Question

below are ages (in years, rounded to the nearest tenth) of randomly selected attendees of a climate strike rally during the year of 2019.
8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5, 10.7, 11.2, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1, 12.4, 12.5, 12.6, 13.0, 13.6, 14.2, 14.4, 14.6, 14.7, 14.8, 14.8, 14.9, 15.0, 15.4, 15.6, 16.4, 17.9, 19.3, 19.4, 19.6, 19.8, 21.9, 22.8, 23.6
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.
hint 1 hint 2 hint 3
frequency table chart with columns: lower class limits, upper class limits, frequency, rel. freq.

Explanation:

Step1: Find min, max, and class width

First, find the minimum value ($\min = 8.1$) and maximum value ($\max = 23.6$). The range is $\max - \min = 23.6 - 8.1 = 15.5$. We need 8 classes, so class width $w = \frac{15.5}{8} \approx 1.9375$. Round up to the next tenths place, so $w = 2.0$.

Step2: Determine class limits

Start with lower class limit = 8.1. Each class has width 2.0. So the classes are:
- 8.1 - 10.0 (since 8.1 + 2.0 - 0.1 = 10.0)
- 10.1 - 12.0
- 12.1 - 14.0
- 14.1 - 16.0
- 16.1 - 18.0
- 18.1 - 20.0
- 20.1 - 22.0
- 22.1 - 24.0

Step3: Count frequencies

Count how many data points fall in each class:
- 8.1 - 10.0: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5? Wait, 10.5 is above 10.0? Wait, no: 8.1 to 10.0 (inclusive? Or exclusive? Wait, class limits: lower inclusive, upper exclusive? Wait, the first class: lower 8.1, upper 10.0 (since 8.1 + 2.0 = 10.1? Wait, no, class width is 2.0, so each class is [lower, lower + width). So first class: 8.1 ≤ x < 10.1? Wait, no, the problem says "rounded to the nearest tenth", so maybe the upper limit is the next tenth. Wait, let's recalculate class width correctly.

Wait, the problem says "class width should be rounded up to the next tenths place". The range is 23.6 - 8.1 = 15.5. Number of classes = 8. So 15.5 / 8 = 1.9375. Rounded up to the next tenths place: 2.0 (since 1.9375 rounded up to tenths is 2.0). So class width is 2.0.

So first class: lower limit = 8.1, upper limit = 8.1 + 2.0 - 0.1 = 10.0? Wait, no, class width is 2.0, so each class is 2.0 wide. So:

Class 1: 8.1 - 10.0 (8.1 ≤ x ≤ 10.0? Or 8.1 < x ≤ 10.1? Wait, the problem says "rounded to the nearest tenth", so the data points are to the nearest tenth. Let's list the data:

Data points: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5, 10.7, 11.2, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1, 12.4, 12.5, 12.6, 13.0, 13.6, 14.2, 14.4, 14.6, 14.7, 14.8, 14.8, 14.9, 15.0, 15.4, 15.6, 16.4, 17.9, 19.3, 19.4, 19.6, 19.8, 21.9, 22.8, 23.6. Wait, let's count the number of data points: 8 + 8 + 8 + 8 + 2 + 4 + 2 + 2? Wait, no, let's count:

First row: 8 (8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is above 10.0? Wait, maybe my class limits are wrong. Let's recalculate:

Minimum = 8.1, maximum = 23.6. Number of classes = 8. So class width = ceil((23.6 - 8.1)/8) = ceil(15.5/8) = ceil(1.9375) = 2.0 (since 1.9375 rounded up to tenths is 2.0). So each class is 2.0 wide. So the classes are:

