1. **State the problem:** We need to estimate the mean length of insects given grouped data with intervals and frequencies. 2. **Formula for mean of grouped data:** $$\text{Mean} = \frac{\sum (f \times x_m)}{\sum f}$$ where $f$ is the frequency and $x_m$ is the midpoint of each class interval. 3. **Calculate midpoints:** - For $0 < x \leq 10$, midpoint $x_m = \frac{0 + 10}{2} = 5$ - For $10 < x \leq 20$, midpoint $x_m = \frac{10 + 20}{2} = 15$ - For $20 < x \leq 30$, midpoint $x_m = \frac{20 + 30}{2} = 25$ 4. **Multiply frequencies by midpoints:** - $7 \times 5 = 35$ - $8 \times 15 = 120$ - $5 \times 25 = 125$ 5. **Sum frequencies and products:** - Total frequency $= 7 + 8 + 5 = 20$ - Sum of products $= 35 + 120 + 125 = 280$ 6. **Calculate mean:** $$\text{Mean} = \frac{280}{20} = 14$$ **Final answer:** The estimated mean length of the insects is 14 mm.