Question: The lengths (in kilometers) of a random sample of 18 rivers on the South Island of New Zealand that flow to the Pacific Ocean are listed in table below.
Data Table for Lengths of Rivers (in km)
Lengths of Rivers (in km)
64
93
209
161
121
98
66
209
72
288
169
138
80
322
208
48
136
64
For the data shown above, find the following.
Do not round any of your answers.
a) Find the 5-number summary:
{
,
,
,
,
}
What units would go on each of your answers above?
b) Compute the IQR.
Put your answer in the first box and the correct units in the second box.
c) What is the lower fence for this data set?
What is the upper fence for this data set?
d) Which of the following is the correct modified boxplot for the data?
1. **State the problem:** We have 18 river lengths in kilometers and need to find the 5-number summary, IQR, fences, and identify the correct modified boxplot.
2. **Sort the data:**
$$48, 64, 64, 66, 72, 80, 93, 98, 121, 136, 138, 161, 169, 208, 209, 209, 288, 322$$
3. **Find the 5-number summary:**
- Minimum ($Q_0$): smallest value = $48$
- Maximum ($Q_4$): largest value = $322$
- Median ($Q_2$): middle value of 18 numbers is average of 9th and 10th values:
$$\frac{121 + 136}{2} = \frac{257}{2} = 128.5$$
- Lower quartile ($Q_1$): median of lower half (first 9 values): 5th value = $72$
- Upper quartile ($Q_3$): median of upper half (last 9 values): 5th value in upper half = $208$
So the 5-number summary is:
$$\{48, 72, 128.5, 208, 322\}$$
Units for all values are kilometers (km).
4. **Compute the IQR (Interquartile Range):**
$$IQR = Q_3 - Q_1 = 208 - 72 = 136$$
Units: kilometers (km).
5. **Calculate fences:**
- Lower fence:
$$Q_1 - 1.5 \times IQR = 72 - 1.5 \times 136 = 72 - 204 = -132$$
- Upper fence:
$$Q_3 + 1.5 \times IQR = 208 + 1.5 \times 136 = 208 + 204 = 412$$
6. **Interpret fences:**
- Any data below $-132$ or above $412$ is an outlier.
- Since minimum is $48$ and maximum is $322$, no outliers exist.
7. **Identify correct modified boxplot:**
- The boxplot must have whiskers at minimum $48$ and maximum $322$.
- The box edges at $72$ and $208$.
- Median line at $128.5$.
From the given options, the boxplot with these values is:
$$\{48, 72, 128.5, 208, 322\}$$
**Final answers:**
- 5-number summary: $\{48, 72, 128.5, 208, 322\}$ km
- IQR: $136$ km
- Lower fence: $-132$
- Upper fence: $412$
- Correct modified boxplot corresponds to the one with these five values.