Question

11/30/23, 7:48 am

1. the box - and - whisker plots below show the distributions of act scores for boys and girls in a certain high school where the same number of boys and girls took the act.

which of the following statements correctly compare the act scores of the boys and girls in this high school?
select all that apply.
a. more girls than boys scored above 34 on the act.
b. more boys than girls scored above 31 on the act.
c. more girls than boys scored above 30 on the act
d. more boys than girls scored above 24 on the act.
e. more girls than boys scored above 22 on the act.

Explanation:

Step1: Analyze Box - Whisker Plot Components

Box - and - whisker plots show the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of a data set. The number of data points (boys and girls) is the same. To compare the number of students scoring above a certain value, we can use the quartiles and the spread of the data.

Step2: Analyze Option A

For the score of 34: The maximum score for boys is less than 34 (from the box - whisker plot of boys), and the maximum for girls is 36. But we need to see the proportion above 34. The upper whisker of girls goes to 36, but the number of girls above 34: since the number of boys and girls is the same, and the boys' data does not reach 34 (their upper whisker is lower), but actually, let's think about the median and quartiles. Wait, the key is that the number of boys and girls is equal. For a score above 34: the boys' data has a maximum less than 34? No, looking at the plot, the boys' upper whisker is around, say, 32? Wait, the girls' upper whisker is at 36, boys' at around 32. So the number of girls above 34: since the girls' data extends to 36, and boys' does not go above 32 (approx), so more girls than boys scored above 34? Wait, no, maybe I misread. Wait, the x - axis is ACT scores: 22,24,26,28,30,32,34,36. The boys' box - plot: the upper whisker is at, let's see, the boys' plot: the right whisker is at around 32? And girls' right whisker at 36. So for score above 34: boys have no one (since their max is below 34), girls have some (from 34 to 36). So more girls than boys scored above 34? Wait, but option A says "More girls than boys scored above 34". But let's check other options.

Step3: Analyze Option B

Score above 31: Boys' third quartile (Q3) is around 31? Wait, boys' box: the median of boys is around 30? Wait, no, let's look at the box - and - whisker plot structure. The box for boys is from, say, 24 to 32, median at 30? Girls' box is from 24 to 30, median at 26? Wait, no, the plot: boys' box is longer, from lower whisker (around 22) to upper whisker (around 32), box from 24 to 30, median at 28? Wait, maybe I should re - interpret. The key is that the number of boys and girls is the same. For a value, the proportion above it can be determined by the position in the data.

For option B: More boys than girls scored above 31. Let's see, the boys' data: the upper half (above median) of boys: their scores are from median to upper whisker. The girls' upper half: from their median to upper whisker. Since the boys' upper whisker is higher than girls' in the range above 31? Wait, boys' upper quartile (Q3) is higher than girls' Q3? Wait, the boys' box is shifted to the right. So boys have higher scores on average. So for score above 31: boys' data has more points above 31 because their Q3 is higher, and the number of data points is the same. So more boys than girls scored above 31? Wait, no, maybe I got it wrong. Wait, the correct way: in a box - and - whisker plot, the number of data points above the third quartile (Q3) is 25% (since Q3 is the 75th percentile). But if the Q3 of boys is higher than girls' Q3, then the number of boys above Q3 (which is above 31) would be more. Wait, let's assume the number of boys and girls is \( n \). For boys, the number above 31: if their Q3 is above 31, and girls' Q3 is below 31, then more boys than girls scored above 31. So option B is correct? Wait, no, let's check the options again.

Wait, the original options:

A. More girls than boys scored above 34 on the ACT.

B. More boys than girls scored above 31 on the ACT.

C. More girls than boys scored above 30 on the ACT.

D. More boys than girls scored above 24 on the ACT.

E. More girls than boys scored above 22 on the ACT.

Wait, let's re - analyze:

- The boys' box - plot is shifted to the right, meaning their scores are higher on average. So:

For score above 31: Boys have a higher proportion above 31 because their data is more spread out to the right. So more boys than girls scored above 31 (option B).

For score above 30: Boys' median is around 30? No, girls' box is from 24 to 30, boys' from 24 to 32. So above 30: boys' data has more points above 30 (since their box goes to 32, girls' to 30). Wait, no, girls' upper whisker is 36, but their box is up to 30. So above 30: girls have some (from 30 to 36), boys have from 30 to 32. But the number of boys and girls is the same. Wait, maybe the correct options are B and D? Wait, no, let's think again.

Wait, the key is that the number of boys and girls is equal. So to compare the number above a value, we can use the fact that in a box - and - whisker plot, the median divides the data into two equal parts (50% below, 50% above). The first quartile (Q1) is 25% below, 75% above; third quartile (Q3) is 75% below, 25% above.

- Option B: More boys than girls scored above 31. Since boys' Q3 is above 31 (because their box is from, say, 24 to 32, so Q3 is 32), and girls' Q3 is below 31 (their box is from 24 to 30, Q3 is 30). So 25% of boys are above 32 (which is above 31), and 25% of girls are above 30 (which is below 31). So more boys than girls scored above 31. So B is correct.

- Option D: More boys than girls scored above 24. The Q1 for boys: let's say boys' Q1 is 24, girls' Q1 is 24? Wait, no, boys' box starts at 24, girls' box starts at 24? Wait, the boys' lower whisker is at 22, box from 24 to 32; girls' lower whisker at 22, box from 24 to 30. So above 24: for boys, 75% (since Q1 is 24, so 75% above Q1), for girls, 75% above Q1 (if Q1 is 24). But boys' data is more spread out to the right, so actually, the number of boys above 24: since their median is higher, the proportion above 24 is the same? No, wait, the number of data points is the same. Wait, no, Q1 is the 25th percentile. So above Q1 (24), 75% of the data. But if the boys' data has a higher median, the 75% above 24 for boys: their scores are higher, but the number is the same as girls? No, the number of data points is the same. Wait, maybe I made a mistake.

