Question

camerons scout leader organized a contest to see who could build the strongest boat out of aluminum foil. after building their boats, the scouts measured how many pennies the boats could support before sinking. this box plot shows the results. what fraction of the boats could support 12 or more pennies?

Explanation:

Step1: Recall box plot properties

In a box plot, the data is divided into four quartiles (each representing 25% of the data). The median (middle line of the box) splits the data into lower and upper halves. The upper half (values ≥ median) is 50% of the data, but more precisely, the quartiles: Q1 (25th percentile), Q2 (median, 50th percentile), Q3 (75th percentile). The data above Q3 is the top 25%? Wait, no: the box spans from Q1 to Q3, containing the middle 50% (interquartile range, IQR). The whiskers go from minimum to Q1 (lower 25%) and Q3 to maximum (upper 25%)? Wait, no: actually, the box plot has: minimum, Q1 (25th percentile: 25% of data ≤ Q1), Q2 (median, 50th percentile), Q3 (75th percentile: 75% of data ≤ Q3), maximum. So the data from Q3 to maximum is the top 25%? Wait, no: 100% - 75% = 25% of data is above Q3 (since 75% is ≤ Q3). Wait, no: Q3 is the 75th percentile, meaning 75% of the data is less than or equal to Q3, so 25% of the data is greater than Q3. But in the box plot, the right end of the box is Q3. So the data points that are 12 or more: let's look at the plot. The box ends at 12? Wait, the box plot: the box is from, say, let's see the x-axis: 0, 4, 8, 12, 16, 20. The box is from, maybe Q1 at 6, Q2 at 10, Q3 at 12? Wait, the box has two parts, maybe the median is at 10, and Q3 is at 12. Wait, no, the key is: in a box plot, the data is divided into four equal parts (quartiles), each 25% of the data. So the four quartiles are:

- Lower quartile (Q1): 25% of data ≤ Q1

- Median (Q2): 50% of data ≤ Q2

- Upper quartile (Q3): 75% of data ≤ Q3

So the data above Q3 (i.e., > Q3) is 25%? No, 100% - 75% = 25% of data is > Q3. But wait, the question is "12 or more". If Q3 is 12, then data ≥ Q3 is 25%? Wait, no: Q3 is the 75th percentile, so 75% of data is ≤ Q3, so 25% of data is ≥ Q3 (since 100 - 75 = 25). Wait, but maybe the median is at Q2, and the upper half (above median) is 50%, but let's re-express.

Wait, the problem: the box plot shows the results. We need to find the fraction of boats that could support 12 or more pennies. Let's think about quartiles. In a box plot, the data is split into four equal groups (quartiles), each 25% of the data. So:

- 25% of data: ≤ Q1 (lower quartile)

- 25%: Q1 to median (Q2)

- 25%: median (Q2) to Q3 (upper quartile)

- 25%: Q3 to maximum (upper 25%)

Wait, no, that's not correct. The correct division is:

- Q1: 25th percentile (25% of data ≤ Q1)

- Q2: 50th percentile (median, 50% of data ≤ Q2)

- Q3: 75th percentile (75% of data ≤ Q3)

So the data is divided into:

- Below Q1: 25%

- Q1 to Q2: 25% (total 50% ≤ Q2)

- Q2 to Q3: 25% (total 75% ≤ Q3)

- Q3 to maximum: 25% (total 100% ≤ maximum)

Wait, that can't be, because 25% + 25% + 25% + 25% = 100%. So each quartile (the four segments) is 25% of the data. So:

1. Minimum to Q1: 25%

2. Q1 to Q2: 25%

3. Q2 to Q3: 25%

4. Q3 to maximum: 25%

Ah, that's the correct way: the box plot's box is from Q1 to Q3, which is the middle 50% (Q1 to Q2 is 25%, Q2 to Q3 is 25%), and the whiskers are minimum to Q1 (25%) and Q3 to maximum (25%). Wait, no, the box is Q1 to Q3, so that's Q1 (25th) to Q3 (75th), so the length of the box is IQR (Q3 - Q1), containing 50% of the data (from 25th to 75th percentile). Then the lower whisker is minimum to Q1 (25% of data, 0th to 25th percentile), and upper whisker is Q3 to maximum (25% of data, 75th to 100th percentile).

