1. **State the problem:** We are given grouped data for the thickness of books and their frequencies. We need to estimate the mean thickness. 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 $0 < x \leq 2$, midpoint $x_m = \frac{0 + 2}{2} = 1$ - For $2 < x \leq 4$, midpoint $x_m = \frac{2 + 4}{2} = 3$ - For $4 < x \leq 6$, midpoint $x_m = \frac{4 + 6}{2} = 5$ 4. **Multiply midpoints by frequencies:** - $1 \times 5 = 5$ - $3 \times 8 = 24$ - $5 \times 7 = 35$ 5. **Sum of frequencies:** $$\sum f = 5 + 8 + 7 = 20$$ 6. **Sum of $f \times x_m$:** $$\sum (f \times x_m) = 5 + 24 + 35 = 64$$ 7. **Calculate mean:** $$\text{Mean} = \frac{64}{20} = 3.2$$ **Final answer:** The estimated mean thickness is **3.2 mm**.