1. 8.1 - 10.0 (8.1 ≤ x ≤ 10.0)
2. 10.1 - 12.0 (10.1 ≤ x ≤ 12.0)
3. 12.1 - 14.0 (12.1 ≤ x ≤ 14.0)
4. 14.1 - 16.0 (14.1 ≤ x ≤ 16.0)
5. 16.1 - 18.0 (16.1 ≤ x ≤ 18.0)
6. 18.1 - 20.0 (18.1 ≤ x ≤ 20.0)
7. 20.1 - 22.0 (20.1 ≤ x ≤ 22.0)
8. 22.1 - 24.0 (22.1 ≤ x ≤ 24.0)

Now count frequencies:

1. 8.1 - 10.0: 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is in 10.1 - 12.0. So 8.1,8.4,9.3,9.3,9.5,9.7,9.8: that's 7? Wait, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. So first class: 8.1,8.4,9.3,9.3,9.5,9.7,9.8: 7? Wait, no, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. Let's list all data:

Data points:

Row 1: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5 (8)

Row 2: 10.7, 11.2, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1 (8)

Row 3: 12.4, 12.5, 12.6, 13.0, 13.6, 14.2, 14.4, 14.6 (8)

Row 4: 14.7, 14.8, 14.8, 14.9, 15.0, 15.4, 15.6, 16.4 (8)

Row 5: 17.9, 19.3, 19.4, 19.6, 19.8, 21.9, 22.8, 23.6 (8)

Wait, total data points: 5 rows × 8 = 40. Let's confirm:

Row 1: 8

Row 2: 8 (total 16)

Row 3: 8 (total 24)

Row 4: 8 (total 32)

Row 5: 8 (total 40). Correct.

Now, assign each to class:

Class 1: 8.1 - 10.0:

Data in this range: 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is >10.0, so no. Wait, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. So class 1: 8.1,8.4,9.3,9.3,9.5,9.7,9.8: that's 7? Wait, 10.5 is 10.5, so 10.5 is in class 2. Wait, row 1 has 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5: 8.1-10.0: 8.1,8.4,9.3,9.3,9.5,9.7,9.8 (7), and 10.5 (1) in class 2.

Class 2: 10.1 - 12.0:

Data: 10.5,10.7,11.2,11.6,11.7,11.8,11.9,12.0. Wait, 10.5 is 10.5 (yes), 10.7,11.2,11.6,11.7,11.8,11.9,12.0: that's 8? Wait, row 1: 10.5 (1), row 2: 10.7,11.2,11.6,11.7,11.8,11.9,12.0 (7), total 8. Correct.

Class 3: 12.1 - 14.0:

Data: 12.1,12.4,12.5,12.6,13.0,13.6,14.2? Wait, 14.2 is 14.2, which is in 14.1 - 16.0. Wait, 12.1,12.4,12.5,12.6,13.0,13.6: that's 6? Wait, row 2: 12.1 (1), row 3: 12.4,12.5,12.6,13.0,13.6 (5), total 6? No, row 3 has 8 data points: 12.4,12.5,12.6,13.0,13.6,14.2,14.4,14.6. So 12.1 - 14.0: 12.1,12.4,12.5,12.6,13.0,13.6 (6), and 14.2,14.4,14.6 in class 4.

Class 4: 14.1 - 16.0:

Data: 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6,16.4? Wait, 16.4 is 16.4, which is in 16.1 - 18.0. So 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6: that's 10? No, row 3: 14.2,14.4,14.6 (3), row 4: 14.7,14.8,14.8,14.9,15.0,15.4,15.6,16.4 (8). So 14.1 - 16.0: 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6 (10). Wait, but total data is 40, so let's check:

Class 1: 7

Class 2: 8 (7+8=15)

Class 3: 6 (15+6=21)

Class 4: 10 (21+10=31)

Class 5: 16.1 - 18.0: 16.4,17.9 (2) (31+2=33)

Class 6: 18.1 - 20.0: 19.3,19.4,19.6,19.8 (4) (33+4=37)

Class 7: 20.1 - 22.0: 21.9 (1) (37+1=38)

Class 8: 22.1 - 24.0: 22.8,23.6 (2) (38+2=40). Perfect, that adds up.

