1. **Problem Statement:** Find the 75th percentile manually for the given frequency distribution with $n=33$ and $p=0.75$. 2. **Step 1: Calculate the position of the percentile** Use the formula for the percentile position: $$L = n \times p = 33 \times 0.75 = 24.75$$ This means the 75th percentile lies between the 24th and 25th data points. 3. **Step 2: Identify the class interval containing the 75th percentile** From the cumulative frequency (CF) column: - CF just before 24.75 is 25 at $X=30$ - CF just after 24.75 is 25 at $X=30$ So the 75th percentile lies in the class where $X=30$. 4. **Step 3: Use the percentile formula for grouped data** $$P_p = L + \left(\frac{\frac{n \times p - F}{f}}{w}\right)$$ Where: - $L$ = lower boundary of the class containing the percentile - $F$ = cumulative frequency before the class - $f$ = frequency of the class - $w$ = class width From the table: - $L = 29.5$ (lower boundary of class 30) - $F = 19$ (CF before class 30) - $f = 5$ (frequency of class 30) - $w = 1$ (class width) 5. **Step 4: Calculate the percentile value** $$P_{75} = 29.5 + \left(\frac{24.75 - 19}{5}\right) \times 1 = 29.5 + \frac{5.75}{5} = 29.5 + 1.15 = 30.65$$ 6. **Step 5: Interpretation** The 75th percentile is approximately $30.65$, meaning 75% of the data lies below this value. **Final answer:** $$\boxed{30.65}$$