1. **State the problem:** We need to find the 73rd percentile, $P_{73}$, of the given data set: $$2, 6, 9, 22, 23, 27, 33, 48, 49, 50, 55, 56, 63, 67, 68, 77, 80, 87, 89, 98$$ 2. **Formula for percentile position:** The position $L$ of the $k$th percentile in an ordered data set of size $n$ is given by: $$L = \frac{k}{100} (n + 1)$$ where $k=73$ and $n=20$ (since there are 20 data points). 3. **Calculate the position:** $$L = \frac{73}{100} (20 + 1) = 0.73 \times 21 = 15.33$$ This means the 73rd percentile lies between the 15th and 16th data points. 4. **Identify the data points:** - 15th data point: $68$ - 16th data point: $77$ 5. **Interpolate to find $P_{73}$:** $$P_{73} = x_{15} + (L - 15)(x_{16} - x_{15})$$ $$P_{73} = 68 + (15.33 - 15)(77 - 68)$$ $$P_{73} = 68 + 0.33 \times 9 = 68 + 2.97 = 70.97$$ 6. **Final answer:** The 73rd percentile $P_{73}$ is approximately **70.97**. This means that about 73% of the data values are below 70.97.