Question

if done correctly, your frequency table should have appeared as follows:
lower\tupper\tfreq.\trel. freq.
0.5\t2.1\t2\t0.050
2.2\t3.8\t4\t0.100
3.9\t5.5\t5\t0.125
5.6\t7.2\t15\t0.375
7.3\t8.9\t5\t0.125
9.0\t10.6\t5\t0.125
10.7\t12.3\t2\t0.050
12.4\t14.0\t2\t0.050
use this information as you continue.
enter the class boundaries below.
#1: 0.45 ✔\t#2: 2.15 ✔\t#3: 2.15 ×
#4: 3.85 ×\t#5: 3.85 ×\t#6: 5.55 ×
#7: 5.55 ×\t#8: 7.25 ×\t#9: 7.25 ×

Explanation:

To find class boundaries, we first determine the gap between the upper limit of one class and the lower limit of the next. The decimal precision here is one decimal place, so the gap between classes (e.g., between 2.1 and 2.2) is \(2.2 - 2.1 = 0.1\). The half - gap is \(\frac{0.1}{2}=0.05\).

Step 1: First class boundary

The first lower class boundary: For the first class with lower limit 0.5, we subtract the half - gap. So, \(0.5−0.05 = 0.45\) (which is correct as given).
The first upper class boundary: For the first class with upper limit 2.1, we add the half - gap. So, \(2.1 + 0.05=2.15\) (which is correct as given).

Step 2: Second class boundary (lower)

The second class has a lower limit of 2.2. The lower class boundary of the second class should be equal to the upper class boundary of the first class. So, it is 2.15 (but the original #3 was wrong, it should be 2.15? Wait, no, let's re - examine. Wait, the classes are:
1. \(0.5 - 2.1\)
2. \(2.2 - 3.8\)
3. \(3.9 - 5.5\)
4. \(5.6 - 7.2\)
5. \(7.3 - 8.9\)
6. \(9.0 - 10.6\)
7. \(10.7 - 12.3\)
8. \(12.4 - 14.0\)

To find the class boundaries, we use the formula:

For a class with lower limit \(L\) and upper limit \(U\), the lower class boundary \(= L - 0.05\) (since the measurement is to one decimal place, the unit of measurement is \(0.1\), so half - unit is \(0.05\)) and the upper class boundary \(= U+ 0.05\)

Wait, no, actually, when the classes are consecutive (with a gap of \(0.1\) between \(2.1\) and \(2.2\)), the class boundaries are calculated as follows:

The boundary between the first class (\(0.5 - 2.1\)) and the second class (\(2.2 - 3.8\)) is \(\frac{2.1 + 2.2}{2}=\frac{4.3}{2}=2.15\)

The boundary between the second class (\(2.2 - 3.8\)) and the third class (\(3.9 - 5.5\)) is \(\frac{3.8+3.9}{2}=\frac{7.7}{2} = 3.85\)

The boundary between the third class (\(3.9 - 5.5\)) and the fourth class (\(5.6 - 7.2\)) is \(\frac{5.5 + 5.6}{2}=\frac{11.1}{2}=5.55\)

The boundary between the fourth class (\(5.6 - 7.2\)) and the fifth class (\(7.3 - 8.9\)) is \(\frac{7.2+7.3}{2}=\frac{14.5}{2}=7.25\)

The boundary between the fifth class (\(7.3 - 8.9\)) and the sixth class (\(9.0 - 10.6\)) is \(\frac{8.9 + 9.0}{2}=\frac{17.9}{2}=8.95\)

The boundary between the sixth class (\(9.0 - 10.6\)) and the seventh class (\(10.7 - 12.3\)) is \(\frac{10.6+10.7}{2}=\frac{21.3}{2}=10.65\)

The boundary between the seventh class (\(10.7 - 12.3\)) and the eighth class (\(12.4 - 14.0\)) is \(\frac{12.3+12.4}{2}=\frac{24.7}{2}=12.35\)

The upper boundary of the last class (\(12.4 - 14.0\)) is \(14.0 + 0.05=14.05\)

Now, let's list all class boundaries:

1. Lower boundary of first class: \(0.5 - 0.05 = 0.45\) (correct)
2. Upper boundary of first class (and lower boundary of second class): \(\frac{2.1 + 2.2}{2}=2.15\) (correct)
3. Upper boundary of second class (and lower boundary of third class): \(\frac{3.8+3.9}{2}=3.85\) (so #3 should be 3.85)
4. Upper boundary of third class (and lower boundary of fourth class): \(\frac{5.5 + 5.6}{2}=5.55\) (so #4 should be 5.55)
5. Upper boundary of fourth class (and lower boundary of fifth class): \(\frac{7.2+7.3}{2}=7.25\) (so #5 should be 7.25)
6. Upper boundary of fifth class (and lower boundary of sixth class): \(\frac{8.9 + 9.0}{2}=8.95\) (so #6 should be 8.95)
7. Upper boundary of sixth class (and lower boundary of seventh class): \(\frac{10.6+10.7}{2}=10.65\) (so #7 should be 10.65)
8. Upper boundary of seventh class (and lower boundary of eighth class): \(\frac{12.3+12.4}{2}=12.35\) (so #8 should be 12.35)
9. Upper boundary of eighth class: \(14.0+0.05 = 14.05\) (so #9 should be 14.05)

Answer:

The correct class boundaries are:
#1: \(0.45\)
#2: \(2.15\)
#3: \(3.85\)
#4: \(5.55\)
#5: \(7.25\)
#6: \(8.95\)
#7: \(10.65\)
#8: \(12.35\)
#9: \(14.05\)