Question

hours of practice this week
the dot plot shows the number of hours students in the
band practiced their instruments.
which describe the dot plot? select all that apply.
□ there is a cluster from 3 to 4.
□ there is a gap between 1 and 3.
□ there is a peak at 4.
□ the data shows symmetry.
□ the data is skewed left.
□ the data is skewed right.

Explanation:

Step1: Analyze "Cluster from 3 to 4"

Check dots between 3 and 4. At 3, there are dots; at 4, more dots. So cluster from 3 - 4? Wait, 3 and 4 have dots, but is that a cluster? Wait, the values: 0 (1 dot), 1 (2 dots), 2 (0 dots), 3 (4 dots), 4 (7 dots). So between 3 - 4, high density? Wait, no, 3 and 4 are adjacent. Wait, the option says 3 to 4. But let's check other options.

Step2: Analyze "Gap between 1 and 3"

Between 1 (value 1) and 3 (value 3), the value 2 has 0 dots. So gap between 1 and 3 (since 2 is between 1 and 3, no dots). So this is true.

Step3: Analyze "Peak at 4"

Value 4 has the most dots (7), so peak at 4. True.

Step4: Analyze "Symmetry"

Check if left and right mirror. 0:1, 1:2; 3:4, 4:7. Not symmetric. So false.

Step5: Analyze "Skewed left"

Left skew means tail on left. But here, more dots on right (4 has more than 0,1). Wait, left skew: mean < median, tail left. But dots: 0(1),1(2),2(0),3(4),4(7). The data is skewed left? Wait, no: left skew is when most data on right, tail on left. Wait, 0 and 1 have few dots, 3 and 4 have many. So tail on left (0,1), so skewed left? Wait, no: skewed left means the tail is on the left, so the bulk is on the right. So yes, skewed left? Wait, but let's recheck. Wait, the dots: 0 (1), 1 (2), 2 (0), 3 (4), 4 (7). So the data increases from 0 to 4, with a gap at 2. So the tail is on the left (0,1), so skewed left. Wait, but also, gap between 1 and 3 (since 2 has 0), peak at 4, gap between 1 and 3, cluster from 3 - 4? Wait, 3 and 4 have dots, so cluster from 3 - 4? Wait, 3 has 4, 4 has 7, so yes, cluster. Wait, but let's check each option:

1. Cluster from 3 to 4: 3 and 4 have high dot counts, so yes.
2. Gap between 1 and 3: 2 has 0 dots, so between 1 (value 1) and 3 (value 3), gap at 2. Yes.
3. Peak at 4: 4 has most dots, yes.
4. Symmetry: No.
5. Skewed left: Tail on left (0,1 have few), bulk on right (3,4). So skewed left? Wait, left skew is when the tail is on the left, so the distribution has a longer tail on the left. So yes, because 0 and 1 have few dots (tail), and 3,4 have many (bulk on right). So skewed left? Wait, but earlier thought symmetry was false. Wait, maybe I made a mistake. Wait, let's count the number of dots:

- 0: 1
- 1: 2
- 2: 0
- 3: 4
- 4: 7

So the data points: 0,1,1,3,3,3,3,4,4,4,4,4,4,4 (wait, no: 0:1 (1 dot), 1:2 (2 dots: total 3), 3:4 (total 7), 4:7 (total 14). Wait, 0(1),1(2),3(4),4(7). So the values are 0,1,1,3,3,3,3,4,4,4,4,4,4,4. So the distribution: left side (0,1) has 3 dots, right side (3,4) has 11 dots. So tail on left (0,1), so skewed left. So skewed left is true? Wait, but also gap between 1 and 3, peak at 4, cluster from 3 - 4.

Wait, but let's re-express:

- Cluster from 3 to 4: 3 and 4 have many dots, so yes.
- Gap between 1 and 3: 2 has 0, so yes.
- Peak at 4: 4 has most, yes.
- Skewed left: yes, because tail on left.

Wait, but the option "There is a cluster from 3 to 4" – 3 and 4 are consecutive, so cluster. "Gap between 1 and 3" – 2 has no dots, so gap. "Peak at 4" – most dots. "Skewed left" – tail on left.

Wait, but let's check the dot plot again. The x-axis: 0,1,2,3,4. Dots:

- 0: 1 dot
- 1: 2 dots
- 2: 0 dots
- 3: 4 dots
- 4: 7 dots

So:

- Cluster from 3 to 4: 3 and 4 have high density (4 and 7 dots), so yes.
- Gap between 1 and 3: 2 has 0 dots, so between 1 (value 1) and 3 (value 3), gap (since 2 is in between, no dots). Yes.
- Peak at 4: 4 has most dots, yes.
- Symmetry: No, as 0(1),1(2) vs 3(4),4(7) – not mirror.
- Skewed left: Yes, because the tail is on the left (0,1 have few dots), and the bulk is on the right (3,4 have many). So skewed left.

Wait, but maybe I made a mistake with skewed left. Let's recall: skewed left (negative skew) has the tail on the left, so the mean is less than the median. Let's calculate median: 14 data points, median is average of 7th and 8th. Data points in order: 0,1,1,3,3,3,3,4,4,4,4,4,4,4. 7th:3, 8th:4. Median: 3.5. Mean: (0*1 + 1*2 + 3*4 + 4*7)/(1+2+4+7) = (0 + 2 + 12 + 28)/14 = 42/14 = 3. So mean (3) < median (3.5), so skewed left. Yes.

So the correct options are:

- There is a cluster from 3 to 4.
- There is a gap between 1 and 3.
- There is a peak at 4.
- The data is skewed left.

Wait, but let's check again. The option "There is a cluster from 3 to 4" – 3 and 4 are adjacent, so the dots are clustered around 3 - 4. Yes. "Gap between 1 and 3" – 2 has no dots, so gap. "Peak at 4" – most dots. "Skewed left" – yes.

But maybe the initial analysis was wrong. Let's confirm each:

1. Cluster from 3 to 4: 3 and 4 have the most dots, so they form a cluster. Correct.
2. Gap between 1 and 3: Value 2 (between 1 and 3) has 0 dots. Correct.
3. Peak at 4: Value 4 has the highest frequency (most dots). Correct.
4. Symmetry: The left side (0,1) and right side (3,4) do not mirror. Incorrect.
5. Skewed left: The tail is on the left (fewer dots at 0,1), and the bulk is on the right (more dots at 3,4). Correct.

So the correct options are:

- There is a cluster from 3 to 4.
- There is a gap between 1 and 3.
- There is a peak at 4.
- The data is skewed left.

Wait, but maybe the "cluster from 3 to 4" – 3 and 4 are consecutive, so cluster. Yes. So these four? Wait, no, let's count the dots:

0:1, 1:2, 2:0, 3:4, 4:7.

So:

- Cluster from 3 to 4: 3 (4) and 4 (7) have high counts, so cluster. Correct.
- Gap between 1 and 3: 2 has 0, so gap. Correct.
- Peak at 4: 4 has most. Correct.
- Skewed left: Yes, because the left (0,1) has few, right (3,4) has many, tail on left. Correct.

So the correct options are the ones with these checked.

Answer:

• There is a cluster from 3 to 4.
• There is a gap between 1 and 3.
• There is a peak at 4.
• The data is skewed left.