If mean of the following distribution is 211, then what is the value of (4p - 3q) ?

Class  100 - 150 150 - 200 200 - 250 250 - 300 300 - 350 Total
Frequency 4 p 12 q 2 25

 

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  1. 12
  2. 13
  3. 14
  4. 15

Answer (Detailed Solution Below)

Option 3 : 14
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Detailed Solution

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Formula used:

Mean (x̅) = (Summation of fi xi) / (Summation of fi)

Where xi is the class mark and fi is the frequency.

Class Mark (xi) = (Lower Limit + Upper Limit) / 2

Calculation:

First, calculate the class mark (xi) for each class and then fixi:

Class Frequency (fi) Class Mark (xi) fixi
100 - 150 4 (100+150)/2 = 125 4 × 125 = 500
150 - 200 p (150+200)/2 = 175 p × 175 = 175p
200 - 250 12 (200+250)/2 = 225 12 × 225 = 2700
250 - 300 q (250+300)/2 = 275 q × 275 = 275q
300 - 350 2 (300+350)/2 = 325 2 × 325 = 650
Total 25   500 + 175p + 2700 + 275q + 650 = 3850 + 175p + 275q

From the total frequency, we have:

4 + p + 12 + q + 2 = 25

⇒ 18 + p + q = 25

⇒ p + q = 25 - 18

⇒ p + q = 7 ......(Equation 1)

Given Mean = 211

x̅  = (Summation of fixi) / (Summation of fi)

⇒ 211 = (3850 + 175p + 275q) / 25

⇒ 211 × 25 = 3850 + 175p + 275q

⇒ 5275 = 3850 + 175p + 275q

⇒ 5275 - 3850 = 175p + 275q

⇒ 1425 = 175p + 275q

⇒ 1425/25 = 175p/25 + 275q/25

⇒ 57 = 7p + 11q ......(Equation 2)

From Equation 1, p = 7 - q.

Substitute p in Equation 2:

57 = 7(7 - q) + 11q

⇒ 57 = 49 - 7q + 11q

⇒ 57 - 49 = 4q

⇒ 8 = 4q

⇒ q = 8/4

⇒ q = 2

Substitute q = 2 in Equation 1:

p + 2 = 7

⇒ p = 7 - 2

⇒ p = 5

Now, find the value of (4p - 3q):

4p - 3q = 4 × 5 - 3 × 2

⇒ 4p - 3q = 20 - 6

⇒ 4p - 3q = 14

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