Question

the test scores of students in a biology class are listed below.
16 17 19 36 46 50 51 56 59 60
60 61 61 62 66 69 70 70 70 70
70 70 71 76 76 80 81 83 98 100
(a) find the following
$q_1$ 56
$q_2$ 67.5
$q_3$
iqr
lower outlier boundary
upper outlier boundary
list the outliers separated by commas

Explanation:

Step1: Find total data points

There are 30 data points, $n=30$.

Step2: Split data for Q3

Upper half: values after $Q_2$ (15th-30th): 70,70,70,70,70,71,76,76,80,81,83,98,100

Step3: Calculate Q3 position

$Q_3$ = median of upper half. For 15 values, 8th value: $Q_3=76$

Step4: Calculate IQR

$IQR = Q_3 - Q_1$
$IQR = 76 - 56 = 20$

Step5: Find lower outlier boundary

Lower boundary = $Q_1 - 1.5\times IQR$
$56 - 1.5\times20 = 56 - 30 = 26$

Step6: Find upper outlier boundary

Upper boundary = $Q_3 + 1.5\times IQR$
$76 + 1.5\times20 = 76 + 30 = 106$

Step7: Identify outliers

Values <26 or >106: 16,17,19

Answer:

$Q_3$: 76
IQR: 20
Lower Outlier Boundary: 26
Upper Outlier Boundary: 106
Outliers: 16, 17, 19