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19a) Calculate the value of 'w'. 19b) Determine the median score.

Mathematics

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Question 19a)
To find the value of 'w', we use the formula for the average (mean) score:

Average=∑(Mark×No. of pupils)∑(No. of pupils)\text{Average} = \frac{\sum (\text{Mark} \times \text{No. of pupils})}{\sum (\text{No. of pupils})}

  1. Calculate the sum of (Mark × No. of pupils).

    ∑(Mark×No. of pupils)=(45×2)+(60×w)+(50×4)+(70×1)\sum (\text{Mark} \times \text{No. of pupils}) = (45 \times 2) + (60 \times w) + (50 \times 4) + (70 \times 1) =90+60w+200+70= 90 + 60w + 200 + 70 =360+60w= 360 + 60w

  2. Calculate the total number of pupils.

    ∑(No. of pupils)=2+w+4+1\sum (\text{No. of pupils}) = 2 + w + 4 + 1 =7+w= 7 + w

  3. Substitute the given average score (54) and the sums into the average formula.

    54=360+60w7+w54 = \frac{360 + 60w}{7 + w}

  4. Solve for w.

    54(7+w)=360+60w54(7 + w) = 360 + 60w 378+54w=360+60w378 + 54w = 360 + 60w 378−360=60w−54w378 - 360 = 60w - 54w 18=6w18 = 6w w=186w = \frac{18}{6} w=3w = \boxed{3}

    Question 19b)
    To find the median score, we first need to update the table with the value of w=3w=3.

    MarksNo. of pupils
    452
    603
    504
    701

  5. Find the total number of pupils (N).

    N=2+3+4+1=10N = 2 + 3 + 4 + 1 = 10

  6. Order the marks from lowest to highest and list the scores.

    The ordered scores are:
    45, 45 (from 2 pupils scoring 45)
    50, 50, 50, 50 (from 4 pupils scoring 50)
    60, 60, 60 (from 3 pupils scoring 60)
    70 (from 1 pupil scoring 70)

  7. Determine the position of the median.

    Since N = 10 (an even number), the median is the average of the two middle values. These are the N2\frac{N}{2}-th and (N2+1)(\frac{N}{2} + 1)-th values.
    The positions are the 102=5\frac{10}{2} = 5-th score and the (102+1)=6(\frac{10}{2} + 1) = 6-th score.

    From the ordered list:
    The 5th score is 50.
    The 6th score is 50.

  8. Calculate the median.

    Median=5th score+6th score2\text{Median} = \frac{\text{5th score} + \text{6th score}}{2} Median=50+502\text{Median} = \frac{50 + 50}{2} Median=1002\text{Median} = \frac{100}{2} Median=50\text{Median} = \boxed{50}
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