Question

CI.
0-30
30-6060-9090-120 120-150150-180
Frequency
12
21
X
52
y y
।।
यदि उपरोक्त बंटन का माध्य । और बारम्बारता का योग 150 हो तो अज्ञात
बारम्बारता तथा y का मान ज्ञात कीजिए।​

Answer 1

Correct Statement is

If mean of the following data is 95 and sum of all the frequencies is 150, find the values of x and y.

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$\large\underline\purple{\bold{Solution :- }}$

$⠀⠀⠀⠀</p><p>\begin{gathered}\boxed{\begin{array}{cccccc}\sf Class\: interval&\sf F_i&\sf Mid\:Value&\sf u_i = \dfrac{x_i - 75}{30}&\sf f_i u_i\\\frac{\quad \qquad\qquad}{}&\frac{\quad \qquad \qquad}{}&\\\sf 0-30&\sf 12&\sf 15&\sf -2&\sf -24\\\\\sf 30-60 &\sf 21&\sf 45&\sf -1 &\sf -21 \\\\\sf 60-90 &\sf x&\sf A = 75&\sf 0&\sf 0\\\\\sf 90-120&\sf 52&\sf 105&\sf 1 &\sf 52\\\\\sf 120-150 &\sf y &\sf 135&\sf 2y&\sf 8\\\\\sf 150-180 &\sf 11&\sf 165&\sf 3&\sf 33\\\frac{\qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}&\frac{\qquad\qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}&\frac{\qquad \qquad\qquad}{}\\\sf Total& \sf \sum f = 150&&& \sf 40 + 24\end{array}}\end{gathered}$

☆ Now, we have

$\tt :  ⟼A = 75 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \\ \tt :  ⟼h = 30 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt :  ⟼ \sum{f_i} = 150 \: \: \: \: \: \: \: \: \: \: \: \: \\ \tt :  ⟼ \sum \: f_i u_i = 40 + 2y$

$\tt :  ⟼ \: mean \: = \: \overline {x} \: = 95$

$\begin{gathered}\tt\red {According \: to \: statement}\end{gathered}$

☆ Sum of frequency = 150

$\bf\implies \: \sum \: f_i \: = \: 150$

$\tt :  ⟼12 + 21 + x + 52 + y + 11 = 150$

$\tt :  ⟼96 + x + y = 150$

$\tt :  ⟼x + y = 150 - 96$

$\tt :  ⟼x + y = 54 - - - (1)$

$\begin{gathered}\tt\red{According \: to \: statement}\end{gathered}$

$\begin{gathered}\bf\red{mean \: = 95}\end{gathered}$

$\begin{gathered}\bf\red{\tt :  ⟼ \: \overline {x} \: = A + \: h \times \dfrac{ \sum \: f_i u_i}{ \sum \:f_i } }\end{gathered}$

$\tt :  ⟼ \: 95 = 75 + 30 \times \dfrac{40 + 2y}{150}$

$\tt :  ⟼ \: 95 - 75 = \dfrac{40 + 2y}{5}$

$\tt :  ⟼ \: 20 = \dfrac{40 + 2y}{5}$

$\tt :  ⟼ \: 2y = 100 - 40$

$\tt :  ⟼ \: 2y = 60$

$\bf\implies \:y = 30$

☆ On substituting y = 30 in equation (1), we get

$\tt :  ⟼ \: x + 30 = 54$

$\bf\implies \:x = 24$

Answer 2

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Seat: Prime Minister's Office, South Block, Central Secretariat, New Delhi, India; Camp ...

Precursor: Vice President of the Executive Councilj