So in the given box plot, the right end of the box is at 12 (Q3), and the maximum is at 16? Wait, the whisker goes from Q3 (12) to maximum (16). So the data from Q3 (12) to maximum is the upper 25%? Wait, no: 75th percentile is Q3, so 75% of data is ≤ Q3, so 25% of data is > Q3 (i.e., ≥ Q3 + 1? No, ≥ Q3). Wait, no: 75% of data is ≤ Q3, so 25% of data is ≥ Q3 (since 100% - 75% = 25%). So if Q3 is 12, then 25% of the data is ≥ 12? Wait, no, that's not right. Wait, percentile: 25th percentile (Q1): 25% of data is ≤ Q1. 50th percentile (Q2): 50% ≤ Q2. 75th percentile (Q3): 75% ≤ Q3. So the percentage of data ≥ Q3 is 100% - 75% = 25%. So if the Q3 is at 12, then the number of boats that can support 12 or more pennies is 25%? Wait, but maybe the median is at 10, and Q3 is at 12. Wait, the box plot: the box has two parts, maybe the first part (left of median) is Q1 to Q2, and right is Q2 to Q3. So the median is the middle line, splitting the box into two equal parts (each 25% of the data, since the box is 50% total). So the right half of the box (Q2 to Q3) is 25% of the data, and the whisker from Q3 to maximum is another 25%? No, that would be 50% above Q2, but we need above 12. Wait, maybe the Q3 is at 12, so the data from Q3 to maximum is 25% (since 75% is ≤ Q3, 25% is ≥ Q3). But wait, the problem is asking for 12 or more. So if Q3 is 12, then 25% of the data is ≥ 12? Wait, no, 75% is ≤ Q3, so 25% is > Q3. But if Q3 is 12, then data ≥ 12 includes the 25% above Q3 (since Q3 is 12, so data > 12 is 25%, and data = 12 is part of the 75%? Wait, no, percentile is ≤ Q3. So Q3 is the value where 75% of data is ≤ Q3, so data = Q3 is included in that 75%. So data ≥ Q3 would be data > Q3 (25%) plus data = Q3 (which is part of the 75%). Wait, this is confusing. Let's re-express with an example. Suppose we have 100 data points. Q1 is the 25th data point (when sorted), Q2 is 50th, Q3 is 75th. So:

- Data points 1 - 25: ≤ Q1 (25% of data)

- Data points 26 - 50: between Q1 and Q2 (25% of data)

- Data points 51 - 75: between Q2 and Q3 (25% of data)

- Data points 76 - 100: ≥ Q3 (25% of data)

Ah! There we go. So Q3 is the 75th data point (when sorted), so data points 76 - 100 (25% of data) are ≥ Q3 (since Q3 is the 75th, so 75th data point is Q3, so data points 76 - 100 are greater than Q3? Wait, no: the 75th percentile is the value such that 75% of the data is less than or equal to it. So if the 75th data point (in sorted order) is, say, 12, then 75 data points are ≤ 12, and 25 data points are > 12. Wait, no: the 75th percentile is the value at the 75th position (if sorted) when there are 100 data points. So data points 1 - 75: ≤ Q3, data points 76 - 100: > Q3. So the percentage of data > Q3 is 25%, and data ≥ Q3 is 25% (if Q3 is not equal to any data points above it) or more. But in the box plot, the right end of the box is Q3, so the data from Q3 to maximum is the upper 25% (data > Q3) plus any data equal to Q3? Wait, no, the box plot's Q3 is the 75th percentile, so the data above Q3 (i.e., > Q3) is 25% of the data. But in the problem, the question is "12 or more". Let's look at the box plot: the box ends at 12 (Q3 is 12), so the data points that are 12 or more would include the data from Q3 (12) to maximum. Since Q3 is the 75th percentile, the data from Q3 to maximum is 25% of the data (as 75% is ≤ Q3, so 25% is ≥ Q3? Wait, no, 75% is ≤ Q3, so 25% is > Q3. But if Q3 is 12, then data ≥ 12 would be data = 12 (which is part of the 75%) plus data > 12 (25%). Wait, this is a mistake. Let's correct:

