Statistics Examples

STATISTICS EXAMPLES

Example 1 : If mean of the seies \(x_1\), \(x^2\), ..... , \(x_n\) is \(\bar{x}\), then the mean of the series \(x_i\) + 2i, i = 1, 2, ......, n will be-

Solution : As given \(\bar{x}\) = \(x_1 + x_2 + .... + x_n\over n\)

If the mean of the series \(x_i\) + 2i, i = 1, 2, ....., n be \(\bar{X}\), then

\(\bar{X}\) = \((x_1+2) + (x_2+2.2) + (x_3+2.3) + .... + (x_n + 2.n)\over n\)

    = \(x_1 + x_2 + .... + x_n\over n\) + \(2(1+2+3+....+n)\over n\)

    = \(\bar{x}\) + \(2n(n+1)\over 2n\)

    = \(\bar{x}\) + n + 1.



Example 2 : A student obtained 75%, 80%, 85% marks in three subjects. If the marks of another subject are added then his average marks can not be less than-

Solution : Total marks obtained from three subjects out of 300 = 75 + 80 + 85 = 240

if the marks of another subject is added then the total marks obtained out of 400 is greater than 240

if marks obtained in fourth subject is 0 then

    minimum average marks = \(240\over 400\)\(\times\)100 = 60%



Example 3 : The mean and variance of 5 observations of an experiment are 4 and 5.2 respectively. If from these observations three are 1, 2 and 6, then remaining will be-

Solution : As given \(\bar{x}\) = 4, n = 5 and \({\sigma}^2\) = 5.2. If the remaining observations are \(x_1\), \(x_2\) then

\({\sigma}^2\) = \(\sum{(x_i - \bar{x})}^2\over n\) = 5.2

\(\implies\) \({(x_1-4)}^2 + {(x_2-4)}^2 + {(1-4)}^2 + {2-4)}^2 + {(6-4)}^2\over 5\) = 5.2

\(\implies\) \({(x_1-4)}^2 + {(x_2-4)}^2\) = 9     .....(1)

Also \(\bar{x}\) = 4 \(\implies\) \(x_1 + x_2 + 1 + 2 + 6\over 5\) = 4 \(\implies\) \(x_1 + x_2\) = 11     .....(2)

from eq.(1), (2)     \(x_1\), \(x_2\) = 4, 7


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