Now, relative frequency = frequency / total (40).

So:

Class 1: Lower=8.1, Upper=10.0, Frequency=7, Rel. Freq=7/40=0.175

Class 2: Lower=10.1, Upper=12.0, Frequency=8, Rel. Freq=8/40=0.200

Class 3: Lower=12.1, Upper=14.0, Frequency=6, Rel. Freq=6/40=0.150

Class 4: Lower=14.1, Upper=16.0, Frequency=10, Rel. Freq=10/40=0.250

Class 5: Lower=16.1, Upper=18.0, Frequency=2, Rel. Freq=2/40=0.050

Class 6: Lower=18.1, Upper=20.0, Frequency=4, Rel. Freq=4/40=0.100

Class 7: Lower=20.1, Upper=22.0, Frequency=1, Rel. Freq=1/40=0.025

Class 8: Lower=22.1, Upper=24.0, Frequency=2, Rel. Freq=2/40=0.050

Let's verify the frequencies again:

Class 1 (8.1-10.0): 8.1,8.4,9.3,9.3,9.5,9.7,9.8 (7) – correct.

Class 2 (10.1-12.0): 10.5,10.7,11.2,11.6,11.7,11.8,11.9,12.0 (8) – correct.

Class 3 (12.1-14.0): 12.1,12.4,12.5,12.6,13.0,13.6 (6) – correct.

Class 4 (14.1-16.0): 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6 (10) – correct.

Class 5 (16.1-18.0): 16.4,17.9 (2) – correct.

Class 6 (18.1-20.0): 19.3,19.4,19.6,19.8 (4) – correct.

Class 7 (20.1-22.0): 21.9 (1) – correct.

Class 8 (22.1-24.0): 22.8,23.6 (2) – correct.

Total: 7+8+6+10+2+4+1+2=40 – correct.

Relative frequencies:

7/40=0.175, 8/40=0.200, 6/40=0.150, 10/40=0.250, 2/40=0.050, 4/40

Answer:

Step1: Find min, max, and class width

First, find the minimum value ($\min = 8.1$) and maximum value ($\max = 23.6$). The range is $\max - \min = 23.6 - 8.1 = 15.5$. We need 8 classes, so class width $w = \frac{15.5}{8} \approx 1.9375$. Round up to the next tenths place, so $w = 2.0$.

Step2: Determine class limits

Start with lower class limit = 8.1. Each class has width 2.0. So the classes are:

• 8.1 - 10.0 (since 8.1 + 2.0 - 0.1 = 10.0)
• 10.1 - 12.0
• 12.1 - 14.0
• 14.1 - 16.0
• 16.1 - 18.0
• 18.1 - 20.0
• 20.1 - 22.0
• 22.1 - 24.0

Step3: Count frequencies

Count how many data points fall in each class:

• 8.1 - 10.0: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5? Wait, 10.5 is above 10.0? Wait, no: 8.1 to 10.0 (inclusive? Or exclusive? Wait, class limits: lower inclusive, upper exclusive? Wait, the first class: lower 8.1, upper 10.0 (since 8.1 + 2.0 = 10.1? Wait, no, class width is 2.0, so each class is [lower, lower + width). So first class: 8.1 ≤ x < 10.1? Wait, no, the problem says "rounded to the nearest tenth", so maybe the upper limit is the next tenth. Wait, let's recalculate class width correctly.

Wait, the problem says "class width should be rounded up to the next tenths place". The range is 23.6 - 8.1 = 15.5. Number of classes = 8. So 15.5 / 8 = 1.9375. Rounded up to the next tenths place: 2.0 (since 1.9375 rounded up to tenths is 2.0). So class width is 2.0.

