Question

create a box plot of devoirys transportation costs for two weeks. her expenses were: {13, 18, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 18, 19}. min: lower quartile: median: upper quartile: max: use the box plot of transportation costs to answer the questions. were there any clear outliers in devoirys transportation costs? the interquartile range (iqr) of a data set is the difference between the upper and lower quartiles. what is the iqr for devoirys transportation costs?

Explanation:

Step1: Sort the data

First, sort the data set $\{13, 18, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 18, 19\}$ in ascending - order: $\{13, 13, 14, 15, 16, 17, 18, 18, 18, 19, 19, 20, 21, 22\}$.

Step2: Find the minimum and maximum

The minimum value (Min) of the sorted data set is $13$, and the maximum value (Max) is $22$.

Step3: Calculate the median

Since there are $n = 14$ data points, the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values. $\frac{n}{2}=7$ and $\frac{n}{2}+1 = 8$. The median is $\frac{18 + 18}{2}=18$.

Step4: Calculate the lower quartile

The lower half of the data set is $\{13, 13, 14, 15, 16, 17, 18\}$. Since there are $n_1=7$ data points in the lower - half, the lower quartile ($Q_1$) is the $(\frac{n_1 + 1}{2})$ - th value. $\frac{7+1}{2}=4$, so $Q_1 = 15$.

Step5: Calculate the upper quartile

The upper half of the data set is $\{18, 18, 19, 19, 20, 21, 22\}$. Since there are $n_2 = 7$ data points in the upper - half, the upper quartile ($Q_3$) is the $(\frac{n_2+1}{2})$ - th value. $\frac{7 + 1}{2}=4$, so $Q_3=19$.

Step6: Check for outliers

The inter - quartile range $IQR=Q_3 - Q_1=19 - 15 = 4$. The lower fence for outliers is $Q_1-1.5\times IQR=15-1.5\times4=15 - 6 = 9$. The upper fence for outliers is $Q_3 + 1.5\times IQR=19+1.5\times4=19 + 6 = 25$. Since all data points are within the range $[9,25]$, there are no outliers.

Answer:

Min: $13$
Lower Quartile: $15$
Median: $18$
Upper Quartile: $19$
Max: $22$
Were there any clear outliers in Devoiry’s transportation costs? No
What is the IQR for Devoiry’s transportation costs? $4$