Question

the 10 members of the swim team completed the following numbers of laps at todays practice. 76, 77, 77, 79, 82, 83, 85, 89, 91, 156 complete the parts below to identify any outliers. (a) let ( q_1 ) be the lower quartile and ( q_3 ) be the upper quartile of the data set. find ( q_1 ) and ( q_3 ) for the data set. ( q_1=square ) ( q_3=square ) (b) find the interquartile range (iqr) of the data set. ( mathrm{iqr}=square ) (c) calculate a lower boundary using ( q_1 - 1.5cdotmathrm{iqr} ). calculate an upper boundary using ( q_3 + 1.5cdotmathrm{iqr} ). (note that ( 1.5cdotmathrm{iqr} ) means 1.5 times the iqr.) lower boundary: ( square ) upper boundary: ( square )

Explanation:

Step1: Find the median of the lower half (Q1)

The data set is \(76,77,77,79,82,83,85,89,91,156\). The lower half is \(76,77,77,79,82\). The median of this lower half is \(77\).

Step2: Find the median of the upper half (Q3)

The upper half is \(83,85,89,91,156\). The median of this upper half is \(89\).

Step3: Calculate the inter - quartile range (IQR)

\(IQR=Q_3 - Q_1\). Substituting \(Q_1 = 77\) and \(Q_3 = 89\), we get \(IQR=89 - 77=12\).

Step4: Calculate the lower boundary

Lower boundary \(=Q_1-1.5\times IQR\). Substituting \(Q_1 = 77\) and \(IQR = 12\), we have \(77-1.5\times12=77 - 18 = 59\).

Step5: Calculate the upper boundary

Upper boundary \(=Q_3 + 1.5\times IQR\). Substituting \(Q_3 = 89\) and \(IQR = 12\), we get \(89+1.5\times12=89 + 18=107\).

Answer:

a. \(Q_1 = 77\), \(Q_3 = 89\)
b. \(IQR = 12\)
c. Lower boundary: \(59\), Upper boundary: \(107\)