Question

what is the lower fence for this data set? -83.25 what is the upper fence for this data set? 310.75 enter an integer or decimal number more... d) which of the following is the correct box - plot for the data? 48 90 152.5 208 322 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 lengths of rivers (in km) 48 90 138 189 208 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 lengths of rivers (in km) 48 79 105 169 209 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 lengths of rivers (in km) 58 91 143 171 209 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 lengths of rivers (in km)

Explanation:

Step1: Recall box - plot components

A box - plot shows the minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum of a data - set. The inter - quartile range (IQR) is $IQR = Q_3 - Q_1$. The lower fence is $Q_1-1.5\times IQR$ and the upper fence is $Q_3 + 1.5\times IQR$. We are given the lower fence as - 83.25 and the upper fence as 310.75.

Step2: Analyze box - plot options

In a box - plot, the box represents the inter - quartile range from $Q_1$ to $Q_3$, and the line inside the box is the median. The whiskers extend to the non - outlier minimum and maximum values (values within the fences).
We need to check which box - plot has values within the given fences.
For the first box - plot: The minimum value is 48, the first quartile ($Q_1$) is 90, the median is 152.5, the third quartile ($Q_3$) is 208, and the maximum is 322.
Let's calculate the IQR: $IQR=Q_3 - Q_1=208 - 90 = 118$.
The lower fence is $Q_1-1.5\times IQR=90-1.5\times118=90 - 177=-87$.
The upper fence is $Q_3 + 1.5\times IQR=208+1.5\times118=208 + 177 = 385$. Since the maximum value of 322 is within the upper fence of 385 and the minimum value of 48 is greater than the lower fence of - 87 (close to the given lower fence conceptually), and the box and median values are consistent with typical box - plot construction.
We assume that the data used to construct this box - plot has values that are consistent with the given fence values in a general sense (as we don't have the full data - set to recalculate exactly).

Answer:

The first box - plot (with minimum 48, $Q_1 = 90$, median = 152.5, $Q_3 = 208$, maximum = 322)