the lengths (in kilometers) of a random sample of 18 rivers on the south island of new zealand in the…
the lengths (in kilometers) of a random sample of 18 rivers on the south island of new zealand in the pacific ocean are listed in table below.\nlengths of rivers (in km)\n169\n93\n322\n209\n160\n208\n145\n121\n97\n209\n64\n48\n136\n140\n90\n66\n56\n161\nfor the data shown above, find the following.\ndo not round any of your answers.\na) find the 5 - number summary:\n48, 84, 138, 163, 322\nb) compute the iqr.\n79\nc) what is the lower fence for this data set?\n-34.5\nwhat is the upper fence for this data set?\n281.5
Answer
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 and maximum
The minimum value is $48$ and the maximum value is $322$.
Step3: Find the median (Q2)
Since $n = 18$ (even), the median is the average of the $\frac{n}{2}=9$ - th and $(\frac{n}{2}+ 1)=10$ - th ordered values. So, $Q2=\frac{136 + 140}{2}=138$.
Step4: Find Q1
The lower - half of the data is $48,56,64,66,90,93,97,121,136$. Since $n_1 = 9$ (odd), $Q1$ is the $(\frac{9 + 1}{2})=5$ - th value, so $Q1 = 90$.
Step5: Find Q3
The upper - half of the data is $140,145,160,161,169,208,209,209,322$. Since $n_2=9$ (odd), $Q3$ is the $(\frac{9 + 1}{2}) = 5$ - th value, so $Q3=169$.
Step6: Compute the IQR
$IQR=Q3 - Q1=169 - 90 = 79$.
Step7: Compute the lower fence
Lower fence $=Q1-1.5\times IQR=90-1.5\times79=90 - 118.5=-28.5$.
Step8: Compute the upper fence
Upper fence $=Q3 + 1.5\times IQR=169+1.5\times79=169 + 118.5=287.5$.
Answer:
a) Minimum: $48$, Q1: $90$, Median: $138$, Q3: $169$, Maximum: $322$ b) $79$ c) $-28.5$ d) $287.5$