The lower quartile (Q1) for the homework completion time is rounded up to 5 minutes, and the upper quartile (Q3) is rounded up to 25 minutes. The estimated interquartile range (IQR) is 20 minutes.
Given the data for the histogram, we can calculate the cumulative frequencies as follows:
Time (minutes) Frequency density Cumulative frequency
0-5 1.4 1.4
5-15 0.8 2.2
15-25 1.4 3.6
25-40 1 4.6
The lower quartile (Q1) is the time at which 25% of the students have completed their homework. In this case, the cumulative frequency of 3.6 corresponds to 75% of the students, so Q1 is the time at which the cumulative frequency is 0.75 of 4.6, or 3.45. Since the time intervals are 5 minutes, we round Q1 up to the nearest interval, which is 5 minutes.
The upper quartile (Q3) is the time at which 75% of the students have completed their homework. In this case, the cumulative frequency of 4.6 corresponds to 100% of the students, so Q3 is the time at which the cumulative frequency is 0.25 of 4.6, or 1.15. Since the time intervals are 5 minutes, we round Q3 up to the nearest interval, which is 25 minutes.
Therefore, the estimated IQR is Q3 - Q1 = 25 minutes - 5 minutes = 20 minutes.