1. **Problem Statement:** We are given ungrouped data and want to find the $k^{th}$ percentile using the formula $$P_k = \frac{kN}{100}$$ where $N$ is the sample size and $k$ is the desired percentile. 2. **Given Data:** $\{39,15,6,7,41,41,49,43,43,47,36\}$ with $N=11$. 3. **Step 1: Sort the data in ascending order:** $$6,7,15,36,39,41,41,43,43,47,49$$ 4. **Step 2: Calculate positions for $P_{25}$ and $P_{75}$:** - For $P_{25}$: $$P_{25} = \frac{25 \times 11}{100} = 2.75$$ - For $P_{75}$: $$P_{75} = \frac{75 \times 11}{100} = 8.25$$ 5. **Step 3: Interpret positions:** - Position 2.75 means the $P_{25}$ lies between the 2nd and 3rd data points. - Position 8.25 means the $P_{75}$ lies between the 8th and 9th data points. 6. **Step 4: Find values at these positions using interpolation:** - $P_{25}$ between 2nd (7) and 3rd (15) values: $$P_{25} = 7 + 0.75 \times (15 - 7) = 7 + 6 = 13$$ - $P_{75}$ between 8th (43) and 9th (43) values: $$P_{75} = 43 + 0.25 \times (43 - 43) = 43$$ 7. **Step 5: Compare with given values:** - Given $P_{25}$ position is 3rd with value 15, but correct position is 2.75 with value 13. - Given $P_{75}$ position is 9th with value 43, which matches our calculation. **Final corrected percentiles:** $$P_{25} = 13$$ $$P_{75} = 43$$