Question: Find the third quartile for the given quantitative data.
78 17
56 63
77 32
51 75
7 30
83 43
70 41
40
1. **State the problem:** We need to find the third quartile ($Q_3$) of the given quantitative data set.
2. **List the data:** The data points are $78, 17, 56, 63, 77, 32, 51, 75, 7, 30, 83, 43, 70, 41, 40$.
3. **Sort the data in ascending order:**
$$7, 17, 30, 32, 40, 41, 43, 51, 56, 63, 70, 75, 77, 78, 83$$
4. **Number of data points:** There are $n = 15$ data points.
5. **Formula for the third quartile position:**
$$Q_3 = \text{value at position } \left(\frac{3(n+1)}{4}\right)$$
6. **Calculate the position:**
$$\frac{3(15+1)}{4} = \frac{3 \times 16}{4} = \frac{48}{4} = 12$$
7. **Find the value at position 12:** The 12th value in the sorted list is $75$.
8. **Conclusion:** The third quartile $Q_3$ is $75$.
This means that 75% of the data values are less than or equal to $75$.