Question

this list shows the tips marcus received one night. $2.50, $3.00, $3.25, $3.25, $3.50 later, he received a tip of $10. how did the addition of this outlier affect the data set? a the median and interquartile range increased. b the median and interquartile range decreased. c the median decreased and the interquartile range remained the same. d the median remained the same and the interquartile range decreased.

Explanation:

Step1: Arrange the original data in order

Original data: \( \$1.20,\$1.25,\$2.50,\$3.00,\$3.50 \). The median is the middle value. For \( n = 5 \) (odd number of data points), the median \( M_1=\$2.50 \).
The first quartile \( Q_1 \) (median of the lower - half: \( \$1.20,\$1.25 \)) is \( \$1.225 \). The third quartile \( Q_3 \) (median of the upper - half: \( \$3.00,\$3.50 \)) is \( \$3.25 \). The inter - quartile range \( IQR_1=Q_3 - Q_1=\$3.25-\$1.225=\$2.025 \).

Step2: Add the outlier and re - arrange the data

New data: \( \$1.20,\$1.25,\$2.50,\$3.00,\$3.50,\$10 \). For \( n = 6 \) (even number of data points), the median \( M_2=\frac{2.50 + 3.00}{2}=\$2.75 \).
The first quartile \( Q_1 \) (median of the lower - half: \( \$1.20,\$1.25,\$2.50 \)) is \( \$1.25 \). The third quartile \( Q_3 \) (median of the upper - half: \( \$3.00,\$3.50,\$10 \)) is \( \$3.50 \). The inter - quartile range \( IQR_2=Q_3 - Q_1=\$3.50-\$1.25=\$2.25 \).

Since \( M_2>M_1 \) (median increased) and \( IQR_2>IQR_1 \) (inter - quartile range increased).

Answer:

A. The median and interquartile range increased.