Wait, the correct answer is:

Brief Explanations:

To solve this, we analyze the box - and - whisker plots (same number of boys and girls).
- **Option A**: Boys’ max score is below 34 (from the plot), but girls’ max is 36. However, the number of girls above 34 is not more (boys have no one above 34, but girls’ upper whisker is short). Incorrect.
- **Option B**: Boys’ third quartile (Q3) is above 31 (their box extends to ~32), while girls’ Q3 is below 31 (box to ~30). Since 25% of data is above Q3, more boys (25% of \( n \)) than girls (25% of \( n \) with Q3 < 31) scored above 31. Correct.
- **Option C**: Boys’ median is above 30 (box shows median ~30 - 32), girls’ median is below 30. So more boys than girls scored above 30. Incorrect.
- **Option D**: Boys’ first quartile (Q1) is 24 (box starts at 24), so 75% of boys scored above 24. Girls’ Q1 is also 24, but boys’ scores are more spread to the right (higher median/quartiles). Wait, no—since the number of boys and girls is equal, and boys’ scores are higher on average, more boys than girls scored above 24. Correct? Wait, no, Q1 is 25% below, so 75% above. If both have Q1 = 24, but boys’ scores are higher, the number above 24 should be the same? No, the plot shows boys’ box is longer to the right, meaning more boys have higher scores (above 24, the spread is larger for boys). So more boys than girls scored above 24. Correct.
- **Option E**: Both have minimum 22, so 100% scored above 22. Incorrect.

Wait, but the initial analysis might have errors. Let's re - check the options. The correct options are **B** and **D**? Wait, no, the original options:

Wait, the correct answer (after proper analysis of box - and - whisker plots, same sample size, comparing proportions above scores):

- **Option B**: More boys than girls scored above 31 (boys’ data is more to the right, so higher proportion above 31).
- **Option D**: More boys than girls scored above 24 (boys’ scores are higher on average, so more above 24).

But let's confirm with the plot:

- Boys’ box: from ~24 to ~32, median ~30.
- Girls’ box: from ~24 to ~30, median ~26.

So:
- Above 31: Boys’ Q3 is ~32, so 25% of boys are above 32 (above 31). Girls’ Q3 is ~30, so 25% of girls are above 30 (below 31). Thus, more boys above 31 (B correct).
- Above 24: Q1 for both is 24, so 75% of each scored above 24. But boys’ scores are higher (median ~30 vs. girls’ ~26), so the number of boys with scores above 24 (and higher) is more (since their scores are more spread to the right). So more boys than girls scored above 24 (D correct).

But the problem says "Select all that apply". So the correct options are **B. More boys than girls scored above 31 on the ACT** and **D. More boys than girls scored above 24 on the ACT**.

To solve this, we analyze the box - and - whisker plots (same number of boys and girls).

• Option A: Boys’ max score is below 34 (from the plot), but girls’ max is 36. However, the number of girls above 34 is not more (boys have no one above 34, but girls’ upper whisker is short). Incorrect.
• Option B: Boys’ third quartile (Q3) is above 31 (their box extends to ~32), while girls’ Q3 is below 31 (box to ~30). Since 25% of data is above Q3, more boys (25% of \( n \)) than girls (25% of \( n \) with Q3 < 31) scored above 31. Correct.
• Option C: Boys’ median is above 30 (box shows median ~30 - 32), girls’ median is below 30. So more boys than girls scored above 30. Incorrect.
• Option D: Boys’ first quartile (Q1) is 24 (box starts at 24), so 75% of boys scored above 24. Girls’ Q1 is also 24, but boys’ scores are more spread to the right (higher median/quartiles). Wait, no—since the number of boys and girls is equal, and boys’ scores are higher on average, more boys than girls scored above 24. Correct? Wait, no, Q1 is 25% below, so 75% above. If both have Q1 = 24, but boys’ scores are higher, the number above 24 should be the same? No, the plot shows boys’ box is longer to the right, meaning more boys have higher scores (above 24, the spread is larger for boys). So more boys than girls scored above 24. Correct.
• Option E: Both have minimum 22, so 100% scored above 22. Incorrect.

Wait, but the initial analysis might have errors. Let's re - check the options. The correct options are B and D? Wait, no, the original options:

Wait, the correct answer (after proper analysis of box - and - whisker plots, same sample size, comparing proportions above scores):

• Option B: More boys than girls scored above 31 (boys’ data is more to the right, so higher proportion above 31).
• Option D: More boys than girls scored above 24 (boys’ scores are higher on average, so more above 24).

But let's confirm with the plot:

• Boys’ box: from ~24 to ~32, median ~30.
• Girls’ box: from ~24 to ~30, median ~26.

So:

• Above 31: Boys’ Q3 is ~32, so 25% of boys are above 32 (above 31). Girls’ Q3 is ~30, so 25% of girls are above 30 (below 31). Thus, more boys above 31 (B correct).
• Above 24: Q1 for both is 24, so 75% of each scored above 24. But boys’ scores are higher (median ~30 vs. girls’ ~26), so the number of boys with scores above 24 (and higher) is more (since their scores are more spread to the right). So more boys than girls scored above 24 (D correct).

But the problem says "Select all that apply". So the correct options are B. More boys than girls scored above 31 on the ACT and D. More boys than girls scored above 24 on the ACT.

Answer:

B. More boys than girls scored above 31 on the ACT, D. More boys than girls scored above 24 on the ACT