Question

below are ages (in years, rounded to the nearest tenth) of 30 randomly selected children at a summer camp in july, 2018.
7.2 7.5 7.7 7.8 8.0 8.3
8.3 8.8 8.9 9.0 9.6 9.7
9.9 9.9 10.0 10.2 10.3 10.3
10.4 11.1 11.1 11.1 11.2 11.4
11.7 11.8 11.9 13.1 13.3 13.3
construct a frequency distribution table for this information using 5 classes. enter relative frequencies as well, rounded to three places after the decimal.
the first lower class limit should be the minimum value. the class width should be rounded up to the next tenths place.
lower class limits upper class limits frequency rel. freq.

Explanation:

Step1: Find min, max, and class width

Min age = 7.2, Max age = 13.3.
Range = \( 13.3 - 7.2 = 6.1 \).
Class width = \( \lceil \frac{6.1}{5}
ceil = \lceil 1.22
ceil = 1.3 \) (rounded up to tenths).

Step2: Define class limits

1st class: Lower = 7.2, Upper = \( 7.2 + 1.3 - 0.1 = 8.4 \) (subtract 0.1 for inclusive upper limit).
2nd class: Lower = 8.5, Upper = \( 8.5 + 1.3 - 0.1 = 9.7 \).
3rd class: Lower = 9.8, Upper = \( 9.8 + 1.3 - 0.1 = 11.0 \).
4th class: Lower = 11.1, Upper = \( 11.1 + 1.3 - 0.1 = 12.3 \).
5th class: Lower = 12.4, Upper = \( 12.4 + 1.3 - 0.1 = 13.6 \) (covers max 13.3).

Step3: Count frequencies

• 7.2–8.4: 7.2,7.5,7.7,7.8,8.0,8.3,8.3,8.8,8.9 → Wait, no: 7.2,7.5,7.7,7.8,8.0,8.3,8.3 (7 values? Wait, original data:

Row1: 7.2,7.5,7.7,7.8,8.0,8.3 (6)
Row2: 8.3,8.8,8.9,9.0,9.6,9.7 (8.3,8.8,8.9 ≤8.4? 8.3 yes, 8.8 yes, 8.9 yes; 9.0 no. So row1: 6, row2: 3 (8.3,8.8,8.9) → total 6+3=9? Wait, let's list all data:
7.2,7.5,7.7,7.8,8.0,8.3,
8.3,8.8,8.9,9.0,9.6,9.7,
9.9,9.9,10.0,10.2,10.3,10.3,
10.4,11.1,11.1,11.1,11.2,11.4,
11.7,11.8,11.9,13.1,13.3,13.3

Now, class 7.2–8.4: values ≤8.4.
Check each:
7.2,7.5,7.7,7.8,8.0,8.3,8.3,8.8,8.9 → 9 values (8.8,8.9 ≤8.4? 8.8 is 8.8 ≤8.4? No! 8.4 is upper limit. So 8.8 >8.4, 8.9>8.4. Oops, mistake. Class width is 1.3, so 7.2 to 7.2+1.3=8.5? Wait, maybe I messed up class limits. Let's recalculate class width correctly.

Range = 13.3 -7.2 =6.1. Number of classes=5. So class width = 6.1/5=1.22, round up to 1.3 (tenths place). So each class is 1.3 wide.

First class: 7.2 – 7.2+1.3=8.5 (exclusive upper? Or inclusive? The problem says "upper class limits" – typically, for continuous data, we use exclusive, but here ages are rounded to tenths, so maybe inclusive. Wait, the hint says "first lower class limit is minimum, class width rounded up to next tenths place". So class width is 1.3 (since 1.22 rounded up to tenths is 1.3).

So classes:

1. 7.2 – 7.2+1.3 - 0.1? No, better: lower limit, upper limit, with width 1.3. So:

Class 1: 7.2 ≤ x < 8.5 (since 7.2 +1.3=8.5)
Class 2: 8.5 ≤ x < 9.8
Class 3: 9.8 ≤ x < 11.1
Class 4: 11.1 ≤ x < 12.4
Class 5: 12.4 ≤ x < 13.7 (covers 13.3)

Now count frequencies:

Class 1 (7.2–8.5):
7.2,7.5,7.7,7.8,8.0,8.3,8.3,8.8? Wait 8.8 <8.5? No, 8.8 is 8.8 ≥8.5. So:
7.2,7.5,7.7,7.8,8.0,8.3,8.3 → 7 values? Wait 8.8 is 8.8, which is in class 2 (8.5–9.8). Let's list all:

Data points:
7.2,7.5,7.7,7.8,8.0,8.3,
8.3,8.8,8.9,9.0,9.6,9.7,
9.9,9.9,10.0,10.2,10.3,10.3,
10.4,11.1,11.1,11.1,11.2,11.4,
11.7,11.8,11.9,13.1,13.3,13.3

Class 1 (7.2–8.5):
7.2,7.5,7.7,7.8,8.0,8.3,8.3 → 7 (since 8.8,8.9 are ≥8.5)

Class 2 (8.5–9.8):
8.8,8.9,9.0,9.6,9.7 → 5 (8.8,8.9,9.0,9.6,9.7)

Class 3 (9.8–11.1):
9.9,9.9,10.0,10.2,10.3,10.3,10.4? Wait 10.4 <11.1? Yes. Wait 9.9,9.9,10.0,10.2,10.3,10.3,10.4 → 7? Wait 10.4 is 10.4 <11.1, yes. Wait 9.8 ≤x <11.1: 9.9,9.9,10.0,10.2,10.3,10.3,10.4 → 7? Wait original data: row3: 9.9,9.9,10.0,10.2,10.3,10.3 (6), row4:10.4 (1) → total 7.

Class 4 (11.1–12.4):
11.1,11.1,11.1,11.2,11.4,11.7? No, 11.7 ≥12.4? 11.7 <12.4, yes. Wait 11.1,11.1,11.1,11.2,11.4,11.7,11.8,11.9? No, 11.7,11.8,11.9 <12.4? Yes. Wait data: row4:11.1,11.1,11.1,11.2,11.4 (5), row5:11.7,11.8,11.9 (3) → total 8.

Class 5 (12.4–13.7):
13.1,13.3,13.3 → 3.

Now check total: 7+5+7+8+3=30. Correct.

Step4: Calculate relative frequencies (frequency/30)

• Class1: \( 7/30 ≈ 0.233 \)
• Class2: \( 5/30 ≈ 0.167 \)
• Class3: \( 7/30 ≈ 0.233 \)
• Class4: \( 8/30 ≈ 0.267 \)
• Class5: \( 3/30 = 0.100 \)

Answer:

Lower Class Limits	Upper Class Limits	Frequency	Rel. Freq.
8.5	9.8	5	0.167
9.8	11.1	7	0.233
11.1	12.4	8	0.267
12.4	13.7	3	0.100