1. **State the problem:** We have grouped data of student ages and their frequencies. We need to calculate the mean age and analyze the histogram to find the modal age. 2. **Calculate the mean age:** - Intervals and frequencies are: - 18-<19: 24 students - 19-<20: 70 students - 20-<24: 76 students - 24-<26: 48 students - 26-<30: 16 students - 30-<32: 6 students - Find midpoints of each age group: - 18-<19 midpoint = $\frac{18 + 19}{2} = 18.5$ - 19-<20 midpoint = $\frac{19 + 20}{2} = 19.5$ - 20-<24 midpoint = $\frac{20 + 24}{2} = 22$ - 24-<26 midpoint = $\frac{24 + 26}{2} = 25$ - 26-<30 midpoint = $\frac{26 + 30}{2} = 28$ - 30-<32 midpoint = $\frac{30 + 32}{2} = 31$ - Calculate sum of $\text{midpoint} \times \text{frequency}$: $$(18.5)(24) + (19.5)(70) + (22)(76) + (25)(48) + (28)(16) + (31)(6)$$ $$= 444 + 1365 + 1672 + 1200 + 448 + 186 = 5315$$ - Total number of students: $$24 + 70 + 76 + 48 + 16 + 6 = 240$$ - Mean age $= \frac{5315}{240} = 22.15$ years (approx) 3. **Draw a histogram:** (b)(i) - Horizontal axis: Age groups 18-<19, 19-<20, 20-<24, 24-<26, 26-<30, 30-<32 - Vertical axis: Number of students as bar heights - Bars heights: 24, 70, 76, 48, 16, 6 respectively 4. **Estimate the modal age from histogram:** (b)(ii) - The tallest bar corresponds to age group 20-<24 with 76 students. - Mode is estimated as the midpoint of this group: $22$ years approximately. **Final answers:** - Mean age $\approx 22.15$ years - Estimated modal age $\approx 22$ years