In a box plot, the quartiles are defined as:

- Q1: 25th percentile (25% of data ≤ Q1)

- Q2: 50th percentile (median, 50% of data ≤ Q2)

- Q3: 75th percentile (75% of data ≤ Q3)

So the proportion of data ≤ Q3 is 75%, so the proportion of data > Q3 is 25%. The proportion of data ≥ Q3 is 25% (if Q3 is not equal to any data points above it) or more (if there are data points equal to Q3 above the 75th percentile). But in the box plot, the right whisker is from Q3 to maximum, which represents the data from Q3 to maximum, which is 25% of the data (since 75% is ≤ Q3, so 25% is ≥ Q3? Wait, no, 75% ≤ Q3, so 25% > Q3. So the whisker from Q3 to maximum is the data > Q3, which is 25% of the data. But if Q3 is 12, then data ≥ 12 would include data = 12 (which is part of the 75%) and data > 12 (25%). Wait, that can't be. Wait, no: the 75th percentile is the value such that 75% of the data is less than or equal to it. So if the 75th percentile is 12, that means 75% of the boats supported 12 or fewer pennies, and 25% supported more than 12 pennies. But the question is "12 or more". So we need to find the fraction of boats that supported 12 or more. So if 75% supported ≤12, then 25% supported >12. But wait, maybe the median is at 10, and Q3 is at 12. Wait, the box plot has two parts, maybe the first part (left of median) is Q1 to Q2, and right is Q2 to Q3. So the median is the middle line, splitting the box into two equal parts (each 25% of the data, since the box is 50% total). So the right half of the box (Q2 to Q3) is 25% of the data, and the whisker from Q3 to maximum is another 25%? No, that would be 50% above Q2, but we need above 12. Wait, maybe the key is that in a box plot, the data is divided into four equal parts (quartiles), each 25% of the data. So:

- Lower quartile (Q1): 25% of data (minimum to Q1)

- Middle lower (Q1 to Q2): 25% of data

- Middle upper (Q2 to Q3): 25% of data

- Upper quartile (Q3 to maximum): 25% of data

Ah! This is the correct way to think about it when the data is evenly divided into four quartiles (each 25%). So the box is from Q1 to Q3, which is the middle 50% (Q1 to Q2: 25%, Q2 to Q3: 25%). Then the upper quartile (Q3 to maximum) is the top 25% of the data. So if the Q3 is at 12, then the data from Q3 (12) to maximum is the top 25% of the data. But the question is "12 or more". So does Q3 include 12? Yes, because Q3 is the 75th percentile, so the data from Q3 to maximum includes Q3 (12) and above. Wait, no: if the upper quartile is Q3 to maximum, that's 25% of the data, which is the data greater than Q3? No, that's the confusion. Let's take an example with 4 data points (simplified):

Data: [1, 2, 3, 4]

Q1: 1.5 (25th percentile), Q2: 2.5 (median), Q3: 3.5 (75th percentile). The box is from 1.5 to 3.5, whiskers from 1 to 1.5 and 3.5 to 4. The upper quartile (Q3 to maximum) is 3.5 to 4, which is 25% of the data (the last data point, 4, which is 25% of 4 data points). So in this case, the upper quartile is the data greater than Q3 (3.5), which is 4. So the upper quartile is the top 25% of the data, which is greater than Q3.

But in the problem, the box plot's Q3 is at 12, so the data greater than Q3 (12) is the top 25% of the data. But the question is "12 or more". So if Q3 is 12, then data ≥ 12 would include the data equal to Q3 (12) and above. But in the quartile division, the middle upper (Q2 to Q3) is 25% of the data (from Q2 to Q3), and upper quartile (Q3 to maximum) is 25% of the data (from Q3 to maximum). Wait, no, that can't be, because Q2 to Q3 is 25% (middle upper), and Q3 to maximum is 25% (upper), so total above Q2 is 50% (middle upper + upper). But we need above 12.

