High School

The mean of 20 observations is 38. If two observations are taken as 84 and 36 instead of 48 and 63, find the new means.

a. 38.45
b. 41.15
c. 37.55
d. 40.05

The average weight of 8 persons increases by 1.5 kg if a person weighing 65 kg is replaced by a new person. What would be the weight of the new person?

a. 76 kg
b. 80 kg
c. 77 kg
d. None of these

Answer :

Let's solve each part of the question step-by-step.

Part 1: Finding the New Mean of 20 Observations

Initially, the mean of 20 observations is given as 38.

  1. Calculate the original total sum of observations:
    [tex]\text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 38 \times 20 = 760[/tex]

  2. Adjust the total sum for changes in observations:

    • Originally, the two observations were 48 and 63, which sum to 111.
    • They are replaced by 84 and 36, which sum to 120.
    • Difference in sums: [tex]120 - 111 = 9[/tex]

  3. Calculate the new total sum:
    [tex]\text{New Total Sum} = 760 + 9 = 769[/tex]

  4. Find the new mean:
    [tex]\text{New Mean} = \frac{\text{New Total Sum}}{\text{Number of Observations}} = \frac{769}{20} = 38.45[/tex]

The correct answer for the first part is: a. 38.45

Part 2: Finding the Weight of the New Person

The average weight of 8 persons increases by 1.5 kg.

  1. Find the original total weight of 8 persons:
    Let the original total weight be [tex]x[/tex].
    Original average weight is [tex]\frac{x}{8}[/tex].

  2. Determine the new total weight after the change:

    • New average weight is [tex]\frac{x}{8} + 1.5[/tex].
    • New total weight = New average weight [tex]\times[/tex] number of persons = [tex](\frac{x}{8} + 1.5) \times 8[/tex]
    • [tex]x + 12[/tex]

  3. Find the weight of the new person:

    • Original weight of the replaced person = 65 kg
    • New weight of the person = [tex]x + 12 - x + 65 = 65 + 12 = 77[/tex] kg

The correct answer for the second part is: c. 77 kg

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