Question

the lengths (in kilometers) of a random sample of 18 rivers on the south island of new zealand the pacific ocean are listed in table below. lengths of rivers (in km) 169 93 322 209 160 208 145 121 97 209 64 48 136 140 90 66 56 161 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? what is the upper fence for this data set?

Explanation:

Step1: Sort the data

48, 56, 64, 66, 90, 93, 97, 121, 136, 140, 145, 160, 161, 169, 208, 209, 209, 322

Step2: Find the minimum

The minimum value is 48.

Step3: Find Q1 (first - quartile)

There are $n = 18$ data points. The position of Q1 is $\frac{n+1}{4}=\frac{18 + 1}{4}=4.75$.
Q1 is $0.25\times(66 - 64)+64=64.5$

Step4: Find the median (Q2)

The position of the median is $\frac{n + 1}{2}=\frac{18+1}{2}=9.5$.
Median is $\frac{136+140}{2}=138$

Step5: Find Q3 (third - quartile)

The position of Q3 is $\frac{3(n + 1)}{4}=\frac{3\times(18 + 1)}{4}=14.25$.
Q3 is $0.25\times(169 - 161)+161=163$

Step6: Find the maximum

The maximum value is 322.

Step7: Compute the IQR

IQR = Q3 - Q1 = $163-64.5 = 98.5$

Step8: Compute the lower fence

Lower fence=Q1 - 1.5×IQR=$64.5-1.5\times98.5=64.5 - 147.75=- 83.25$

Step9: Compute the upper fence

Upper fence=Q3 + 1.5×IQR=$163+1.5\times98.5=163 + 147.75=310.75$

Answer:

a) [48, 64.5, 138, 163, 322]
b) 98.5
c) -83.25
d) 310.75