Question

below are ages (in years, rounded to the nearest tenth) of 30 randomly selected children at a summer camp in july, 2018.

6.7\t6.8\t7.7\t8.2\t8.3\t8.6
8.8\t9.0\t9.0\t9.2\t9.3\t9.3
9.5\t9.5\t9.9\t9.9\t10.1\t10.3
10.4\t10.8\t11.0\t11.0\t11.0\t11.1
11.2\t11.8\t11.8\t12.3\t13.7\t15.1

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.

hint 1 hint 2 hint 3

lower class limits\tupper class limits\tfrequency\trel. freq.
\t\t\t
\t\t\t
\t\t\t
\t\t\t
\t\t\t

Explanation:

Step1: Find min, max, and class width

Min = 6.7, Max = 15.1. Range = \( 15.1 - 6.7 = 8.4 \). Class width = \( \frac{8.4}{5} = 1.68 \), rounded up to next tenths: 1.7.

Step2: Determine class limits

1st class: Lower = 6.7, Upper = \( 6.7 + 1.7 - 0.1 = 8.3 \) (subtract 0.1 for upper limit of previous class)
2nd: Lower = 8.4, Upper = \( 8.3 + 1.7 = 10.0 \)
3rd: Lower = 10.1, Upper = \( 10.0 + 1.7 = 11.7 \)
4th: Lower = 11.8, Upper = \( 11.7 + 1.7 = 13.4 \)
5th: Lower = 13.5, Upper = \( 13.4 + 1.7 = 15.1 \)

Step3: Count frequencies

- 6.7 - 8.3: Values: 6.7, 6.8, 7.7, 8.2, 8.3, 8.6, 8.8? Wait, no: 6.7,6.8,7.7,8.2,8.3,8.6,8.8? Wait, 8.8 is 8.8 >8.3? Wait, 1st class upper limit is 8.3? Wait, no, class width is 1.7, so 6.7 to \( 6.7 + 1.7 = 8.4 \)? Wait, maybe I messed up. Let's recalculate class width: range 8.4, 5 classes: 8.4/5=1.68, round up to 1.7. So classes:
1. 6.7 - 6.7 + 1.7 - 0.1 = 8.3? No, better: lower limit, upper limit (inclusive). So first class: 6.7 - 8.3 (since 6.7 +1.7=8.4, but we round to tenths, so 8.3? Wait, no, let's list all data:

Data: 6.7,6.8,7.7,8.2,8.3,8.6,8.8,9.0,9.0,9.2,9.3,9.3,9.5,9.5,9.9,9.9,10.1,10.3,10.4,10.8,11.0,11.0,11.0,11.1,11.2,11.8,11.8,12.3,13.7,15.1

Now, class width 1.7. So:

Class 1: 6.7 - 8.3 (6.7 +1.7=8.4, but upper limit is 8.3? Wait, no, 6.7 to 8.3 (inclusive), next 8.4 to 10.0 (8.3+0.1=8.4, 8.4+1.7=10.1? No, 8.4 +1.7=10.1, but upper limit 10.0? Wait, maybe better to use lower and upper limits as:

Class 1: 6.7 - 8.3 (width 1.7: 8.3 -6.7=1.6? No, 8.4 -6.7=1.7. So upper limit should be 8.4? Wait, the problem says "rounded up to the next tenths place" for class width. 1.68 rounded up to tenths is 1.7. So class width is 1.7. So first class: 6.7 to 6.7 +1.7 =8.4. Then next: 8.4 to 8.4 +1.7=10.1, then 10.1 to 10.1+1.7=11.8, then 11.8 to 11.8+1.7=13.5, then 13.5 to 13.5+1.7=15.2. But max is 15.1, so adjust last class to 13.5 -15.1.

Now, count frequencies:

Class 1: 6.7 -8.4: values 6.7,6.8,7.7,8.2,8.3,8.6,8.8? Wait, 8.6 and 8.8 are >8.4? No, 8.4 is upper limit. So 6.7,6.8,7.7,8.2,8.3: that's 5? Wait 6.7,6.8,7.7,8.2,8.3,8.6? 8.6 is 8.6 >8.4? Yes, 8.6>8.4. So class 1: 6.7 -8.3 (since 8.4 is next class). Wait, maybe the problem's "rounded up to next tenths" for class width: 1.68 rounded up to tenths is 1.7, so class width 1.7. So lower limits: 6.7, 8.4, 10.1, 11.8, 13.5. Upper limits: 8.3 (6.7+1.7-0.1), 10.0 (8.4+1.7-0.1), 11.7 (10.1+1.7-0.1), 13.4 (11.8+1.7-0.1), 15.1 (13.5+1.7-0.1=15.1). Yes, that works.

Now count:

Class 1: 6.7 -8.3: values 6.7,6.8,7.7,8.2,8.3,8.6? No, 8.6 is 8.6>8.3? Yes. Wait, 6.7,6.8,7.7,8.2,8.3: that's 5? Wait 8.6 is 8.6, which is in next class (8.4-10.0). Let's list all data with their classes:

6.7: 6.7-8.3

6.8: 6.7-8.3

7.7: 6.7-8.3

8.2: 6.7-8.3

8.3: 6.7-8.3

8.6: 8.4-10.0

8.8: 8.4-10.0

9.0: 8.4-10.0

9.0: 8.4-10.0

9.2: 8.4-10.0

9.3: 8.4-10.0

9.3: 8.4-10.0

9.5: 10.1-11.7? No, 9.5 is <10.1. Wait, 8.4-10.0: 8.4 to 10.0 (inclusive). So 9.5 is in 8.4-10.0.

9.5: 8.4-10.0

9.9: 8.4-10.0

9.9: 8.4-10.0

10.1: 10.1-11.7

10.3: 10.1-11.7

10.4: 10.1-11.7

10.8: 10.1-11.7

11.0: 10.1-11.7

11.0: 10.1-11.7

11.0: 10.1-11.7

11.1: 10.1-11.7

11.2: 10.1-11.7

11.8: 11.8-13.4

11.8: 11.8-13.4

12.3: 11.8-13.4

13.7: 13.5-15.1

15.1: 13.5-15.1

Now count each class:

Class 1 (6.7-8.3): 6.7,6.8,7.7,8.2,8.3 → 5

Class 2 (8.4-10.0): 8.6,8.8,9.0,9.0,9.2,9.3,9.3,9.5,9.5,9.9,9.9 → 11

Class 3 (10.1-11.7): 10.1,10.3,10.4,10.8,11.0,11.0,11.0,11.1,11.2 → 9

Class 4 (11.8-13.4): 11.8,11.8,12.3 → 3

Class 5 (13.5-15.1): 13.7,15.1 → 2

Now check total: 5+11+9+3+2=30, correct.

Step4: Calculate relative frequencies

Rel. Freq. = Frequency / 30

Class 1: \( 5/30 ≈ 0.167 \)

Class 2: \( 11/30 ≈ 0.367 \)

Class 3: \( 9/30 = 0.300 \)

Class 4: \( 3/30 = 0.100 \)

Class 5: \( 2/30 ≈ 0.067 \)

Answer:

Lower Class Limits	Upper Class Limits	Frequency	Rel. Freq.
8.4	10.0	11	0.367
10.1	11.7	9	0.300
11.8	13.4	3	0.100
13.5	15.1	2	0.067