2^2+3^2+a^2+11^2 = 4+9+a^2+121 = a^2+134.22+32+a2+112=4+9+a2+121=a2+134.

Thus,

a2+1344−(a+164)2=494.\frac{a^2+134}{4} - \left(\frac{a+16}{4}\right)^2 = \frac{49}{4}.4a2+134​−(4a+16​)2=449​.
  1. Simplify the equation
a2+1344−(a+16)216=494.\frac{a^2+134}{4} - \frac{(a+16)^2}{16} = \frac{49}{4}.4a2+134​−16(a+16)2​=449​.

Multiply throughout by 161616:

4(a2+134)−(a+16)2=196.4(a^2+134) - (a+16)^2 = 196.4(a2+134)−(a+16)2=196.

Expand:

4a2+536−(a2+32a+256)=196.4a^2 + 536 - (a^2 + 32a + 256) = 196.4a2+536−(a2+32a+256)=196. 4a2+536−a2−32a−256=196.4a^2 + 536 - a^2 - 32a - 256 = 196.4a2+536−a2−32a−256=196. 3a2−32a+280=196.3a^2 - 32a + 280 = 196.3a2−32a+280=196. 3a2−32a+84=0.3a^2 - 32a + 84 = 0.3a2−32a+84=0.
  1. Match with the options

We obtained

3a2−32a+84=0.3a^2 - 32a + 84 = 0.3a2−32a+84=0.

This matches Option B.

  1. Verification with stored answer

Stored correct answer: B

Derived answer: B

So the derived answer agrees with the stored answer.

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