Wait, maybe the key is that the median (Q2) is the middle, so 50% of the data is above the median, and 50% below. But in the box plot, the right half of the box (Q2 to Q3) is 25% of the data, and the whisker (Q3 to maximum) is 25% of the data. So total above Q2 is 50%

Answer:

Step1: Recall box plot properties

In a box plot, the data is divided into four quartiles (each representing 25% of the data). The median (middle line of the box) splits the data into lower and upper halves. The upper half (values ≥ median) is 50% of the data, but more precisely, the quartiles: Q1 (25th percentile), Q2 (median, 50th percentile), Q3 (75th percentile). The data above Q3 is the top 25%? Wait, no: the box spans from Q1 to Q3, containing the middle 50% (interquartile range, IQR). The whiskers go from minimum to Q1 (lower 25%) and Q3 to maximum (upper 25%)? Wait, no: actually, the box plot has: minimum, Q1 (25th percentile: 25% of data ≤ Q1), Q2 (median, 50th percentile), Q3 (75th percentile: 75% of data ≤ Q3), maximum. So the data from Q3 to maximum is the top 25%? Wait, no: 100% - 75% = 25% of data is above Q3 (since 75% is ≤ Q3). Wait, no: Q3 is the 75th percentile, meaning 75% of the data is less than or equal to Q3, so 25% of the data is greater than Q3. But in the box plot, the right end of the box is Q3. So the data points that are 12 or more: let's look at the plot. The box ends at 12? Wait, the box plot: the box is from, say, let's see the x-axis: 0, 4, 8, 12, 16, 20. The box is from, maybe Q1 at 6, Q2 at 10, Q3 at 12? Wait, the box has two parts, maybe the median is at 10, and Q3 is at 12. Wait, no, the key is: in a box plot, the data is divided into four equal parts (quartiles), each 25% of the data. So the four quartiles are:

• Lower quartile (Q1): 25% of data ≤ Q1

• Median (Q2): 50% of data ≤ Q2

• Upper quartile (Q3): 75% of data ≤ Q3

So the data above Q3 (i.e., > Q3) is 25%? No, 100% - 75% = 25% of data is > Q3. But wait, the question is "12 or more". If Q3 is 12, then data ≥ Q3 is 25%? Wait, no: Q3 is the 75th percentile, so 75% of data is ≤ Q3, so 25% of data is ≥ Q3 (since 100 - 75 = 25). Wait, but maybe the median is at Q2, and the upper half (above median) is 50%, but let's re-express.

Wait, the problem: the box plot shows the results. We need to find the fraction of boats that could support 12 or more pennies. Let's think about quartiles. In a box plot, the data is split into four equal groups (quartiles), each 25% of the data. So:

• 25% of data: ≤ Q1 (lower quartile)

• 25%: Q1 to median (Q2)

• 25%: median (Q2) to Q3 (upper quartile)

• 25%: Q3 to maximum (upper 25%)

Wait, no, that's not correct. The correct division is:

• Q1: 25th percentile (25% of data ≤ Q1)

• Q2: 50th percentile (median, 50% of data ≤ Q2)

• Q3: 75th percentile (75% of data ≤ Q3)

So the data is divided into:

• Below Q1: 25%

• Q1 to Q2: 25% (total 50% ≤ Q2)

• Q2 to Q3: 25% (total 75% ≤ Q3)

• Q3 to maximum: 25% (total 100% ≤ maximum)

Wait, that can't be, because 25% + 25% + 25% + 25% = 100%. So each quartile (the four segments) is 25% of the data. So:

1. Minimum to Q1: 25%

1. Q1 to Q2: 25%

1. Q2 to Q3: 25%

1. Q3 to maximum: 25%

Ah, that's the correct way: the box plot's box is from Q1 to Q3, which is the middle 50% (Q1 to Q2 is 25%, Q2 to Q3 is 25%), and the whiskers are minimum to Q1 (25%) and Q3 to maximum (25%). Wait, no, the box is Q1 to Q3, so that's Q1 (25th) to Q3 (75th), so the length of the box is IQR (Q3 - Q1), containing 50% of the data (from 25th to 75th percentile). Then the lower whisker is minimum to Q1 (25% of data, 0th to 25th percentile), and upper whisker is Q3 to maximum (25% of data, 75th to 100th percentile).

