JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 6)
Let the mean and variance of the frequency distribution
$$\matrix{ {x:} & {{x_1} = 2} & {{x_2} = 6} & {{x_3} = 8} & {{x_4} = 9} \cr {f:} & 4 & 4 & \alpha & \beta \cr } $$
be 6 and 6.8 respectively. If x3 is changed from 8 to 7, then the mean for the new data will be :
$$\matrix{ {x:} & {{x_1} = 2} & {{x_2} = 6} & {{x_3} = 8} & {{x_4} = 9} \cr {f:} & 4 & 4 & \alpha & \beta \cr } $$
be 6 and 6.8 respectively. If x3 is changed from 8 to 7, then the mean for the new data will be :
4
5
$${{17} \over 3}$$
$${{16} \over 3}$$
Explanation
Given 32 + 8$$\alpha$$ + 9$$\beta$$ = (8 + $$\alpha$$ + $$\beta$$) $$\times$$ 6
$$\Rightarrow$$ 2$$\alpha$$ + 3$$\beta$$ = 16 ..... (i)
Also, 4 $$\times$$ 16 + 4 $$\times$$ $$\alpha$$ + 9$$\beta$$ = (8 + $$\alpha$$ + $$\beta$$) $$\times$$ 6.8
$$\Rightarrow$$ 640 + 40$$\alpha$$ + 90$$\beta$$ = 544 + 68$$\alpha$$ + 68$$\beta$$
$$\Rightarrow$$ 28$$\alpha$$ $$-$$ 22$$\beta$$ = 96
$$\Rightarrow$$ 14$$\alpha$$ $$-$$ 11$$\beta$$ = 48 ..... (ii)
from (i) & (ii)
$$\alpha$$ = 5 & $$\beta$$ = 2
So, new mean = $${{32 + 35 + 18} \over {15}} = {{85} \over {15}} = {{17} \over 3}$$
$$\Rightarrow$$ 2$$\alpha$$ + 3$$\beta$$ = 16 ..... (i)
Also, 4 $$\times$$ 16 + 4 $$\times$$ $$\alpha$$ + 9$$\beta$$ = (8 + $$\alpha$$ + $$\beta$$) $$\times$$ 6.8
$$\Rightarrow$$ 640 + 40$$\alpha$$ + 90$$\beta$$ = 544 + 68$$\alpha$$ + 68$$\beta$$
$$\Rightarrow$$ 28$$\alpha$$ $$-$$ 22$$\beta$$ = 96
$$\Rightarrow$$ 14$$\alpha$$ $$-$$ 11$$\beta$$ = 48 ..... (ii)
from (i) & (ii)
$$\alpha$$ = 5 & $$\beta$$ = 2
So, new mean = $${{32 + 35 + 18} \over {15}} = {{85} \over {15}} = {{17} \over 3}$$
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