SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Statistics, STD2 2026 HSC 36

A group of students participated in a coding challenge. The times taken to complete the challenge are normally distributed with a mean of 60 minutes and a standard deviation of 10 minutes.

  1. In a random sample of 2000 students, how many students would be expected to take between 40 and 70 minutes to complete the challenge?   (2 marks)
  2. Only the fastest three students of the 2000 students qualified for a national challenge.
  3. What was the maximum time needed to qualify?   (2 marks)
Show Answers Only

a.    \(\text{Approximately }1630\text{ students}\)

b.    \(30\ \text{minutes}\)

Show Worked Solution

a.    \(z\text{-score (40)} = \dfrac{40-\mu}{\sigma} = \dfrac{40-60}{10} = -2\)

\(z\text{-score (70)} = \dfrac{40-\mu}{\sigma} = \dfrac{70-60}{10} = 1\)

\(\text{68% of times have a $z$-score between $-1$ and $1$.}\)

\(\text{13.5% of times have a $z$-score between 1 and 2.}\)

\(\Rightarrow \text{% of times with a $z$-score between $-1$ and $2$} = 68+13.5 = 81.5\%\)
 

\(\text{Expected number}=2000\times 0.815 = 1630\ \text{students}\)
 

b.    \(\dfrac{3}{2000}\times100\%=0.15\%\)

\(\text{Since 99.7% of times have a $z$-score between $-3$ and $3$,}\)

\(\text{0.15% of times are below a $z$-score of $-3$.}\)

\(\text{Find time with $z$-score = $-3$:}\)

\(-3\) \(=\dfrac{x-60}{10}\)  
\(x-60\) \(=-3 \times 10\)  
\(x\) \(=30\)  

 
\(\therefore\ \text{Maximum qualifying time}= 30\ \text{minutes}\)