91585 – Worked Solutions - 2025
QUESTION ONE
3.5 (AS 91577) – 2025 Answers
Quite often, there are different ways to solve a Mathematics problem.
These solutions/strategies show one possible way to solve them.
Question 1
School Sport New Zealand organises and promotes secondary school sports tournaments. Since 2000, they have collected annual data on student participation by sport, region, and gender. The most recent 2023 School Sports Census covered 91 different sports, and surveyed a total of 144 862 students representing their school in at least one sport.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
Question 1(a)
We will use a Venn Diagram to solve this problem.
Question 1(a)
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
Question 1(a)
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
Question 1(a)
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
We will use a Venn Diagram to solve this problem.
Follow-up surveys in some regions have been used to estimate the numbers of students who participated in the following combinations of sports:
• badminton and table tennis: 800
• table tennis and hockey: 600
• badminton and hockey: 1100
• badminton, table tennis, and hockey: 500
We will use a Venn Diagram to solve this problem.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
We will use a Venn Diagram to solve this problem.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
We will use a Venn Diagram to solve this problem.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
We will use a Venn Diagram to solve this problem.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
We will use a Venn Diagram to solve this problem.
The 2023 School Sports Census showed:
• badminton: 11 195 participants
• table tennis: 2125 participants
• hockey: 13 304 participants
We will use a Venn Diagram to solve this problem.
School Sport New Zealand organises and promotes secondary school sports tournaments. Since 2000, they have collected annual data on student participation by sport, region, and gender. The most recent 2023 School Sports Census covered 91 different sports, and surveyed a total of 144 862 students representing their school in at least one sport.
We will use a Venn Diagram to solve this problem.
School Sport New Zealand organises and promotes secondary school sports tournaments. Since 2000, they have collected annual data on student participation by sport, region, and gender. The most recent 2023 School Sports Census covered 91 different sports, and surveyed a total of 144 862 students representing their school in at least one sport.
Total students in the survey = 144 862
Question 1(a)
Estimate the probability that a randomly selected student from the 2023 School Sports Census participates in only one of these three sports (badminton, table tennis, and hockey).
Total students in the survey = 144 862
Question 1(a)
Estimate the probability that a randomly selected student from the 2023 School Sports Census participates in only one of these three sports (badminton, table tennis, and hockey).
Total students in the survey = 144 862
Question 1(a)
Estimate the probability that a randomly selected student from the 2023 School Sports Census participates in only one of these three sports (badminton, table tennis, and hockey).
Total students in the survey = 144 862
Question 1(a)
Estimate the probability that a randomly selected student from the 2023 School Sports Census participates in only one of these three sports (badminton, table tennis, and hockey).
Total students in the survey = 144 862
Fraction, decimal, percentage are accepted
Question 1(b)
A student is randomly selected from those who play hockey.
Estimate the probability that they play both hockey and table tennis, but not badminton.
Total students in the survey = 144 862
Question 1(b)
A student is randomly selected from those who play hockey.
Estimate the probability that they play both hockey and table tennis, but not badminton.
Total students in the survey = 144 862
Question 1(b)
A student is randomly selected from those who play hockey.
Estimate the probability that they play both hockey and table tennis, but not badminton.
Total students in the survey = 144 862
Question 1(b)
A student is randomly selected from those who play hockey.
Estimate the probability that they play both hockey and table tennis, but not badminton.
Total students in the survey = 144 862
Question 1(b)
A student is randomly selected from those who play hockey.
Estimate the probability that they play both hockey and table tennis, but not badminton.
Total students in the survey = 144 862
Fraction, decimal, percentage are accepted
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
These are close, but still different enough to suggest they are not independent events, hence they are dependent events.
Question 1(c)
Interpret the relationship between the events ‘student plays badminton’ and ‘student plays hockey’.
As part of your answer, explain whether these two events are independent.
Use calculations and statistical reasoning to support your answer.
Total students in the survey = 144 862
These are close, but still different enough to suggest they are not independent events, hence they are dependent events.
Question 1(d) i
A school is comparing the risk of football injuries on two different playing surfaces: artificial turf and natural grass. Over a season, data from football matches played at this school showed:
• 80% of the games were played on artificial turf.
• The rest of the games were played on natural grass.
• On artificial turf, the probability of a player getting injured in a game was 0.04.
• On natural grass, the probability of injury was 0.10.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Question 1(d) i
We will use a Tree Diagram
to solve this problem.
Fraction, decimal, percentage are accepted
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Question 1(d) ii
(ii) The school claims that, in general, for students playing football, more injuries occur playing on artificial turf than on natural grass.
Discuss why care should be taken using this data to make this claim.
Care should be taken with generalizing this data to all students playing football in general, because;
Discusses impact of any of these TWO issues on injury rates. Others may be considered.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
Question 1(d) iii
(iii) Estimate the probability that, from a group of three randomly selected football players at this school, only one is injured during the game.
State and justify and assumption(s) that you have made in calculating this probability.
In order for this probability to be true, we do need to assume that there is NO dependence on one player getting injured with other players being injured, ie the events are independent of each other.
Another assumption we need to make is that the injury rate remains the same for each player.