(1 point)
which of the following are true?
a. the median value for d1 is less than the median value for d3.

b. three quarters of the data values for d2 are greater than the median value for d1.

c. at least a quarter of the data values for d3 are less than the median value for d1.

d. at least three quarters of the data values in d1 are less than all of the data values in d2.

e. the data in d3 is skewed right.

f. at least a quarter of the data values in d2 are less than all of the data values in d3.

Question

(1 point)
which of the following are true?
a. the median value for d1 is less than the median value for d3.

b. three quarters of the data values for d2 are greater than the median value for d1.

c. at least a quarter of the data values for d3 are less than the median value for d1.

d. at least three quarters of the data values in d1 are less than all of the data values in d2.

e. the data in d3 is skewed right.

f. at least a quarter of the data values in d2 are less than all of the data values in d3.

Accepted Answer

# Brief Explanations:
- **Option A**: The median of a box - plot is the line inside the box. For D1, the box is around a value, and for D3, the box's median line. Visually, D1's median seems higher than D3's, so A is false.
- **Option B**: The median of D1: Let's assume the box - plot for D1. The median of D2: The box - plot for D2. Three - quarters of D2's data are above the first quartile of D2. The median of D1: Let's see the position. The first quartile of D2 is less than the median of D1? Wait, no. Wait, three - quarters of D2's data are greater than the first quartile of D2. If the median of D1 is less than the third quartile of D2? Wait, maybe better to look at the values. D2's max is 68, D1's min is 10. Wait, no, D1's max is 118, D2's max is 68. Wait, no, the box - plot for D1: min = 10, max = 118. D2: min = 38, max = 68. D3: min = 61, max = 100. The median of D1: inside the box. The median of D2: inside its box. Three - quarters of D2's data are greater than the first quartile of D2. If the median of D1 is, say, let's assume the median of D1 is around, say, 80? No, D1's box is from, say, the first quartile to third quartile. Wait, maybe a better approach: In a box - plot, the median is the middle line. For D2, the median is inside its box. Three - quarters of D2's data are above the first quartile of D2. If the median of D1 is less than the third quartile of D2? Wait, no, D2's max is 68, D1's min is 10. Wait, D1's data goes up to 118, D2's up to 68. Wait, maybe I got D1 and D2 mixed. Wait, D1: min = 10, max = 118. D2: min = 38, max = 68. D3: min = 61, max = 100. So D2's data is between 38 and 68, D1's between 10 and 118, D3's between 61 and 100. The median of D1: let's say the box for D1 is from, say, 50 to 90, median at 70? D2's box: from 40 to 60, median at 50? No, maybe not. Wait, the key is: Three - quarters of D2's data are greater than the first quartile of D2. If the median of D1 is less than the third quartile of D2? Wait, no, D2's third quartile is less than 68. D1's median: let's assume the median of D1 is, say, 80? No, D1's box is longer. Wait, maybe the correct way: For option B, three - quarters of D2's data are greater than the first quartile of D2. If the median of D1 is less than the third quartile of D2? Wait, no, D2's data is all less than D1's max, but D2's third quartile is less than 68, and D1's median is, say, 70? Then three - quarters of D2's data (which are greater than D2's first quartile) would be greater than the median of D1? No, maybe I'm wrong. Wait, let's check the other options.
- **Option C**: At least a quarter of D3's data are less than the median of D2. D3's min is 61, D2's median: let's say D2's median is 50? No, D2's data is from 38 to 68, so median is between 38 and 68. D3's min is 61, which is greater than D2's median (if D2's median is, say, 50). So at least a quarter of D3's data (the first quartile) is 61 or more, which is greater than D2's median. So C is false.
- **Option D**: At least three - quarters of D1's data are less than all of D2's data? All of D2's data is less than 68 (max of D2 is 68). Three - quarters of D1's data are less than the third quartile of D1. If the third quartile of D1 is less than 68? No, D1's max is 118, so third quartile of D1 is less than 118, but D2's max is 68. So three - quarters of D1's data are less than the third quartile of D1, but the third quartile of D1 could be greater than 68. So D is false.
- **Option E**: The data in D3 is skewed right? In a box - plot, if the right whisker is longer or the box is shifted left, it's skewed right. D3's box: min = 61, the box is from, say, 61 to 100, with the median in the middle? Wait, no, the box - plot for D3: the left part (from min to median) is shorter than the right part (median to max)? Wait, D3's min is 61, and the box is divided into two parts. If the left part (from 61 to median) is shorter than the right part (median to 100), then it's skewed right. So E is true? Wait, no, skewed right means the tail is on the right. Wait, the box - plot for D3: min = 61, the box is from 61 to 100, with the median line. If the left side of the box (from min to median) is shorter than the right side (median to max), then it's skewed right. So E is true? Wait, no, maybe I'm wrong. Wait, skewed right: mean > median, and the tail is on the right. In the box - plot, the right whisker is longer or the box is shifted left. For D3, the min is 61, and the box starts at 61, so the left whisker is non - existent (since min is at the start of the box? No, the box is from first quartile to third quartile. Wait, D3's box: first quartile is 61? No, min is 61, so first quartile is greater than or equal to 61. So the box is from Q1 to Q3, with median inside. If Q1 = 61, median is, say, 80, Q3 = 100. Then the left part (Q1 to median) is 19, right part (median to Q3) is 20. So almost symmetric? Wait, maybe E is false.
- **Option F**: At least a quarter of D2's data are less than all of D3's data. All of D3's data is greater than or equal to 61. D2's data is from 38 to 68. So the first quartile of D2: at least a quarter of D2's data is less than the first quartile of D2. If the first quartile of D2 is less than 61, then at least a quarter of D2's data is less than 61, which is less than all of D3's data (since D3's min is 61). So F is true? Wait, D3's min is 61, so all of D3's data is ≥ 61. D2's data is from 38 to 68. So the first quartile of D2 is less than 61 (since D2's min is 38 and max is 68, the first quartile is at most 68, but likely less than 61? Wait, D2's min is 38, max is 68. The first quartile of D2: 25% of data is less than Q1. If Q1 of D2 is less than 61, then at least a quarter of D2's data is less than 61, which is less than all of D3's data (since D3's data starts at 61). So F is true? Wait, no, D3's min is 61, so D3's data is ≥ 61. D2's data: 25% of D2's data is less than Q1 of D2. If Q1 of D2 < 61, then 25% of D2's data is < 61, hence less than all of D3's data (since D3's data is ≥ 61). So F is true. Wait, but let's check again.
Wait, maybe I made a mistake. Let's re - evaluate:

