Question

the following are the distances (in miles) to the nearest airport for 12 families.
5, 13, 14, 14, 19, 20, 26, 36, 36, 42, 43, 46
notice that the numbers are ordered from least to greatest.
give the five - number summary and the interquartile range for the data set.
five - number summary
minimum:
lower quartile:
median:
upper quartile:
maximum:
interquartile
range:

Explanation:

Step1: Find Minimum and Maximum

The data is ordered: \(5, 13, 14, 14, 19, 20, 26, 36, 36, 42, 43, 46\). Minimum is the first value (\(5\)), Maximum is the last (\(46\)).

Step2: Find Median (Q2)

For \(n = 12\) (even), median is average of \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th values. \(\frac{12}{2}=6\)-th (\(20\)) and \(7\)-th (\(26\)) values. Median \(=\frac{20 + 26}{2}=23\).

Step3: Find Lower Quartile (Q1)

Lower half: \(5, 13, 14, 14, 19, 20\) (first 6 values). For \(n = 6\) (even), Q1 is average of 3rd and 4th values: \(\frac{14 + 14}{2}=14\).

Step4: Find Upper Quartile (Q3)

Upper half: \(26, 36, 36, 42, 43, 46\) (last 6 values). Q3 is average of 3rd and 4th values: \(\frac{36 + 42}{2}=39\).

Step5: Find Interquartile Range (IQR)

IQR \(= Q3 - Q1 = 39 - 14 = 25\).

Answer:

Five - number summary:
Minimum: \(5\)
Lower quartile: \(14\)
Median: \(23\)
Upper quartile: \(39\)
Maximum: \(46\)
Interquartile range: \(25\)