So in the given box plot, the right end of the box is at 12 (Q3), and the maximum is at 16? Wait, the whisker goes from Q3 (12) to maximum (16). So the data from Q3 (12) to maximum is the upper 25%? Wait, no: 75th percentile is Q3, so 75% of data is ≤ Q3, so 25% of data is > Q3 (i.e., ≥ Q3 + 1? No, ≥ Q3). Wait, no: 75% of data is ≤ Q3, so 25% of data is ≥ Q3 (since 100% - 75% = 25%). So if Q3 is 12, then 25% of the data is ≥ 12? Wait, no, that's not right. Wait, percentile: 25th percentile (Q1): 25% of data is ≤ Q1. 50th percentile (Q2): 50% ≤ Q2. 75th percentile (Q3): 75% ≤ Q3. So the percentage of data ≥ Q3 is 100% - 75% = 25%. So if the Q3 is at 12, then the number of boats that can support 12 or more pennies is 25%? Wait, but maybe the median is at 10, and Q3 is at 12. Wait, the box plot: the box has two parts, maybe the first part (left of median) is Q1 to Q2, and right is Q2 to Q3. So the median is the middle line, splitting the box into two equal parts (each 25% of the data, since the box is 50% total). So the right half of the box (Q2 to Q3) is 25% of the data, and the whisker from Q3 to maximum is another 25%? No, that would be 50% above Q2, but we need above 12. Wait, maybe the Q3 is at 12, so the data from Q3 to maximum is 25% (since 75% is ≤ Q3, 25% is ≥ Q3). But wait, the problem is asking for 12 or more. So if Q3 is 12, then 25% of the data is ≥ 12? Wait, no, 75% is ≤ Q3, so 25% is > Q3. But if Q3 is 12, then data ≥ 12 includes the 25% above Q3 (since Q3 is 12, so data > 12 is 25%, and data = 12 is part of the 75%? Wait, no, percentile is ≤ Q3. So Q3 is the value where 75% of data is ≤ Q3, so data = Q3 is included in that 75%. So data ≥ Q3 would be data > Q3 (25%) plus data = Q3 (which is part of the 75%). Wait, this is confusing. Let's re-express with an example. Suppose we have 100 data points. Q1 is the 25th data point (when sorted), Q2 is 50th, Q3 is 75th. So:

• Data points 1 - 25: ≤ Q1 (25% of data)

• Data points 26 - 50: between Q1 and Q2 (25% of data)

• Data points 51 - 75: between Q2 and Q3 (25% of data)

• Data points 76 - 100: ≥ Q3 (25% of data)

Ah! There we go. So Q3 is the 75th data point (when sorted), so data points 76 - 100 (25% of data) are ≥ Q3 (since Q3 is the 75th, so 75th data point is Q3, so data points 76 - 100 are greater than Q3? Wait, no: the 75th percentile is the value such that 75% of the data is less than or equal to it. So if the 75th data point (in sorted order) is, say, 12, then 75 data points are ≤ 12, and 25 data points are > 12. Wait, no: the 75th percentile is the value at the 75th position (if sorted) when there are 100 data points. So data points 1 - 75: ≤ Q3, data points 76 - 100: > Q3. So the percentage of data > Q3 is 25%, and data ≥ Q3 is 25% (if Q3 is not equal to any data points above it) or more. But in the box plot, the right end of the box is Q3, so the data from Q3 to maximum is the upper 25% (data > Q3) plus any data equal to Q3? Wait, no, the box plot's Q3 is the 75th percentile, so the data above Q3 (i.e., > Q3) is 25% of the data. But in the problem, the question is "12 or more". Let's look at the box plot: the box ends at 12 (Q3 is 12), so the data points that are 12 or more would include the data from Q3 (12) to maximum. Since Q3 is the 75th percentile, the data from Q3 to maximum is 25% of the data (as 75% is ≤ Q3, so 25% is ≥ Q3? Wait, no, 75% is ≤ Q3, so 25% is > Q3. But if Q3 is 12, then data ≥ 12 would be data = 12 (which is part of the 75%) plus data > 12 (25%). Wait, this is a mistake. Let's correct:

In a box plot, the quartiles are defined as:

