Question

the following are the ages of 12 physics teachers in a school district.
28, 31, 33, 35, 41, 41, 43, 44, 45, 45, 50, 54
notice that the ages 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

Minimum is the smallest value: \(28\).
Maximum is the largest value: \(54\).

Step2: Find Median (Second Quartile)

For \(n = 12\) (even), median is average of \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th terms.
\(\frac{12}{2}=6\)-th term: \(41\); \(7\)-th term: \(43\).
Median \(=\frac{41 + 43}{2}=42\).

Step3: Find Lower Quartile (\(Q_1\))

Lower half: first \(6\) terms: \(28, 31, 33, 35, 41, 41\).
For \(n = 6\) (even), \(Q_1\) is average of \(3\)-rd and \(4\)-th terms.
\(3\)-rd: \(33\); \(4\)-th: \(35\).
\(Q_1=\frac{33 + 35}{2}=34\).

Step4: Find Upper Quartile (\(Q_3\))

Upper half: last \(6\) terms: \(43, 44, 45, 45, 50, 54\).
For \(n = 6\) (even), \(Q_3\) is average of \(3\)-rd and \(4\)-th terms.
\(3\)-rd: \(45\); \(4\)-th: \(45\).
\(Q_3=\frac{45 + 45}{2}=45\).

Step5: Find Interquartile Range (IQR)

IQR \(= Q_3 - Q_1 = 45 - 34 = 11\).

Answer:

Five - number summary:
Minimum: \(28\)
Lower quartile: \(34\)
Median: \(42\)
Upper quartile: \(45\)
Maximum: \(54\)
Interquartile range: \(11\)