Question

a cherry farmer gathered a random sample 30 cherries and weighed them. below are the weights, in grams.
8.3 9.4 9.5 10.0 10.0 10.0
10.2 10.2 10.3 10.6 10.7 11.1
11.2 11.4 11.4 11.6 11.9 11.9
12.0 12.2 12.3 12.4 12.5 12.6
12.7 12.8 13.5 13.9 14.1 14.6
construct a frequency distribution (with relative frequencies) table for this information using 5 classes.
enter relative frequencies as well, rounded to two 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.
8.3 9.5 3 0.10
9.6 10.8 8 0.27
10.9 12.1 7 0.23
12.2 13.4 8 0.27

Explanation:

Step 1: Find Min, Max, and Class Width

Min = 8.3, Max = 14.6. Range = \( 14.6 - 8.3 = 6.3 \). Class width = \( \frac{6.3}{5} = 1.26 \), rounded up to 1.3 (next tenths place).

Step 2: Define Class Limits

1. Class 1: 8.3 – 8.3 + 1.3 - 0.1 = 9.5 (since upper limit = lower + width - 0.1 for continuous data)
2. Class 2: 9.6 – 9.6 + 1.3 - 0.1 = 10.8
3. Class 3: 10.9 – 10.9 + 1.3 - 0.1 = 12.1
4. Class 4: 12.2 – 12.2 + 1.3 - 0.1 = 13.4
5. Class 5: 13.5 – 13.5 + 1.3 - 0.1 = 14.7 (covers max 14.6)

Step 3: Count Frequencies

- Class 1 (8.3–9.5): 8.3, 9.4, 9.5 → Frequency = 3
- Class 2 (9.6–10.8): 10.0,10.0,10.0,10.2,10.2,10.3,10.6,10.7 → Frequency = 8
- Class 3 (10.9–12.1): 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → Wait, 12.0 is ≤12.1? 12.0 ≤12.1: yes. Wait, original data: 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → 8? Wait no, let's list all data:

Data points: 8.3,9.4,9.5,10.0,10.0,10.0,10.2,10.2,10.3,10.6,10.7,11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0,12.2,12.3,12.4,12.5,12.6,12.7,12.8,13.5,13.9,14.1,14.6.

Class 3 (10.9–12.1): 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → 8? Wait 12.0 is in 10.9–12.1 (12.1 is upper limit). Then Class 4 (12.2–13.4): 12.2,12.3,12.4,12.5,12.6,12.7,12.8 → 7 (since 13.5 is next class). Class 5 (13.5–14.7): 13.5,13.9,14.1,14.6 → 4.

Wait, total should be 30. Let's check:

Class 1: 3, Class 2: 8, Class 3: 8? Wait no, let's recount:

Class 1 (8.3 ≤ x < 9.6): 8.3,9.4,9.5 → 3 (correct, since 9.5 < 9.6? Wait, class width is 1.3, so 8.3 to 8.3+1.3=9.6? Wait, maybe class limits are inclusive of lower, exclusive of upper (continuous). So:

Class 1: 8.3 – 9.6 (exclusive upper), so 8.3,9.4,9.5 (all <9.6) → 3.

Class 2: 9.6 – 10.9 (exclusive upper): 10.0,10.0,10.0,10.2,10.2,10.3,10.6,10.7 → 8 (all <10.9).

Class 3: 10.9 – 12.2 (exclusive upper): 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0,12.2? Wait 12.2 is equal to upper limit? No, 10.9 +1.3=12.2, so upper limit is 12.2 (exclusive). So 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → 8? Wait 12.0 <12.2: yes. Then 12.2 is in next class.

Class 4: 12.2 – 13.5 (exclusive upper): 12.2,12.3,12.4,12.5,12.6,12.7,12.8 → 7 (13.5 is next class).

Class 5: 13.5 – 14.8 (exclusive upper): 13.5,13.9,14.1,14.6 → 4.

Now check total: 3+8+8+7+4=30. Correct.

Step 4: Calculate Relative Frequencies

Relative Frequency = Frequency / 30.

- Class 1: \( \frac{3}{30} = 0.10 \)
- Class 2: \( \frac{8}{30} \approx 0.27 \)
- Class 3: \( \frac{8}{30} \approx 0.27 \) (wait, earlier mistake: class 3 frequency is 8, not 7)
- Class 4: \( \frac{7}{30} \approx 0.23 \)
- Class 5: \( \frac{4}{30} \approx 0.13 \)

Wait, let's redo class limits with correct width (1.3):

Min = 8.3, so:

Class 1: 8.3 – 8.3 + 1.3 = 9.6 (lower:8.3, upper:9.6)
Class 2: 9.6 – 9.6 + 1.3 = 10.9
Class 3: 10.9 – 10.9 + 1.3 = 12.2
Class 4: 12.2 – 12.2 + 1.3 = 13.5
Class 5: 13.5 – 13.5 + 1.3 = 14.8

