Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is…………..

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alok kumar singh | Contributor-Level 10

2 months ago

$\frac{1}{7} \sum_{i = 1}^{7} {(x_{i} - 62)}^{2} = 20$ $\bar{x} = 62$

$\sum_{i = 1}^{7} {(x_{i} - 62)}^{2}$ = 140……………… (i)

If any one student get less 50 marks then ${(x_{i} - 62)}^{2} \geq 144$

but $\sum_{i = 1}^{7} {(x_{i} - 62)}^{2} = 140$

$∴$ both condition cannot satisfy together hence no students fails.

<math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>7</mn> </mrow> </mfrac> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>7</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>6</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> <mo>=</mo> <mn>2</mn> <mn>0</mn> </mrow> </math> <math> <mrow> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>6</mn> <mn>2</mn> </mrow> </math>     <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1814711041"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>7</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>6</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </math> = 140……………… (i) If any one student get less 50 marks then <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1814711043"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>6</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>1</mn> <mn>4</mn> <mn>4</mn> </mrow> </math> but <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1029" DrawAspect="Content" ObjectID="_1814711044"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>7</mn> </mrow> </munderover> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>6</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> <mo>=</mo> <mn>1</mn> <mn>4</mn> <mn>0</mn> </mrow> </math> <math> <mrow> <mo>∴</mo> </mrow> </math> both condition cannot satisfy together hence no students fails.