Question

below are weights of mollusks randomly gathered on the coast of southern california.
0.5 0.5 2.6 3.4 3.5 3.6 4.1 4.4
5.2 5.5 5.5 5.7 5.7 5.9 6.0 6.1
6.1 6.3 6.4 6.4 6.5 6.9 6.9 7.0
7.1 7.2 7.4 7.8 8.3 8.4 8.5 9.0
9.7 9.7 10.1 10.4 10.8 10.8 13.1 13.4
construct a frequency distribution table for this information using 8 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.
hint 1 hint 2 hint 3

Explanation:

Step1: Find min, max, and class width

Min = 0.5, Max = 13.4. Number of classes = 8.
Range = \( 13.4 - 0.5 = 12.9 \).
Class width = \( \frac{12.9}{8} \approx 1.6125 \), rounded up to next tenths: \( 1.7 \).

Step2: Define class limits

Start with lower limit 0.5.
Class 1: \( 0.5 - 0.5 + 1.7 - 0.1 = 0.5 - 2.1 \) (since upper limit = lower + width - 0.1 for continuous data).
Class 2: \( 2.2 - 3.8 \) (2.1 + 0.1 = 2.2; 2.2 + 1.7 - 0.1 = 3.8).
Class 3: \( 3.9 - 5.5 \)
Class 4: \( 5.6 - 7.2 \)
Class 5: \( 7.3 - 8.9 \)
Class 6: \( 9.0 - 10.6 \)
Class 7: \( 10.7 - 12.3 \)
Class 8: \( 12.4 - 14.0 \) (covers max 13.4).

Step3: Count frequencies

Data points: 40 (8 columns × 5 rows).
Class 1 (0.5 - 2.1): 0.5, 0.5, 2.6? Wait, 0.5, 0.5, 2.6? No, 0.5, 0.5, 2.6 is 2.6 > 2.1? Wait, 0.5 - 2.1: values ≤2.1. So 0.5, 0.5, 2.6? No, 2.6 >2.1. So 0.5,0.5: frequency=2.
Class 2 (2.2 - 3.8): 2.6, 3.4, 3.5, 3.6: frequency=4.
Class 3 (3.9 - 5.5): 4.1,4.4,5.2,5.5,5.5: frequency=5.
Class 4 (5.6 - 7.2): 5.7,5.7,5.9,6.0,6.1,6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0,7.1,7.2? Wait, 5.6 -7.2: 5.7,5.7,5.9,6.0,6.1,6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0,7.1,7.2? Wait, 7.2 is ≤7.2? Yes. Wait, 5.6 to 7.2: let's list all data:
Row1: 0.5,0.5,2.6,3.4,3.5,3.6,4.1,4.4
Row2:5.2,5.5,5.5,5.7,5.7,5.9,6.0,6.1
Row3:6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0
Row4:7.1,7.2,7.4,7.8,8.3,8.4,8.5,9.0
Row5:9.7,9.7,10.1,10.4,10.8,10.8,13.1,13.4

So Class 1 (0.5-2.1): 0.5,0.5 → 2
Class 2 (2.2-3.8): 2.6,3.4,3.5,3.6 → 4
Class 3 (3.9-5.5): 4.1,4.4,5.2,5.5,5.5 → 5
Class 4 (5.6-7.2): 5.7,5.7,5.9,6.0,6.1,6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0,7.1,7.2 → Wait, 7.2 is ≤7.2? Yes. Let's count: 5.7(2),5.9,6.0,6.1(2),6.3,6.4(2),6.5,6.9(2),7.0,7.1,7.2 → 2+1+1+2+1+2+1+1+1= 13? Wait, row2: 5.7,5.7,5.9,6.0,6.1 → 5; row3:6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0 → 8; row4:7.1,7.2 → 2. Total 5+8+2=15? Wait, no, 5.6-7.2: 5.7,5.7,5.9,6.0,6.1,6.1,6.3,6.4,6.4,6.5,6.9,6.9,7.0,7.1,7.2 → 15? Wait, 2 (5.7) +1(5.9)+1(6.0)+2(6.1)+1(6.3)+2(6.4)+1(6.5)+2(6.9)+1(7.0)+1(7.1)+1(7.2) = 2+1+1+2+1+2+1+2+1+1+1=15.
Class 5 (7.3-8.9): 7.4,7.8,8.3,8.4,8.5 → 5 (row4:7.4,7.8,8.3,8.4,8.5; 9.0 is next class).
Class 6 (9.0-10.6): 9.0,9.7,9.7,10.1,10.4 → 5 (row4:9.0; row5:9.7,9.7,10.1,10.4; 10.8 is next class).
Class 7 (10.7-12.3): 10.8,10.8 → 2 (row5:10.8,10.8; 13.1 is next class).
Class 8 (12.4-14.0): 13.1,13.4 → 2.

Wait, check total: 2+4+5+15+5+5+2+2=40. Correct.

Step4: Calculate relative frequencies

Rel. Freq. = Frequency / 40.
Class1: \( 2/40 = 0.050 \)
Class2: \( 4/40 = 0.100 \)
Class3: \( 5/40 = 0.125 \)
Class4: \( 15/40 = 0.375 \)
Class5: \( 5/40 = 0.125 \)
Class6: \( 5/40 = 0.125 \)
Class7: \( 2/40 = 0.050 \)
Class8: \( 2/40 = 0.050 \)

Lower Class Limits	Upper Class Limits	Frequency	Rel. Freq.
2.2	3.8	4	0.100
3.9	5.5	5	0.125
5.6	7.2	15	0.375
7.3	8.9	5	0.125
9.0	10.6	5	0.125
10.7	12.3	2	0.050
12.4	14.0	2	0.050

Answer:

The frequency distribution table is as above.