1. **State the problem:** Find the 34th percentile, $P_{34}$, from the given data set. 2. **List the data:** The data set has 38 values: $$140, 150, 170, 190, 200, 210, 250, 270, 340, 360, 370, 380, 410, 420, 430, 440, 450, 460, 480, 490, 520, 530, 540, 550, 560, 570, 630, 640, 650, 690, 720, 780, 820, 840, 850, 950, 990, 1000$$ 3. **Formula for percentile position:** $$n = 38$$ $$L = \frac{P}{100} \times (n + 1)$$ where $P=34$ for the 34th percentile. 4. **Calculate the position:** $$L = \frac{34}{100} \times (38 + 1) = 0.34 \times 39 = 13.26$$ 5. **Interpretation:** The 34th percentile lies between the 13th and 14th data points when data is sorted. 6. **Identify the 13th and 14th data points:** Sorted data (already sorted): - 13th value = 410 - 14th value = 420 7. **Interpolate to find $P_{34}$:** $$P_{34} = x_{13} + (L - 13) \times (x_{14} - x_{13})$$ $$= 410 + (13.26 - 13) \times (420 - 410)$$ $$= 410 + 0.26 \times 10 = 410 + 2.6 = 412.6$$ 8. **Final answer:** $$\boxed{412.6}$$