Now count frequencies:

Class 1 (8.3 ≤ x < 9.6): 8.3,9.4,9.5 → 3
Class 2 (9.6 ≤ x < 10.9): 10.0,10.0,10.0,10.2,10.2,10.3,10.6,10.7 → 8 (10.0 ≥9.6, <10.9)
Class 3 (10.9 ≤ x < 12.2): 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → 8 (11.1 ≥10.9, <12.2; 12.0 <12.2)
Class 4 (12.2 ≤ x < 13.5): 12.2,12.3,12.4,12.5,12.6,12.7,12.8 → 7 (12.2 ≥12.2, <13.5; 12.8 <13.5)
Class 5 (13.5 ≤ x < 14.8): 13.5,13.9,14.1,14.6 → 4 (13.5 ≥13.5, <14.8; 14.6 <14.8)

Now relative frequencies:

- Class 1: \( 3/30 = 0.10 \)
- Class 2: \( 8/30 ≈ 0.27 \)
- Class 3: \( 8/30 ≈ 0.27 \) (previously wrong as 7, correct is 8)
- Class 4: \( 7/30 ≈ 0.23 \)
- Class 5: \( 4/30 ≈ 0.13 \)

Wait, the original table had class 3 as 10.9–12.1 (maybe upper limit is 12.1, not 12.2). Let's check the original hint's class limits:

First class: 8.3–9.5 (width 1.2, but rounded up to 1.3? Wait 9.5 -8.3=1.2, so maybe class width is 1.2? Wait the problem says "class width should be rounded up to the next tenths place". Range is 14.6-8.3=6.3. 6.3/5=1.26, rounded up to 1.3 (next tenths: 1.26 → 1.3). So class width 1.3.

But original table's first class is 8.3–9.5 (width 1.2), which is 9.5-8.3=1.2. Maybe the problem's "next tenths place" is after the decimal, so 1.26 rounded up to 1.3? No, 1.26 rounded to tenths is 1.3? Wait 1.26: tenths place is 2, next tenths is 3? Wait, rounding up 1.26 to the next tenths place: 1.26 → 1.3 (since 0.26 ≥0.2, round up to 0.3? Wait, no: "rounded up to the next tenths place" means if the decimal part is 0.26, we round up to 0.3, so class width 1.3. But original table's first class is 8.3–9.5 (width 1.2), which is 9.5-8.3=1.2. Maybe the problem's "next tenths place" is misinterpreted. Let's recalculate with class width 1.2 (since 6.3/5=1.26, maybe "rounded up to the next tenths" is 1.3, but original table has 9.5-8.3=1.2. Let's check the data:

First class: 8.3,9.4,9.5 → 3 (correct). Second class: 9.6–10.8 (9.6=8.3+1.3? No, 8.3+1.2=9.5, so maybe class width 1.2. Then range 6.3, 5 classes: 6.3/5=1.26, rounded up to 1.3? No, maybe the problem's "rounded up to the next tenths place" is 1.3, but original table uses 1.2. Let's proceed with correct frequency count.

Correct frequencies:

Class 1 (8.3–9.5): 3 (8.3,9.4,9.5)
Class 2 (9.6–10.8): 10.0,10.0,10.0,10.2,10.2,10.3,10.6,10.7 → 8
Class 3 (10.9–12.1): 11.1,11.2,11.4,11.4,11.6,11.9,11.9,12.0 → 8 (12.0 ≤12.1)
Class 4 (12.2–13.4): 12.2,12.3,12.4,12.5,12.6,12.7,12.8 → 7 (12.2 ≥12.2, 12.8 ≤13.4)
Class 5 (13.5–14.7): 13.5,13.9,14.1,14.6 → 4 (13.5 ≥13.5, 14.6 ≤14.7)

Relative frequencies:

- Class 1: 3/30 = 0.10
- Class 2: 8/30 ≈ 0.27
- Class 3: 8/30 ≈ 0.27 (previously marked wrong, correct is 8)
- Class 4: 7/30 ≈ 0.23
- Class 5: 4/30 ≈ 0.13

So the correct frequency distribution table is:

| Lower Class Limits | Upper Class Limits | Frequency | Rel. Freq. |
|--------------------|--------------------|-----------|------------|
| 8.3 | 9.5 | 3 | 0.10 |
| 9.6 | 10.8 | 8 | 0.27 |
| 10.9 | 12.1 | 8 | 0.27 |
| 12.2 | 13.4 | 7 | 0.23 |
| 13.5 | 14.7 | 4 | 0.13 |

Answer:

Lower Class Limits	Upper Class Limits	Frequency	Rel. Freq.
9.6	10.8	8	0.27
10.9	12.1	8	0.27
12.2	13.4	7	0.23
13.5	14.7	4	0.13