Question

the lengths (in kilometers) of a random sample of 20 rivers on the south island of new zealand that flow to the pacific ocean are listed in table below.
lengths of rivers (in km)
138
48
209
56
136
64
140
169
145
66
121
208
322
97
64
80
72
167
209
98
for the data shown above, find the following.
do not round any of your answers.
a) find the 5 - number summary:
{}
b) compute the iqr.
c) what is the lower fence for this data set?

Explanation:

Step1: Sort the data

48, 56, 64, 64, 66, 72, 80, 97, 98, 121, 136, 138, 140, 145, 167, 169, 208, 209, 209, 322

Step2: Find the minimum

The minimum value is 48.

Step3: Find Q1

There are \(n = 20\) data - points. The position of Q1 is \(\frac{n + 1}{4}=\frac{20+1}{4}=5.25\). So, \(Q1=66+(0.25)\times(72 - 66)=67.5\).

Step4: Find the median (Q2)

The position of the median is \(\frac{n + 1}{2}=\frac{20 + 1}{2}=10.5\). So, \(Q2=\frac{121+136}{2}=128.5\).

Step5: Find Q3

The position of Q3 is \(\frac{3(n + 1)}{4}=\frac{3\times(20 + 1)}{4}=15.75\). So, \(Q3=169+(0.75)\times(208 - 169)=198.25\).

Step6: Find the maximum

The maximum value is 322.

Step7: Compute the IQR

\(IQR=Q3 - Q1=198.25-67.5 = 130.75\).

Step8: Find the lower - fence

The lower - fence is \(Q1-1.5\times IQR=67.5-1.5\times130.75=67.5 - 196.125=-128.625\). Since we are dealing with non - negative lengths, in the context of the data, we can consider the lower - bound as 0 (as length cannot be negative). But based on the formula calculation, it is \(-128.625\).

Answer:

a) 48, 67.5, 128.5, 198.25, 322
b) 130.75
c) - 128.625