1. **Stating the problem:** We are given a grouped frequency distribution and need to find the 30th percentile class and related values. 2. **Formula for percentile class:** The formula to find the percentile class is: $$P_k = \frac{k}{100} \times n$$ where $k$ is the percentile and $n$ is the total number of observations. 3. **Given data:** - Total number of observations, $n = 27$ - Percentile to find, $k = 30$ 4. **Calculate the position of the 30th percentile:** $$P_{30} = \frac{30}{100} \times 27 = 8.1$$ This means the 30th percentile lies in the class where the cumulative frequency just exceeds 8.1. 5. **Identify the 30th percentile class:** From the cumulative frequencies (cf): - 6-10: cf = 2 - 11-15: cf = 3 - 16-20: cf = 6 - 21-25: cf = 7 - 26-30: cf = 12 Since 8.1 lies between 7 and 12, the 30th percentile class is 26-30. 6. **Values for the 30th percentile class:** - Lower class boundary, $X_{lb} = 25.5$ - Frequency of the class, $f = 5$ - Cumulative frequency before the class, $cf_b = 7$ 7. **Calculate the 30th percentile value using the formula:** $$P_k = X_{lb} + \left( \frac{P_k - cf_b}{f} \right) \times \text{class width}$$ Class width = upper boundary - lower boundary = 30.5 - 25.5 = 5 Substitute values: $$P_{30} = 25.5 + \left( \frac{8.1 - 7}{5} \right) \times 5 = 25.5 + (0.22) \times 5 = 25.5 + 1.1 = 26.6$$ **Final answer:** The 30th percentile is approximately **26.6**.