Question

the five - number summary for the lengths of songs, in minutes, performed by a popular music group is given below.
minimum: 1.8 q₁: 3.3 median: 3.8 q₃: 4.5 maximum: 9.2
the shortest song performed by the group is 1.8 minutes while the longest three songs are 5.7, 7.5, and 9.2
minutes. are any of these song lengths outliers?
no, none of the lengths are outliers.
yes, 7.5 and 9.2 are outliers.
yes, 1.8 and 9.2 are outliers.
yes, 5.7, 7.5, and 9.2 are outliers.

Explanation:

Step1: Calculate the inter - quartile range (IQR)

The inter - quartile range \(IQR = Q_{3}-Q_{1}\). Given \(Q_{1}=3.3\) and \(Q_{3}=4.5\), so \(IQR=4.5 - 3.3=1.2\)

Step2: Calculate the lower and upper bounds

The lower bound \(=Q_{1}-1.5\times IQR\). Substitute \(Q_{1}=3.3\) and \(IQR = 1.2\) into the formula: \(3.3-1.5\times1.2=3.3 - 1.8 = 1.5\)
The upper bound \(=Q_{3}+1.5\times IQR\). Substitute \(Q_{3}=4.5\) and \(IQR = 1.2\) into the formula: \(4.5+1.5\times1.2=4.5+1.8 = 6.3\)

Step3: Check for outliers

For the lower bound: Since \(1.8>1.5\), the minimum value \(1.8\) is not an outlier.
For the values \(5.7\), \(7.5\) and \(9.2\): Since \(5.7<6.3\), \(5.7\) is not an outlier. Since \(7.5>6.3\) and \(9.2>6.3\), \(7.5\) and \(9.2\) are outliers.

Answer:

Yes, 7.5 and 9.2 are outliers.