So first class: lower limit = 8.1, upper limit = 8.1 + 2.0 - 0.1 = 10.0? Wait, no, class width is 2.0, so each class is 2.0 wide. So:

Class 1: 8.1 - 10.0 (8.1 ≤ x ≤ 10.0? Or 8.1 < x ≤ 10.1? Wait, the problem says "rounded to the nearest tenth", so the data points are to the nearest tenth. Let's list the data:

Data points: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5, 10.7, 11.2, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1, 12.4, 12.5, 12.6, 13.0, 13.6, 14.2, 14.4, 14.6, 14.7, 14.8, 14.8, 14.9, 15.0, 15.4, 15.6, 16.4, 17.9, 19.3, 19.4, 19.6, 19.8, 21.9, 22.8, 23.6. Wait, let's count the number of data points: 8 + 8 + 8 + 8 + 2 + 4 + 2 + 2? Wait, no, let's count:

First row: 8 (8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is above 10.0? Wait, maybe my class limits are wrong. Let's recalculate:

Minimum = 8.1, maximum = 23.6. Number of classes = 8. So class width = ceil((23.6 - 8.1)/8) = ceil(15.5/8) = ceil(1.9375) = 2.0 (since 1.9375 rounded up to tenths is 2.0). So each class is 2.0 wide. So the classes are:

1. 8.1 - 10.0 (8.1 ≤ x ≤ 10.0)
2. 10.1 - 12.0 (10.1 ≤ x ≤ 12.0)
3. 12.1 - 14.0 (12.1 ≤ x ≤ 14.0)
4. 14.1 - 16.0 (14.1 ≤ x ≤ 16.0)
5. 16.1 - 18.0 (16.1 ≤ x ≤ 18.0)
6. 18.1 - 20.0 (18.1 ≤ x ≤ 20.0)
7. 20.1 - 22.0 (20.1 ≤ x ≤ 22.0)
8. 22.1 - 24.0 (22.1 ≤ x ≤ 24.0)

Now count frequencies:

1. 8.1 - 10.0: 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is in 10.1 - 12.0. So 8.1,8.4,9.3,9.3,9.5,9.7,9.8: that's 7? Wait, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. So first class: 8.1,8.4,9.3,9.3,9.5,9.7,9.8: 7? Wait, no, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. Let's list all data:

Data points:

Row 1: 8.1, 8.4, 9.3, 9.3, 9.5, 9.7, 9.8, 10.5 (8)

Row 2: 10.7, 11.2, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1 (8)

Row 3: 12.4, 12.5, 12.6, 13.0, 13.6, 14.2, 14.4, 14.6 (8)

Row 4: 14.7, 14.8, 14.8, 14.9, 15.0, 15.4, 15.6, 16.4 (8)

Row 5: 17.9, 19.3, 19.4, 19.6, 19.8, 21.9, 22.8, 23.6 (8)

Wait, total data points: 5 rows × 8 = 40. Let's confirm:

Row 1: 8

Row 2: 8 (total 16)

Row 3: 8 (total 24)

Row 4: 8 (total 32)

Row 5: 8 (total 40). Correct.

Now, assign each to class:

Class 1: 8.1 - 10.0:

Data in this range: 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5? Wait, 10.5 is 10.5, which is >10.0, so no. Wait, 10.5 is 10.5, so 10.5 is in 10.1 - 12.0. So class 1: 8.1,8.4,9.3,9.3,9.5,9.7,9.8: that's 7? Wait, 10.5 is 10.5, so 10.5 is in class 2. Wait, row 1 has 8.1,8.4,9.3,9.3,9.5,9.7,9.8,10.5: 8.1-10.0: 8.1,8.4,9.3,9.3,9.5,9.7,9.8 (7), and 10.5 (1) in class 2.

Class 2: 10.1 - 12.0:

Data: 10.5,10.7,11.2,11.6,11.7,11.8,11.9,12.0. Wait, 10.5 is 10.5 (yes), 10.7,11.2,11.6,11.7,11.8,11.9,12.0: that's 8? Wait, row 1: 10.5 (1), row 2: 10.7,11.2,11.6,11.7,11.8,11.9,12.0 (7), total 8. Correct.

