Question: The lengths (in kilometers) of a random sample of 20 rivers on the South Island of New Zealand that flow to the Tasman Sea are listed in table below.
Data Table for Lengths of Rivers (in km)
Lengths of Rivers (in km)
35
64
37
108
56
64
56
80
64
40
51
68
121
32
72
177
72
48
80
76
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?
20
40
60
80
100
120
140
160
180
Lengths of Rivers (in km)
26
47
64
76
108
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20
40
60
80
100
120
140
160
180
Lengths of Rivers (in km)
32
49.5
64
78
108
[Graphs generated by this script: setBorder(15); initPicture(20,180,-3,6);axes(20,100,1,5,null,1,"off",1);text([92.5,-3],"Lengths of Rivers (in km)");line([32,2],[32,4]); rect([49.5,2],[78,4]); line([64,2],[64,4]);line([108,2],[108,4]); line([32,3],[49.5,3]); line([78,3],[108,3]);fontsize*=.8;fontfill="blue";text([32,4],"32","above");text([49.5,4],"49.5","above");text([64,4],"64","above");text([78,4],"78","above");text([108,4],"108","above");fontfill="black";fontsize*=1.25;dot([121,3],"open");dot([177,3],"open");]
20
40
60
80
100
120
140
160
180
Lengths of Rivers (in km)
35
52
64
80
108
[Graphs generated by this script: setBorder(15); initPicture(20,180,-3,6);axes(20,100,1,5,null,1,"off",1);text([91,-3],"Lengths of Rivers (in km)");line([35,2],[35,4]); rect([52,2],[80,4]); line([64,2],[64,4]);line([108,2],[108,4]); line([35,3],[52,3]); line([80,3],[108,3]);fontsize*=.8;fontfill="blue";text([35,4],"35","above");text([52,4],"52","above");text([64,4],"64","above");text([80,4],"80","above");text([108,4],"108","above");fontfill="black";fontsize*=1.25;dot([135,3],"open");dot([177,3],"open");]
20
40
60
80
100
120
140
160
180
Lengths of Rivers (in km)
26
45
63
72
108
[Graphs generated by this script: setBorder(15); initPicture(20,180,-3,6);axes(20,100,1,5,null,1,"off",1);text([95.5,-3],"Lengths of Rivers (in km)");line([26,2],[26,4]); rect([45,2],[72,4]); line([63,2],[63,4]);line([108,2],[108,4]); line([26,3],[45,3]); line([72,3],[108,3]);fontsize*=.8;fontfill="blue";text([26,4],"26","above");text([45,4],"45","above");text([63,4],"63","above");text([72,4],"72","above");text([108,4],"108","above");fontfill="black";fontsize*=1.25;dot([121,3],"open");dot([177,3],"open");]
4
1. **State the problem:** We have a data set of 20 river lengths (in kilometers) and need to find the 5-number summary, IQR, fences, and identify the correct modified boxplot.
2. **Sort the data:**
$$32, 35, 37, 40, 48, 51, 56, 56, 64, 64, 64, 68, 72, 72, 76, 80, 80, 108, 121, 177$$
3. **Find the 5-number summary:**
- Minimum ($Q_0$): smallest value = $32$
- Maximum ($Q_4$): largest value = $177$
- Median ($Q_2$): middle value of 20 numbers is average of 10th and 11th values:
$$\text{Median} = \frac{64 + 64}{2} = 64$$
- Lower quartile ($Q_1$): median of lower half (first 10 values):
Values: $32, 35, 37, 40, 48, 51, 56, 56, 64, 64$
Median of these 10 values is average of 5th and 6th:
$$Q_1 = \frac{48 + 51}{2} = 49.5$$
- Upper quartile ($Q_3$): median of upper half (last 10 values):
Values: $64, 68, 72, 72, 76, 80, 80, 108, 121, 177$
Median is average of 5th and 6th values:
$$Q_3 = \frac{76 + 80}{2} = 78$$
**5-number summary:**
$$\{32, 49.5, 64, 78, 177\}$$
Units for all values: kilometers (km)
4. **Compute the IQR (Interquartile Range):**
$$IQR = Q_3 - Q_1 = 78 - 49.5 = 28.5$$
Units: kilometers (km)
5. **Calculate fences:**
- Lower fence:
$$\text{Lower fence} = Q_1 - 1.5 \times IQR = 49.5 - 1.5 \times 28.5 = 49.5 - 42.75 = 6.75$$
- Upper fence:
$$\text{Upper fence} = Q_3 + 1.5 \times IQR = 78 + 1.5 \times 28.5 = 78 + 42.75 = 120.75$$
6. **Identify outliers:**
- Values below $6.75$: none
- Values above $120.75$: $121$ and $177$ are outliers
7. **Choose the correct modified boxplot:**
- Box from $Q_1=49.5$ to $Q_3=78$
- Median at $64$
- Whiskers extend to smallest and largest non-outlier values: minimum $32$, maximum $108$
- Outliers at $121$ and $177$
The correct boxplot matches the second option:
- Minimum whisker at $32$
- Box from $49.5$ to $78$
- Median line at $64$
- Maximum whisker at $108$
- Outliers at $121$ and $177$ (shown as open dots)
**Final answers:**
- a) 5-number summary: $\{32, 49.5, 64, 78, 177\}$ km
- b) IQR: $28.5$ km
- c) Lower fence: $6.75$ km, Upper fence: $120.75$ km
- d) Correct modified boxplot is the second one shown in the problem.