1. **Problem Statement:** We have weights of 40 students rounded to the nearest kg. We need to: (i) Construct a frequency table with class intervals of 5 kg starting at 41 kg. (ii) Use the frequency table to find the mean and median weight. 2. **Step 1: Constructing the frequency table** Class intervals: 41-45, 46-50, 51-55, 56-60, 61-65, 66-70, 71-75 (each interval includes the lower bound and excludes the upper bound except the last). Count the number of weights in each interval: - 41-45: 42,44,44 (3) - 46-50: 47,47,48,50 (4) - 51-55: 51,52,52,52,53,54,54,54,55 (9) - 56-60: 56,57,57,57,57,58,58,58,58,59,59,59 (12) - 61-65: 60,62,63,63,64,65 (6) - 66-70: 67,68,68,69 (4) - 71-75: 72,73 (2) Frequency table: | Class Interval | Frequency | |----------------|-----------| | 41 - 45 | 3 | | 46 - 50 | 4 | | 51 - 55 | 9 | | 56 - 60 | 12 | | 61 - 65 | 6 | | 66 - 70 | 4 | | 71 - 75 | 2 | 3. **Step 2: Calculate the mean weight** Use midpoint $x_i$ of each class interval: - 41-45 midpoint = $\frac{41+45}{2} = 43$ - 46-50 midpoint = $48$ - 51-55 midpoint = $53$ - 56-60 midpoint = $58$ - 61-65 midpoint = $63$ - 66-70 midpoint = $68$ - 71-75 midpoint = $73$ Calculate $\sum f_i x_i$: $$ 3 \times 43 + 4 \times 48 + 9 \times 53 + 12 \times 58 + 6 \times 63 + 4 \times 68 + 2 \times 73 = 129 + 192 + 477 + 696 + 378 + 272 + 146 = 2290 $$ Total frequency $\sum f_i = 40$ Mean $= \frac{\sum f_i x_i}{\sum f_i} = \frac{2290}{40} = 57.25$ 4. **Step 3: Calculate the median weight** Median class is where cumulative frequency reaches $\frac{40}{2} = 20$. Cumulative frequencies: - 41-45: 3 - 46-50: 3+4=7 - 51-55: 7+9=16 - 56-60: 16+12=28 (median class) Median class: 56-60 Use median formula: $$ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h $$ Where: - $L=55.5$ (lower boundary of median class, since class is 56-60 rounded to nearest kg, lower boundary is 55.5) - $N=40$ - $F=16$ (cumulative frequency before median class) - $f=12$ (frequency of median class) - $h=5$ (class width) Calculate: $$ \text{Median} = 55.5 + \left( \frac{20 - 16}{12} \right) \times 5 = 55.5 + \frac{4}{12} \times 5 = 55.5 + \frac{5}{3} = 55.5 + 1.6667 = 57.17 $$ **Final answers:** - Mean weight $= 57.25$ kg - Median weight $= 57.17$ kg