1. **State the problem:** We are given a frequency distribution of weights of 50 dogs in intervals and asked to find: (a) The modal class. (b) An estimate for the mean weight. 2. **Modal class:** The modal class is the class interval with the highest frequency. From the table: - 4 < w \leq 8: frequency 5 - 8 < w \leq 12: frequency 17 - 12 < w \leq 16: frequency 15 - 16 < w \leq 20: frequency 9 - 20 < w \leq 24: frequency 4 The highest frequency is 17 for the interval 8 < w \leq 12. **Answer (a):** The modal class is $8 < w \leq 12$ kg. 3. **Estimate the mean weight:** We use the formula for the mean of grouped data: $$\text{Mean} = \frac{\sum (f \times x)}{\sum f}$$ where $f$ is the frequency and $x$ is the midpoint of each class. 4. **Calculate midpoints:** - For 4 < w \leq 8: midpoint $= \frac{4 + 8}{2} = 6$ - For 8 < w \leq 12: midpoint $= \frac{8 + 12}{2} = 10$ - For 12 < w \leq 16: midpoint $= \frac{12 + 16}{2} = 14$ - For 16 < w \leq 20: midpoint $= \frac{16 + 20}{2} = 18$ - For 20 < w \leq 24: midpoint $= \frac{20 + 24}{2} = 22$ 5. **Calculate $f \times x$ for each class:** - $5 \times 6 = 30$ - $17 \times 10 = 170$ - $15 \times 14 = 210$ - $9 \times 18 = 162$ - $4 \times 22 = 88$ 6. **Sum frequencies and $f \times x$ values:** - $\sum f = 5 + 17 + 15 + 9 + 4 = 50$ - $\sum (f \times x) = 30 + 170 + 210 + 162 + 88 = 660$ 7. **Calculate mean:** $$\text{Mean} = \frac{660}{50} = 13.2$$ **Answer (b):** The estimated mean weight is $13.2$ kg.