To solve for the quartiles [tex]\( Q_1, Q_2, \)[/tex] and [tex]\( Q_3 \)[/tex] of the given dataset, we need to follow a series of steps to understand the distribution of the data.
Given:
[tex]\[
\{ 87, 115, 97, 79, 81, 92, 94, 88, 94, 80, 108, 113, 101, 95, 85, 92, 77, 98, 91 \}
\][/tex]
Step 1: Sort the data in ascending order.
The sorted dataset is:
[tex]\[
\{ 77, 79, 80, 81, 85, 87, 88, 91, 92, 92, 94, 94, 95, 97, 98, 101, 108, 113, 115 \}
\][/tex]
Step 2: Determine [tex]\(Q_2\)[/tex] (the median).
[tex]\( Q_2 \)[/tex] can be found by locating the middle value of the sorted dataset. Since there are 19 data points, the 10th value (when indexed starting from 1) is the median.
Thus, [tex]\( Q_2 \)[/tex] is:
[tex]\[
Q_2 = 92
\][/tex]
Step 3: Determine [tex]\(Q_1\)[/tex] (the first quartile).
To find [tex]\( Q_1 \)[/tex], we need to find the median of the first half of the dataset (i.e., the lower 50%). For this dataset, that would be the first 9 values:
[tex]\[
\{ 77, 79, 80, 81, 85, 87, 88, 91, 92 \}
\][/tex]
The median of these 9 numbers is the 5th value (when indexed starting from 1).
Thus, [tex]\( Q_1 \)[/tex] is:
[tex]\[
Q_1 = 86.0
\][/tex]
Step 4: Determine [tex]\(Q_3\)[/tex] (the third quartile).
To find [tex]\( Q_3 \)[/tex], we need to find the median of the second half of the dataset (i.e., the upper 50%). For this dataset, that would be the last 9 values:
[tex]\[
\{ 92, 94, 94, 95, 97, 98, 101, 108, 113, 115 \}
\][/tex]
The median of these 10 numbers is the average of the 5th and 6th values (when indexed starting from 1).
Thus, [tex]\( Q_3 \)[/tex] is:
[tex]\[
Q_3 = 97.5
\][/tex]
So, the values are:
[tex]\[
Q_2 = 92.0
\][/tex]
[tex]\[
Q_1 = 86.0
\][/tex]
[tex]\[
Q_3 = 97.5
\][/tex]
Putting it all together:
a) [tex]\( Q_2 = 92.0 \)[/tex]
b) [tex]\( Q_1 = 86.0 \)[/tex]
c) [tex]\( Q_3 = 97.5 \)[/tex]
