Question

carlos records the number of students in a variety of classes at his school. the results are shown below in order from least to greatest.
20, 23, 23, 24, 26, 28, 30, 31
suppose that the class with 30 students changes to 26 students. answer the following.
(a) what happens to the interquartile range (iqr)? ∘ the iqr increases by \\(\square\\) students. ∘ the iqr decreases by \\(\square\\) students. ∘ the iqr stays the same.
(b) what happens to the range? ∘ the range increases by \\(\square\\) students

Explanation:

Part (a)

Step 1: Find original quartiles

Original data: \(20, 23, 23, 24, 26, 28, 30, 31\) (n = 8)
Median (Q2) is average of 4th and 5th terms: \(\frac{24 + 26}{2}=25\)
Lower half: \(20, 23, 23, 24\) (n = 4), Q1 (median of lower half): \(\frac{23 + 23}{2}=23\)
Upper half: \(26, 28, 30, 31\) (n = 4), Q3 (median of upper half): \(\frac{28 + 30}{2}=29\)
Original IQR: \(Q3 - Q1 = 29 - 23 = 6\)

Step 2: Find new quartiles (30 → 26)

New data: \(20, 23, 23, 24, 26, 26, 28, 31\) (n = 8)
Median (Q2): \(\frac{24 + 26}{2}=25\) (unchanged)
Lower half: \(20, 23, 23, 24\) (unchanged), Q1: \(23\) (unchanged)
Upper half: \(26, 26, 28, 31\) (n = 4), Q3: \(\frac{26 + 28}{2}=27\)
New IQR: \(27 - 23 = 4\)

Step 3: Compare IQR

Original IQR = 6, New IQR = 4. So IQR decreases by \(6 - 4 = 2\)? Wait, no—wait, recalculate upper half. Wait, new data after change: 20,23,23,24,26,26,28,31. Upper half is positions 5 - 8: 26,26,28,31. Median of upper half: (26 + 28)/2 = 27. Original upper half was 26,28,30,31: median (28 + 30)/2 = 29. So Q3 changes from 29 to 27. Q1 is still 23. So IQR was 29 - 23 = 6, now 27 - 23 = 4. Wait, but wait—wait, the original data: when n=8, lower half is first 4, upper half last 4. After changing 30 to 26, the data is 20,23,23,24,26,26,28,31. So upper half is 26 (5th), 26 (6th), 28 (7th), 31 (8th). So Q3 is (26 + 28)/2 = 27. Original Q3 was (28 + 30)/2 = 29. So IQR decreases by 2? Wait, but let's check again. Wait, maybe I made a mistake. Wait, original data: 20,23,23,24,26,28,30,31. Lower half: 20,23,23,24 (Q1 = 23). Upper half: 26,28,30,31 (Q3 = (28 + 30)/2 = 29). IQR = 29 - 23 = 6. New data: 20,23,23,24,26,26,28,31. Lower half: same, Q1 = 23. Upper half: 26,26,28,31. Q3 = (26 + 28)/2 = 27. IQR = 27 - 23 = 4. So IQR decreases by 2? Wait, but the options say "decreases by [ ] students". Wait, but maybe I messed up the upper half. Wait, no—when n is even, the median is between n/2 and n/2 + 1. Then lower half is first n/2, upper half is last n/2. So for n=8, lower half: 4 terms (1 - 4), upper half: 4 terms (5 - 8). So original upper half: 26 (5), 28 (6), 30 (7), 31 (8). Median of upper half: (28 + 30)/2 = 29. New upper half: 26 (5), 26 (6), 28 (7), 31 (8). Median: (26 + 28)/2 = 27. So IQR was 29 - 23 = 6, now 27 - 23 = 4. So IQR decreases by 2? Wait, but the problem's options have "decreases by [ ]"—but maybe I made a mistake. Wait, no—wait, let's check again. Wait, the original data: 20,23,23,24,26,28,30,31. After changing 30 to 26, the data is 20,23,23,24,26,26,28,31. Now, let's find Q1 and Q3 again. Q1 is the median of the first 4: 20,23,23,24. Median is (23 + 23)/2 = 23. Q3 is the median of the last 4: 26,26,28,31. Median is (26 + 28)/2 = 27. So IQR is 27 - 23 = 4. Original IQR was 29 - 23 = 6. So IQR decreases by 2. Wait, but the options have "decreases by [ ]"—so the answer is "The IQR decreases by 2 students."

Part (b)

Step 1: Find original range

Original data: max = 31, min = 20. Range = \(31 - 20 = 11\)

Step 2: Find new range (30 → 26)

New data: max = 31 (unchanged), min = 20 (unchanged)? Wait, no—wait, new data: 20,23,23,24,26,26,28,31. Max is still 31, min is 20. Wait, no—wait, original max was 31, new data's max is 31. Wait, no—wait, the change is 30 to 26, so the new data's values are 20,23,23,24,26,26,28,31. So max is 31, min is 20. Wait, no—wait, original range: 31 - 20 = 11. New range: 31 - 20 = 11? Wait, no—wait, no, the original data had 30, which was less than 31. Wait, original data: 20,23,23,24,26,28,30,31. Max is 31, min is 20. After changing 30 to 26, the data is 20,23,23,24,26,26,28,31. Max is still 31, min is 20. Wait, that can't be. Wait, no—wait, 30 was the 7th term, changed to 26. So the new data's max is still 31 (8th term), min is 20 (1st term). So range is still 31 - 20 = 11? Wait, no—wait, no, original range is 31 - 20 = 11. New range: 31 - 20 = 11. Wait, that's not right. Wait, no—wait, original data: 20,23,23,24,26,28,30,31. Max is 31, min is 20. After changing 30 to 26, the data is 20,23,23,24,26,26,28,31. Max is 31, min is 20. So range remains 11? Wait, no—wait, maybe I made a mistake. Wait, no, the range is max - min. The max is still 31, min is still 20. So range stays the same? Wait, no—wait, original max was 31, new max is 31. Original min is 20, new min is 20. So range is 31 - 20 = 11 in both cases. Wait, but that seems odd. Wait, let's check again. Original data: 20,23,23,24,26,28,30,31. Max: 31, min: 20. Range: 11. New data: 20,23,23,24,26,26,28,31. Max: 31, min: 20. Range: 11. So range stays the same? Wait, but the problem's options for (b) are cut off, but assuming it's similar: "The range increases by...", "decreases by...", "stays the same". So range stays the same.

Answer:

s:
(a) The IQR decreases by \(\boldsymbol{2}\) students.
(b) The range stays the same.