1. **State the problem:** Find the 10th, 25th, 50th, 75th, and 95th percentiles of the data set: 40, 45, 50, 56, 60, 65, 65, 80, 90. 2. **Sort the data:** The data is already sorted: 40, 45, 50, 56, 60, 65, 65, 80, 90. 3. **Number of data points:** $n = 9$. 4. **Percentile formula:** The position $P_k$ of the $k$th percentile is given by: $$P_k = \frac{k}{100} \times (n + 1)$$ 5. **Calculate each percentile position and value:** - 10th percentile position: $$P_{10} = 0.10 \times (9 + 1) = 1$$ Value at position 1 is 40. - 25th percentile position: $$P_{25} = 0.25 \times 10 = 2.5$$ Value is between position 2 and 3: 45 and 50. Interpolate: $$45 + 0.5 \times (50 - 45) = 45 + 2.5 = 47.5$$ - 50th percentile position: $$P_{50} = 0.50 \times 10 = 5$$ Value at position 5 is 60. - 75th percentile position: $$P_{75} = 0.75 \times 10 = 7.5$$ Value is between position 7 and 8: 65 and 80. Interpolate: $$65 + 0.5 \times (80 - 65) = 65 + 7.5 = 72.5$$ - 95th percentile position: $$P_{95} = 0.95 \times 10 = 9.5$$ Position 9 is 90, position 10 does not exist, so take the max value 90. 6. **Final answers:** - 10th percentile = 40 - 25th percentile = 47.5 - 50th percentile = 60 - 75th percentile = 72.5 - 95th percentile = 90 These values represent the data points below which the given percentages of data fall.