which data sets have outliers? check all that apply.\n□ 14,21,24,25,27,32,35\n□ 15,30,35,41,44,50,78\n□…

which data sets have outliers? check all that apply.\n□ 14,21,24,25,27,32,35\n□ 15,30,35,41,44,50,78\n□ 16,32,38,39,41,42,58\n□ 17,23,28,31,39,45,75\n□ 18,30,34,38,43,45,68

which data sets have outliers? check all that apply.\n□ 14,21,24,25,27,32,35\n□ 15,30,35,41,44,50,78\n□ 16,32,38,39,41,42,58\n□ 17,23,28,31,39,45,75\n□ 18,30,34,38,43,45,68

Answer

Answer:

  1. 15, 30, 35, 41, 44, 50, 78
  2. 17, 23, 28, 31, 39, 45, 75

Explanation:

Step1: Calculate quartiles

For a data - set (x_1,x_2,\cdots,x_n) in ascending order, first find the median (Q_2). If (n) is odd, (Q_2=x_{\frac{n + 1}{2}}); if (n) is even, (Q_2=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}). Then find (Q_1) (median of the lower half) and (Q_3) (median of the upper half).

Step2: Calculate the inter - quartile range (IQR)

(IQR = Q_3−Q_1).

Step3: Determine the outlier boundaries

Lower boundary (=Q_1 - 1.5\times IQR), Upper boundary (=Q_3 + 1.5\times IQR). Values outside these boundaries are outliers. For the data - set (15,30,35,41,44,50,78):

  • (n = 7), (Q_2=41), lower half is (15,30,35), (Q_1 = 30), upper half is (44,50,78), (Q_3 = 50).
  • (IQR=Q_3 - Q_1=50 - 30 = 20).
  • Lower boundary (=30-1.5\times20=0), Upper boundary (=50 + 1.5\times20=80). Since all values are within these boundaries, no outliers. For the data - set (17,23,28,31,39,45,75):
  • (n = 7), (Q_2 = 31), lower half is (17,23,28), (Q_1=23), upper half is (39,45,75), (Q_3 = 45).
  • (IQR=45 - 23=22).
  • Lower boundary (=23-1.5\times22=23 - 33=- 10), Upper boundary (=45+1.5\times22=45 + 33 = 78). Since (75) is close to the upper - boundary and there are no extreme values, no outliers. For the data - set (16,32,38,39,41,42,58):
  • (n = 7), (Q_2 = 39), lower half is (16,32,38), (Q_1 = 32), upper half is (41,42,58), (Q_3 = 42).
  • (IQR=42 - 32 = 10).
  • Lower boundary (=32-1.5\times10=17), Upper boundary (=42+1.5\times10=57). Since (58>57), this data - set has an outlier. For the data - set (18,30,34,38,43,45,68):
  • (n = 7), (Q_2 = 38), lower half is (18,30,34), (Q_1 = 30), upper half is (43,45,68), (Q_3 = 45).
  • (IQR=45 - 30 = 15).
  • Lower boundary (=30-1.5\times15=30 - 22.5 = 7.5), Upper boundary (=45+1.5\times15=45 + 22.5 = 67.5). Since (68>67.5), this data - set has an outlier.