Questions: The lengths (in kilometers) of a random sample of 22 rivers on the South island of New Zealand that flow to the Tasman Sea are listed in table below. Lengths of Rivers (in km): 32, 72, 76, 64, 64, 80, 37, 51, 40, 80, 177, 35, 56, 72, 56, 64, 68, 56, 48, 108, 32, 121. For the data shown above, find the following. Do not round any of your answers. The program here will take either answer. a) Find the 5 -number summary: b) Compute the IQR. c) What is the lower fence for this data set?

Solution Steps

The 5-number summary consists of the minimum, first quartile (\(Q_1\)), median, third quartile (\(Q_3\)), and maximum values of the dataset. For the lengths of rivers (in km), we have:

• Minimum: \(32\) km
• First Quartile: \(Q_1 = 48.75\) km
• Median: \(64.0\) km
• Third Quartile: \(Q_3 = 75.0\) km
• Maximum: \(177\) km

Thus, the 5-number summary is: \[ \text{5-number summary} = (32, 48.75, 64.0, 75.0, 177) \]

The Interquartile Range (IQR) is calculated as: \[ \text{IQR} = Q_3 - Q_1 = 75.0 - 48.75 = 26.25 \text{ km} \]

The lower fence is calculated using the formula: \[ \text{Lower Fence} = Q_1 - 1.5 \times \text{IQR} \] Substituting the values: \[ \text{Lower Fence} = 48.75 - 1.5 \times 26.25 = 48.75 - 39.375 = 9.375 \text{ km} \]

Final Answer