Question

a) find the 5 - number summary: {48, 67.5, 128.5, 198.5, 322} b) compute the iqr. 130.75 c) what is the lower fence for this data set? 128.625 enter an integer or decimal number more.. what is the upper fence for this data set? d) which of the following is the correct boxplot for the data? 39 65 137.5 168 248 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 lengths of rivers (in km) 7 64 98 164 248 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 lengths of rivers (in km)

Explanation:

Step1: Recall 5 - number summary

The 5 - number summary consists of minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum. Given: Minimum = 48, $Q_1=67.5$, $Q_2 = 128.5$, $Q_3=198.5$, Maximum = 322.

Step2: Compute IQR

The inter - quartile range (IQR) is $IQR=Q_3 - Q_1$. Substitute $Q_1 = 67.5$ and $Q_3=198.5$ into the formula: $IQR=198.5 - 67.5=130.75$.

Step3: Calculate lower fence

The lower fence is calculated as $Q_1-1.5\times IQR$. Substitute $Q_1 = 67.5$ and $IQR = 130.75$: $67.5-1.5\times130.75=67.5 - 196.125=- 128.625$. There seems to be an error in the provided value of 128.625 for the lower - fence calculation in the question. The correct formula application gives $67.5-1.5\times130.75 = 67.5-196.125=-128.625$. But if we assume the correct values from the 5 - number summary and $IQR = 130.75$, the lower fence is $67.5-1.5\times130.75=-128.625$.

Step4: Calculate upper fence

The upper fence is $Q_3 + 1.5\times IQR$. Substitute $Q_3=198.5$ and $IQR = 130.75$: $198.5+1.5\times130.75=198.5 + 196.125=394.625$.

Step5: Analyze box - plot

A box - plot has the following features: The left - most whisker extends to the minimum value (48), the left side of the box is at $Q_1 = 67.5$, the line inside the box is at the median $Q_2=128.5$, the right side of the box is at $Q_3 = 198.5$, and the right - most whisker extends to the maximum value (322). We would need to visually check the box - plots against these values to determine the correct one.

Answer:

a) Minimum: 48, $Q_1$: 67.5, Median: 128.5, $Q_3$: 198.5, Maximum: 322
b) 130.75
c) - 128.625
d) (No answer provided as box - plots are not fully described in text to choose from)