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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Statistics

NTA Abhyas 2020 · Mathematics · Statistics

The mean and variance of 5 observations are 6 and 6.8 respectively....

The mean and variance of 55 observations are 66 and 6.86.8 respectively. If a number equal to mean is included in the set of observations and the variance of 66 observations is kk, then the value of 34k\dfrac{34}{k} is equal to

Correct answer

6

Step-by-step solution

Let 55 observations are x1,x2,x3,x4,x5{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}

x¯old=6=x1+x2+x3+x4+x55\overset{¯}{x}old = 6 = \dfrac{{x}_{1} + {x}_{2} + {x}_{3} + {x}_{4} + {x}_{5}}{5}

⇒x1+x2+x3+x4+x5=30⇒ {x}_{1} + {x}_{2} + {x}_{3} + {x}_{4} + {x}_{5} = 30 ... \left(1\right)<br/><br/>Var.(old)=x12+x22+....x525−(x¯old)2=6.8<br /><br />Var . \left(old\right) = \dfrac{{x}_{1}^{2} + {x}_{2}^{2} + . . . . {x}_{5}^{2}}{5} - {\left(\overset{¯}{x}old\right)}^{2} = 6.8

=x12+x22+....x52=214= {x}_{1}^{2} + {x}_{2}^{2} + . . . . {x}_{5}^{2} = 214 ... \left(2\right)<br/><br/>Anobservation<br /><br />An observation6isincluded<br/><br/>is included<br /><br />\overset{¯}{x}new = \dfrac{{x}_{1} + {x}_{} + {x}_{3} + {x}_{4} + {x}_{5} + 6}{6} = 6<br/><br/><br /><br />{x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2} . . . . {x}_{5}^{2} + {\left(6\right)}^{2} = 214 + 36 = 250<br/><br/>Var.(new)=2506−(6)2=346<br /><br />Var . \left(new\right) = \dfrac{250}{6} - {\left(6\right)}^{2} = \dfrac{34}{6}
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