Question

the time, in minutes, it took each of 11 students to complete a puzzle was recorded and is shown in the following list. 9,17,20,21,27,29,30,31,32,35,58 one of the students who completed the puzzle claimed that there were two outliers in the data set. based on the 1.5 × iqr rule for outliers, is there evidence to support the student’s claim? a yes, there are two outliers. one outlier is 9 minutes and the other outlier is 58 minutes. b no, there is only one outlier at 9 minutes. c no, there is only one outlier at 58 minutes. d no, there are three outliers. one outlier is 9 minutes, one outlier is 35 minutes, and one outlier is 58 minutes. e no, there are no outliers.

Explanation:

Step1: Order the data

First, we order the data set: \( 9, 17, 20, 21, 27, 29, 30, 31, 32, 35, 58 \)

Step2: Find Q1, Q2 (median), Q3

• The median (Q2) of 11 data points is the 6th value: \( Q2 = 29 \)
• The lower half (first 5 values: \( 9, 17, 20, 21, 27 \)) has median \( Q1 = 20 \)
• The upper half (last 5 values: \( 30, 31, 32, 35, 58 \)) has median \( Q3 = 32 \)

Step3: Calculate IQR

IQR (Interquartile Range) is \( Q3 - Q1 = 32 - 20 = 12 \)

Step4: Find lower and upper bounds

• Lower bound: \( Q1 - 1.5 \times IQR = 20 - 1.5 \times 12 = 20 - 18 = 2 \)
• Upper bound: \( Q3 + 1.5 \times IQR = 32 + 1.5 \times 12 = 32 + 18 = 50 \)

Step5: Identify outliers

Check which data points are outside [2, 50]:

• 9 is above 2 (not an outlier)
• 58 is above 50 (outlier)
• Other points (17,20,21,27,29,30,31,32,35) are within bounds. So only 58 is an outlier? Wait, wait, 9: lower bound is 2, 9 > 2, so not outlier. 58 > 50, so outlier. Wait, but let's recheck. Wait, data is 9,17,20,21,27,29,30,31,32,35,58. Q1 is 20 (since first 5: 9,17,20,21,27; median is 20). Q3 is 32 (last 5: 30,31,32,35,58; median is 32). IQR = 12. Lower bound: 20 - 18 = 2. Upper bound: 32 + 18 = 50. So 58 is above 50 (outlier). 9 is above 2 (not outlier). Wait, but option C says one outlier at 58. Wait, but let's check again. Wait, maybe I made a mistake in Q1 and Q3. Wait, for 11 data points, positions: 1:9, 2:17, 3:20, 4:21, 5:27, 6:29 (Q2), 7:30, 8:31, 9:32, 10:35, 11:58. So lower half is positions 1 - 5: 9,17,20,21,27. Median of lower half (Q1) is position 3: 20. Upper half is positions 7 - 11: 30,31,32,35,58. Median of upper half (Q3) is position 9: 32. So IQR = 32 - 20 = 12. Lower bound: 20 - 1.512 = 2. Upper bound: 32 + 1.512 = 50. So data points: 9 (2 < 9 < 50: not outlier), 17 (same), 20,21,27,29,30,31,32,35 (all <50), 58 (58 >50: outlier). So only 58 is outlier. So option C: No, there is only one outlier at 58 minutes. Wait, but wait, initial data: 9,17,20,21,27,29,30,31,32,35,58. Wait, maybe I messed up Q1. Wait, lower half is first 5 numbers: 9,17,20,21,27. The median is the 3rd number, which is 20. Correct. Upper half is last 5: 30,31,32,35,58. Median is 3rd number, 32. Correct. So IQR 12. Lower bound 2, upper 50. So 58 is outlier, 9 is not. So only one outlier at 58. So option C. Wait, but let's check the options again. Option C: No, there is only one outlier at 58 minutes. So that's correct.

Answer:

C. No, there is only one outlier at 58 minutes.