The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

Option 1 -

$\frac{92}{5}$

Option 2 -

$\frac{134}{5}$

Option 3 -

$\frac{536}{25}$

Option 4 -

$\frac{112}{5}$

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

a month ago

Correct Option - 3

Detailed Solution:

a, b, c, d, e be 5 unknown

n = 7, mean = 8, variance = 16

sum of observations = 7 × 8 = 56

mean of 5 remaining observation = $\frac{56 - 8 - 6}{25} = \frac{42}{5}$

$16 = \frac{\sum x_{i}^{2}}{7} - 64$

$\sum x_{i}^{2} = 560$

$\begin{array}{l} a^{2} + b^{2} + c^{2} + d^{2} + e^{2} = 460 \\ = 460 - {(\frac{42}{5})}^{2} \end{array}$

$= \frac{536}{25}$

a, b, c, d, e be 5 unknownn = 7, mean = 8, variance = 16sum of observations = 7 × 8 = 56mean of 5 remaining observation = <math> <mrow> <mfrac> <mrow> <mn>5</mn> <mn>6</mn> <mo>−</mo> <mn>8</mn> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mn>2</mn> <mn>5</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816075974"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mn>1</mn> <mn>6</mn> <mo>=</mo> <mfrac> <mrow> <mo>∑</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>7</mn> </mrow> </mfrac> <mo>−</mo> <mn>6</mn> <mn>4</mn> </mrow> </math> <math> <mrow> <mo>∑</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>5</mn> <mn>6</mn> <mn>0</mn> </mrow> </math> <math> <mtable columnalign="left"> <mtr> <mtd> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mn>6</mn> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>4</mn> <mn>6</mn> <mn>0</mn> <mo>−</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </math> <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mn>3</mn> <mn>6</mn> </mrow> <mrow> <mn>2</mn> <mn>5</mn> </mrow> </mfrac> </mrow> </math>                                              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816075975"> </o:OLEObject></xml><! [endif]->

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The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

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