Class 3: 12.1 - 14.0:

Data: 12.1,12.4,12.5,12.6,13.0,13.6,14.2? Wait, 14.2 is 14.2, which is in 14.1 - 16.0. Wait, 12.1,12.4,12.5,12.6,13.0,13.6: that's 6? Wait, row 2: 12.1 (1), row 3: 12.4,12.5,12.6,13.0,13.6 (5), total 6? No, row 3 has 8 data points: 12.4,12.5,12.6,13.0,13.6,14.2,14.4,14.6. So 12.1 - 14.0: 12.1,12.4,12.5,12.6,13.0,13.6 (6), and 14.2,14.4,14.6 in class 4.

Class 4: 14.1 - 16.0:

Data: 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6,16.4? Wait, 16.4 is 16.4, which is in 16.1 - 18.0. So 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6: that's 10? No, row 3: 14.2,14.4,14.6 (3), row 4: 14.7,14.8,14.8,14.9,15.0,15.4,15.6,16.4 (8). So 14.1 - 16.0: 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6 (10). Wait, but total data is 40, so let's check:

Class 1: 7

Class 2: 8 (7+8=15)

Class 3: 6 (15+6=21)

Class 4: 10 (21+10=31)

Class 5: 16.1 - 18.0: 16.4,17.9 (2) (31+2=33)

Class 6: 18.1 - 20.0: 19.3,19.4,19.6,19.8 (4) (33+4=37)

Class 7: 20.1 - 22.0: 21.9 (1) (37+1=38)

Class 8: 22.1 - 24.0: 22.8,23.6 (2) (38+2=40). Perfect, that adds up.

Now, relative frequency = frequency / total (40).

So:

Class 1: Lower=8.1, Upper=10.0, Frequency=7, Rel. Freq=7/40=0.175

Class 2: Lower=10.1, Upper=12.0, Frequency=8, Rel. Freq=8/40=0.200

Class 3: Lower=12.1, Upper=14.0, Frequency=6, Rel. Freq=6/40=0.150

Class 4: Lower=14.1, Upper=16.0, Frequency=10, Rel. Freq=10/40=0.250

Class 5: Lower=16.1, Upper=18.0, Frequency=2, Rel. Freq=2/40=0.050

Class 6: Lower=18.1, Upper=20.0, Frequency=4, Rel. Freq=4/40=0.100

Class 7: Lower=20.1, Upper=22.0, Frequency=1, Rel. Freq=1/40=0.025

Class 8: Lower=22.1, Upper=24.0, Frequency=2, Rel. Freq=2/40=0.050

Let's verify the frequencies again:

Class 1 (8.1-10.0): 8.1,8.4,9.3,9.3,9.5,9.7,9.8 (7) – correct.

Class 2 (10.1-12.0): 10.5,10.7,11.2,11.6,11.7,11.8,11.9,12.0 (8) – correct.

Class 3 (12.1-14.0): 12.1,12.4,12.5,12.6,13.0,13.6 (6) – correct.

Class 4 (14.1-16.0): 14.2,14.4,14.6,14.7,14.8,14.8,14.9,15.0,15.4,15.6 (10) – correct.

Class 5 (16.1-18.0): 16.4,17.9 (2) – correct.

Class 6 (18.1-20.0): 19.3,19.4,19.6,19.8 (4) – correct.

Class 7 (20.1-22.0): 21.9 (1) – correct.

Class 8 (22.1-24.0): 22.8,23.6 (2) – correct.

Total: 7+8+6+10+2+4+1+2=40 – correct.

Relative frequencies:

7/40=0.175, 8/40=0.200, 6/40=0.150, 10/40=0.250, 2/40=0.050, 4/40