1. **State the problem:** We are given a frequency distribution table showing thickness intervals of books and their frequencies. We need to estimate the mean thickness of the books. 2. **Formula for mean from grouped data:** $$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is the frequency of the $i$th group and $x_i$ is the midpoint of the $i$th group. 3. **Find midpoints of each class interval:** - For $0 < x \leq 2$, midpoint $x_1 = \frac{0 + 2}{2} = 1$ - For $2 < x \leq 4$, midpoint $x_2 = \frac{2 + 4}{2} = 3$ - For $4 < x \leq 6$, midpoint $x_3 = \frac{4 + 6}{2} = 5$ 4. **Calculate $f_i x_i$ for each group:** - $3 \times 1 = 3$ - $8 \times 3 = 24$ - $9 \times 5 = 45$ 5. **Sum frequencies and sum of $f_i x_i$:** - $\sum f_i = 3 + 8 + 9 = 20$ - $\sum f_i x_i = 3 + 24 + 45 = 72$ 6. **Calculate mean thickness:** $$\text{Mean} = \frac{72}{20} = 3.6$$ **Final answer:** The estimated mean thickness of the books is **3.6 mm**.