- **Option B**: Three - quarters of D2's data are greater than the first quartile of D2. The median of D1: Let's assume D1's median is, say, 70. D2's first quartile: since D2's data is from 38 to 68, Q1 of D2 is, say, 45. Then three - quarters of D2's data (above Q1) are greater than 45, but the median of D1 is 70, which is greater than 45. So B is false.
- **Option F**: At least a quarter of D2's data are less than all of D3's data. All of D3's data is ≥ 61 (since D3's min is 61). D2's data is from 38 to 68. The first quartile of D2 (Q1) is the value where 25% of data is less than Q1. Since D2's min is 38 and max is 68, Q1 of D2 is less than 61 (because 25% of data is below Q1, and the data goes up to 68). So at least 25% of D2's data is less than Q1, which is less than 61, hence less than all of D3's data (which is ≥ 61). So F is true.

Wait, maybe the correct options are F? Wait, no, let's check the other options again.

Wait, the correct answer: Let's analyze each option:

- **A**: Median of D1 vs D3. D1's median: inside its box (which is from, say, low to high, but D3's median: inside its box. D1's median is likely higher than D3's (since D1's data starts lower but goes higher, but D3's data starts at 61). So A is false.
- **B**: Three - quarters of D2's data are greater than median of D1? D2's data is up to 68, D1's median is, say, 80? No, D1's data goes up to 118, but D2's up to 68. Wait, D1's box: the middle line (median) is inside the box. D2's box: middle line (median) inside its box. If D1's median is greater than D2's median, then three - quarters of D2's data (which are above D2's first quartile) would be less than D1's median? So B is false.
- **C**: At least a quarter of D3's data less than median of D2? D3's min is 61, median of D2 is, say, 50 (since D2's data is from 38 to 68). So D3's data is all ≥ 61, which is greater than 50. So no data in D3 is less than median of D2. So C is false.
- **D**: At least three - quarters of D1's data less than all of D2's data? All of D2's data is ≤ 68. Three - quarters of D1's data are less than D1's third quartile. D1's third quartile is greater than 68 (since D1's max is 118), so three - quarters of D1's data are less than a value greater than 68, but not all of D2's data (since D2's data is ≤ 68). So D is false.
- **E**: D3 skewed right? In a box - plot, skewed right means the left whisker is shorter and the box is shifted right, with the right whisker longer. D3's box: min = 61, the box is from 61 to 100, median in the middle. So it's symmetric or slightly skewed left? So E is false.
- **F**: At least a quarter of D2's data less than all of D3's data? All of D3's data is ≥ 61. D2's data is from 38 to 68. The first quartile of D2 (Q1) is the value where 25% of data is below Q1. Since D2's data starts at 38, Q1 is less than 61 (because 25% of data is below Q1, and the data goes up to 68). So at least 25% of D2's data is below Q1, which is less than 61, hence less than all of D3's data (which is ≥ 61). So F is true.

Wait, but maybe I made a mistake. Let's check with the box - plot values:

D1: min = 10, max = 118

D2: min = 38, max = 68

D3: min = 61, max = 100

So D2's data is between 38 and 68, D3's between 61 and 100, D1's between 10 and 118.

- For F: At least a quarter of D2's data are less than all of D3's data. All of D3's data is ≥ 61. So we need to see if at least 25% of D2's data is < 61. Since D2's min is 38 and max is 68, the first quartile (Q1) of D2 is the value where 25% of data is below Q1. Since the data ranges from 38 to 68, Q1 must be less than 61 (because 25% of the data is below Q1, and the upper 75% is from Q1 to 68). So 25% of D2's data is below Q1 (which is < 61), so these data points are less than all of D3's data (which is ≥ 61). So F is true.

# Answer: F. At least a quarter of the data values in D2 are less than all of the data values in D3.

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