To solve this problem, we need to calculate the upper quartile (Q3) for the given data, which represents the weights of students. The quartiles divide the dataset into four equal parts, and Q3 corresponds to the value below which 75% of the data falls.

Step-by-Step Process:

1. Create a Cumulative Frequency Table: We'll first calculate the cumulative frequency (CF) based on the given frequency (F).

So, the total number of students is 30.

2. Find the Position of Q3: The position of the third quartile $Q_3$ is determined using the formula: $Q_3 = \frac{3(N + 1)}{4}$ Where $N$ is the total frequency, which is 30. Therefore: $Q_3 = \frac{3(30 + 1)}{4} = \frac{3 \times 31}{4} = 23.25$ So, $Q_3$ is in the 23.25th position.

3. Identify the Interval Containing Q3: Based on the cumulative frequencies, the 23.25th value falls within the interval 72–79 because the cumulative frequency of the previous class is 19 (from 64–71), and the cumulative frequency of the 72–79 class is 26. Hence, Q3 lies in this interval.

4. Apply the Quartile Formula: The formula for finding the value of $Q_3$ within a grouped data is: $Q_3 = L + \left( \frac{\frac{3(N + 1)}{4} - CF_{\text{before}}}{f} \right) \times h$ Where:

   • $L = 72$ (lower boundary of the interval containing $Q_3$),
   • $N = 30$ (total frequency),
   • $CF_{\text{before}} = 19$ (cumulative frequency of the interval before 72–79),
   • $f = 7$ (frequency of the interval 72–79),
   • $h = 79 - 72 = 7$ (class width).

   Substituting the values: $Q_3 = 72 + \left( \frac{23.25 - 19}{7} \right) \times 7$ $Q_3 = 72 + \left( \frac{4.25}{7} \right) \times 7$ $Q_3 = 72 + 4.25 = 76.25$

So, the upper quartile ($Q_3$) is approximately 76.25, but this does not match any of the given options exactly.

Approximate Closest Option:

The closest option is 77.75 (option D).

Do you want more details or have any questions about this method?

Here are 5 questions related to this topic:

1. How is the median different from the quartiles in a dataset?
2. What other methods exist for finding quartiles in grouped data?
3. Why is interpolation necessary when calculating quartiles for grouped data?
4. How does the number of intervals affect the precision of quartile calculations?
5. Can you explain how cumulative frequency affects the calculation of quartiles?

Tip: When dealing with grouped data, always ensure you correctly determine the cumulative frequency to locate the quartiles accurately.