1. 1
   The times, $t$ minutes, that 80 students spent on homework one evening are grouped as follows: $0 \le t < 20$: 6 students; $20 \le t < 40$: 18 students; $40 \le t < 60$: 30 students; $60 \le t < 80$: 16 students; $80 \le t < 100$: 10 students.

   (a)

   Which class contains the median time?

   [1 mark]

   • A$20 \le t < 40$
   • B$40 \le t < 60$
   • C$60 \le t < 80$
   • D$0 \le t < 20$

   (b)

   Find the number of students who spent at least 60 minutes on homework.

   [1 mark]

   • A16
   • B70
   • C26
   • D54

   (c)

   Use linear interpolation to estimate the percentage of students who spent less than 50 minutes on homework.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A frequency histogram shows the lengths, $L$ cm, of 50 fish caught in a survey. The classes have equal width 5 cm, the first class starting at 10 cm, and the heights of the six bars, in order of increasing length, are 4, 9, 15, 12, 7 and 3.

   (a)

   Which inequality describes the fourth class?

   [1 mark]

   • A$25 \le L < 30$
   • B$25 < L < 30$
   • C$25 \le L \le 30$
   • D$30 \le L < 35$

   (b)

   Find the percentage of the fish that are at least 30 cm long.

   [1 mark]

   • A14%
   • B10%
   • C44%
   • D20%

   (c)

   Two more fish, of lengths 30 cm and 41.5 cm, are added to the data. Describe how the histogram changes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The masses, $m$ grams, of 200 apples were recorded. The cumulative frequencies are: $m < 100$: 8; $m < 120$: 30; $m < 140$: 90; $m < 160$: 150; $m < 180$: 186; $m < 200$: 200. Assume the masses are spread evenly within each class.

   (a)

   Write down the number of apples with a mass of at least 140 g but less than 160 g, and estimate the median mass.

   [3 marks]

   (b)

   Apples heavier than the 90th percentile are sold as 'premium'. Estimate the minimum mass of a premium apple, and explain why your answer is only an estimate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A café records the waiting times, in seconds, of 120 customers at Counter A and 120 customers at Counter B. Counter A: minimum 40, lower quartile 95, median 120, upper quartile 145, maximum 230; the largest waiting time at Counter A other than 230 is 200. Counter B: minimum 35, lower quartile 60, median 80, upper quartile 130, maximum 210.

   (a)

   (i) Show that 230 seconds is an outlier for Counter A.
   (ii) Determine whether Counter B has any outliers.

   [6 marks]

   (b)

   (i) With reference to symmetry, state which counter's waiting times could reasonably be modelled by a normal distribution. Justify your answer.
   (ii) The café manager claims that customers are served faster at Counter B. Compare the two distributions, and discuss the manager's claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions