Question

The Mean of the following Distribution is 50.

$\begin{tabular}{|c|c|}\cline{1-2} \bf x_i & \bf f_i \\\cline{1-2} 10 & 17\\\cline{1-2} 30 & (5a + 3)\\\cline{1-2} 50 & 32\\\cline{1-2} 70 & (7a - 11)\\\cline{1-2}90 & 19\\\cline{1-2}\end{tabular}$

Find the Value of 'a' and Hence, Find the Frequencies of 30 and 70.

Answer 1

Explaination:-

We know that ,

__________________________________

Mean = $\frac{sum \: of \: observation}{number \: of \: observation}$ -----------(1)

___________________________________

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.