A class of 25 students was surveyed to find out how many WhatsApp messages they each sent in a particular week - Junior Cycle Mathematics - Question 9 - 2015

From the previous calculation, we know that there are 22 students who sent 50 or more messages. Among these, the number of students who sent 50 – 70 messages is 10.

Thus, the probability is:

$P(X ext{ sent } 50 - 70) = \frac{10}{22} \approx 0.455$

To estimate the mean using mid-interval values, we calculate as follows:

• Midpoints:

  • For 0 – 30: midpoint = 15
  • For 30 – 50: midpoint = 40
  • For 50 – 70: midpoint = 60
  • For 70 – 100: midpoint = 85
  • For 100 – 160: midpoint = 130

The sum is given by:

$(0 imes 1) + (30 imes 2) + (50 imes 10) + (70 imes 7) + (100 imes 5)${ = 0 + 60 + 500 + 490 + 500 = 1550 }$$

Thus the mean is:

$Mean = \frac{1550}{25} = 62$

Referencing the data from above:

The minimum contribution from each group is used:

• For 0 – 30: use the minimum value 0, with 1 student → $0 imes 1$
• For 30 – 50: use minimum value 30, with 2 students → $30 imes 2$
• For 50 – 70: use minimum value 50, with 10 students → $50 imes 10$
• For 70 – 100: use minimum value 70, with 7 students → $70 imes 7$
• For 100 – 160: use minimum value 100, with 5 students → $100 imes 5$

Calculating this gives:

$0 + 60 + 500 + 490 + 500 = 1550$

Thus, the smallest total value is 1550.