Question: Find the 33rd percentile of the data values below.
62.3, 17.8, 97.9, 44.5, 89, 8.9, 35.6, 71.2, 26.7, 53.4, 80.1
1. **State the problem:** Find the 33rd percentile of the data set: $$62.3, 17.8, 97.9, 44.5, 89, 8.9, 35.6, 71.2, 26.7, 53.4, 80.1$$.
2. **Sort the data in ascending order:**
$$8.9, 17.8, 26.7, 35.6, 44.5, 53.4, 62.3, 71.2, 80.1, 89, 97.9$$
3. **Number of data points:**
$$n = 11$$
4. **Calculate the rank (position) of the 33rd percentile:**
$$P = 33\% = 0.33$$
$$\text{Rank} = P \times (n + 1) = 0.33 \times (11 + 1) = 0.33 \times 12 = 3.96$$
5. **Interpret the rank:**
The rank 3.96 means the 33rd percentile lies between the 3rd and 4th data points.
6. **Identify the 3rd and 4th data points:**
3rd data point = $$26.7$$
4th data point = $$35.6$$
7. **Interpolate to find the 33rd percentile:**
$$\text{Percentile} = 26.7 + (3.96 - 3) \times (35.6 - 26.7)$$
$$= 26.7 + 0.96 \times 8.9$$
$$= 26.7 + 8.544 = 35.244$$
**Final answer:**
$$35.244$$