To find the third quartile (Q3) for the given data set of commute times, follow these steps:
1. Convert the Stem-and-Leaf Plot into a List of Numbers:
The given stem-and-leaf plot needs to be converted into individual commute times:
- Stem 1: Leaves 0, 1, 3, 4, 4, 6, 7
- Commute times: 10, 11, 13, 14, 14, 16, 17
- Stem 2: Leaves 0, 5, 6, 6, 6, 7
- Commute times: 20, 25, 26, 26, 26, 27
- Stem 3: Leaves 0, 3, 3, 3, 5, 8
- Commute times: 30, 33, 33, 33, 35, 38
- Stem 4: Leaves 1, 1, 4, 4, 5, 5, 6
- Commute times: 41, 41, 44, 44, 45, 45, 46
Combined list of commute times:
`10, 11, 13, 14, 14, 16, 17, 20, 25, 26, 26, 26, 27, 30, 33, 33, 33, 35, 38, 41, 41, 44, 44, 45, 45, 46`
2. Order the Commute Times:
The commute times appear to already be in increasing order.
3. Determine the Position of Q3:
The third quartile (Q3) is the value below which 75% of the data fall.
To find Q3, use the formula:
[tex]\[
Q3 = \left(\frac{3(n+1)}{4}\right)^{th} \text{position}
\][/tex]
Where [tex]\( n \)[/tex] is the number of data points.
Here, [tex]\( n = 26 \)[/tex].
Calculating the position:
[tex]\[
Q3 = \left(\frac{3(26+1)}{4}\right)^{th} = \left(\frac{3 \times 27}{4}\right)^{th} = \left(\frac{81}{4}\right)^{th} = 20.25^{th} \text{position}
\][/tex]
4. Find the Value at Q3 Position:
Since 20.25 is not an integer, the third quartile will be between the values at the 20th and 21st positions in the ordered list.
- The 20th value in the list is 38.
- The 21st value in the list is 41.
Interpolating between these positions:
[tex]\[
Q3 = 38 + 0.25 \times (41 - 38) = 38 + 0.25 \times 3 = 38 + 0.75 = 38.75
\][/tex]
Therefore, the third quartile (Q3) of the provided commute times is 40.25 minutes.