• Q1: 25th percentile (25% of data ≤ Q1)

• Q2: 50th percentile (median, 50% of data ≤ Q2)

• Q3: 75th percentile (75% of data ≤ Q3)

So the proportion of data ≤ Q3 is 75%, so the proportion of data > Q3 is 25%. The proportion of data ≥ Q3 is 25% (if Q3 is not equal to any data points above it) or more (if there are data points equal to Q3 above the 75th percentile). But in the box plot, the right whisker is from Q3 to maximum, which represents the data from Q3 to maximum, which is 25% of the data (since 75% is ≤ Q3, so 25% is ≥ Q3? Wait, no, 75% ≤ Q3, so 25% > Q3. So the whisker from Q3 to maximum is the data > Q3, which is 25% of the data. But if Q3 is 12, then data ≥ 12 would include data = 12 (which is part of the 75%) and data > 12 (25%). Wait, that can't be. Wait, no: the 75th percentile is the value such that 75% of the data is less than or equal to it. So if the 75th percentile is 12, that means 75% of the boats supported 12 or fewer pennies, and 25% supported more than 12 pennies. But the question is "12 or more". So we need to find the fraction of boats that supported 12 or more. So if 75% supported ≤12, then 25% supported >12. But wait, maybe the median is at 10, and Q3 is at 12. Wait, the box plot has two parts, maybe the first part (left of median) is Q1 to Q2, and right is Q2 to Q3. So the median is the middle line, splitting the box into two equal parts (each 25% of the data, since the box is 50% total). So the right half of the box (Q2 to Q3) is 25% of the data, and the whisker from Q3 to maximum is another 25%? No, that would be 50% above Q2, but we need above 12. Wait, maybe the key is that in a box plot, the data is divided into four equal parts (quartiles), each 25% of the data. So:

• Lower quartile (Q1): 25% of data (minimum to Q1)

• Middle lower (Q1 to Q2): 25% of data

• Middle upper (Q2 to Q3): 25% of data

• Upper quartile (Q3 to maximum): 25% of data

Ah! This is the correct way to think about it when the data is evenly divided into four quartiles (each 25%). So the box is from Q1 to Q3, which is the middle 50% (Q1 to Q2: 25%, Q2 to Q3: 25%). Then the upper quartile (Q3 to maximum) is the top 25% of the data. So if the Q3 is at 12, then the data from Q3 (12) to maximum is the top 25% of the data. But the question is "12 or more". So does Q3 include 12? Yes, because Q3 is the 75th percentile, so the data from Q3 to maximum includes Q3 (12) and above. Wait, no: if the upper quartile is Q3 to maximum, that's 25% of the data, which is the data greater than Q3? No, that's the confusion. Let's take an example with 4 data points (simplified):

Data: [1, 2, 3, 4]

Q1: 1.5 (25th percentile), Q2: 2.5 (median), Q3: 3.5 (75th percentile). The box is from 1.5 to 3.5, whiskers from 1 to 1.5 and 3.5 to 4. The upper quartile (Q3 to maximum) is 3.5 to 4, which is 25% of the data (the last data point, 4, which is 25% of 4 data points). So in this case, the upper quartile is the data greater than Q3 (3.5), which is 4. So the upper quartile is the top 25% of the data, which is greater than Q3.

But in the problem, the box plot's Q3 is at 12, so the data greater than Q3 (12) is the top 25% of the data. But the question is "12 or more". So if Q3 is 12, then data ≥ 12 would include the data equal to Q3 (12) and above. But in the quartile division, the middle upper (Q2 to Q3) is 25% of the data (from Q2 to Q3), and upper quartile (Q3 to maximum) is 25% of the data (from Q3 to maximum). Wait, no, that can't be, because Q2 to Q3 is 25% (middle upper), and Q3 to maximum is 25% (upper), so total above Q2 is 50% (middle upper + upper). But we need above 12.

Wait, maybe the key is that the median (Q2) is the middle, so 50% of the data is above the median, and 50% below. But in the box plot, the right half of the box (Q2 to Q3) is 25% of the data, and the whisker (Q3 to maximum) is 25% of the data. So total above Q2 is 50%