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91585 – Worked Solutions - 2025

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QUESTION TWO

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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.

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Question 2 (a)

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Question 2 (b)

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Question 2 (c) i

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Question 2 (c) ii

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Question 2 (c) iii

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Question 2 (d)

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

 

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

 

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

 

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Question 2

In the 2023 School Sports Census:

• Netball had 26 950 registered participants.

• Basketball had 26 572 registered participants.

• Among all the students, it was estimated that 6000 played BOTH netball and basketball.

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

 

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

 

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

 

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Question 2 (a)

  1. If a student is selected at random, and is known to play basketball, what is the probability they play netball?

Give ONE reason why this answer can only be an approximation.

 

This can only be an approximation because the number of players playing both Netball and Basketball was an estimation only, not the actual true number.

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Question 2 (b)

(b) Explain whether the events ‘student plays netball’ and ‘student plays basketball’ are mutually exclusive.

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Question 2 (b)

(b) Explain whether the events ‘student plays netball’ and ‘student plays basketball’ are mutually exclusive.

 

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Question 2 (c)

(c) Concussion is a common concern in school sports, especially after head knocks in contact sports. Concussion assessment tests conducted on the sideline give either a positive or negative result for potential concussion, but they are not always accurate.

A study was conducted with 298 secondary school rugby players who experienced a head knock during games. After the sideline test was used, a medical assessment was done to confirm whether each player actually had a concussion or not. Data from this study is shown in the table below.

​

Had a concussion

Did not have a concussion

​

Positive sideline test

0.074

0.124

​

Negative sideline test

0.131

0.671

​

​

​

​

​

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Question 2 (c)

(c) Concussion is a common concern in school sports, especially after head knocks in contact sports. Concussion assessment tests conducted on the sideline give either a positive or negative result for potential concussion, but they are not always accurate.

A study was conducted with 298 secondary school rugby players who experienced a head knock during games. After the sideline test was used, a medical assessment was done to confirm whether each player actually had a concussion or not. Data from this study is shown in the table below.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

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Question 2 (c) i

Calculate the probability that a randomly chosen rugby player had a concussion or had a positive sideline test.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

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Question 2 (c) i

Calculate the probability that a randomly chosen rugby player had a concussion or had a positive sideline test.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

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Question 2 (c) i

Calculate the probability that a randomly chosen rugby player had a concussion or had a positive sideline test.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

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Question 2 (c) ii

A safety guideline states that “... most head knocks in school rugby do not result in concussion.”

Does the data from this study support this claim?

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

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Question 2 (c) ii

A safety guideline states that “... most head knocks in school rugby do not result in concussion.”

Does the data from this study support this claim?

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

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Question 2 (c) iii

A coach says: “If a player gets a positive sideline test result, there’s a ‘one in four’ chance they actually have a concussion.”

Does the data from this study support this claim?

Support your answer with statistical reasoning.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

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Question 2 (c) iii

A coach says: “If a player gets a positive sideline test result, there’s a ‘one in four’ chance they actually have a concussion.”

Does the data from this study support this claim?

Support your answer with statistical reasoning.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

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Question 2 (c) iii

A coach says: “If a player gets a positive sideline test result, there’s a ‘one in four’ chance they actually have a concussion.”

Does the data from this study support this claim?

Support your answer with statistical reasoning.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

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Question 2 (c) iii

A coach says: “If a player gets a positive sideline test result, there’s a ‘one in four’ chance they actually have a concussion.”

Does the data from this study support this claim?

Support your answer with statistical reasoning.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

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Question 2 (c) iii

A coach says: “If a player gets a positive sideline test result, there’s a ‘one in four’ chance they actually have a concussion.”

Does the data from this study support this claim?

Support your answer with statistical reasoning.

​

Had a concussion

Did not have a concussion

Total

Positive sideline test

0.074

0.124

0.198

Negative sideline test

0.131

0.671

0.802

Total

0.205

0.795

1

 

There is a 37.37% chance that a player with a positive sideline test actually has concussion, and this is greater than the “one in four chance” quoted as one in four = 25%.

​

The coach’s statement is not supported by the data.

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Question 2 (d)

(d) The 2023 School Sports Census of 50 000 students recorded 8880 cricket players. A random sample of 1000 New Zealand secondary school students found that 120 played cricket. Amongst secondary school sports coordinators in New Zealand, it is thought that about 25% of students played cricket.

Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023.

Discuss which estimate is the most reliable for predicting future cricket participation.

Use statistical reasoning to support your answer.

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Question 2 (d)

(d) The 2023 School Sports Census of 50 000 students recorded 8880 cricket players. A random sample of 1000 New Zealand secondary school students found that 120 played cricket. Amongst secondary school sports coordinators in New Zealand, it is thought that about 25% of students played cricket.

Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023.

Discuss which estimate is the most reliable for predicting future cricket participation.

Use statistical reasoning to support your answer.

 

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Question 2 (d)

(d) The 2023 School Sports Census of 50 000 students recorded 8880 cricket players. A random sample of 1000 New Zealand secondary school students found that 120 played cricket. Amongst secondary school sports coordinators in New Zealand, it is thought that about 25% of students played cricket.

Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023.

Discuss which estimate is the most reliable for predicting future cricket participation.

Use statistical reasoning to support your answer.

 

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Question 2 (d)

(d) The 2023 School Sports Census of 50 000 students recorded 8880 cricket players. A random sample of 1000 New Zealand secondary school students found that 120 played cricket. Amongst secondary school sports coordinators in New Zealand, it is thought that about 25% of students played cricket.

Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023.

Discuss which estimate is the most reliable for predicting future cricket participation.

Use statistical reasoning to support your answer.

 

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Question 2 (d)

(d) The 2023 School Sports Census of 50 000 students recorded 8880 cricket players. A random sample of 1000 New Zealand secondary school students found that 120 played cricket. Amongst secondary school sports coordinators in New Zealand, it is thought that about 25% of students played cricket.

Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023.

Discuss which estimate is the most reliable for predicting future cricket participation.

Use statistical reasoning to support your answer.