The Mean of the following Distribution is 50.
[tex]\begin{tabular}{|c|c|}\cline{1-2} \bf x_i & \bf f_i \\\cline{1-2} 10 & 17\\\cline{1-2} 30 & (5a + 3)\\\cline{1-2} 50 & 32\\\cline{1-2} 70 & (7a - 11)\\\cline{1-2}90 & 19\\\cline{1-2}\end{tabular}[/tex]
Find the Value of 'a' and Hence, Find the Frequencies of 30 and 70.
[tex]\begin{tabular}{|c|c|}\cline{1-2} \bf x_i & \bf f_i \\\cline{1-2} 10 & 17\\\cline{1-2} 30 & (5a + 3)\\\cline{1-2} 50 & 32\\\cline{1-2} 70 & (7a - 11)\\\cline{1-2}90 & 19\\\cline{1-2}\end{tabular}[/tex]
Find the Value of 'a' and Hence, Find the Frequencies of 30 and 70.
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Explaination:-
We know that ,
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Mean =
-----------(1)
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Sum of the frequencies ,
= 17 + ( 5a + 3 ) + 32 + 7a - 11 +19 = 60 + 12a -------( 2 )
Sum of ( fx )
= 17 × 10 + 30 ( 5a + 3 ) + 50 × 32 + 70 ( 7a - 11 ) + ( 90 + 19 )
= 170 + 150a + 90 + 1600 + 490a - 770 + 1710
= 2800 + 640a -----( 3 )
According to the given problem ,
Mean = 50
( 3 ) / ( 2 ) = 50 [ from ( 1 ) ]
[ 2800 + 640a ] / ( 60 + 12a ) = 50
2800 + 640a = 50 ( 60 + 12a )
2800 + 640 a = 3000 + 600a
640 a - 600a = 3000 - 2800
40a = 200
a = 200 / 40
a = 5
Therefore,
Frequeny of 30 = 5a + 3 = 5 × 5 + 3 = 25 + 3 = 28
Frequecy of 70 = 7a - 11 = 7 × 5 - 11 = 35 -11 = 24
I hope this helps you.