1. 1
   A school has 600 Diploma Programme students: 330 in DP1 and 270 in DP2. The head of year wants a sample of 40 of these students to answer a questionnaire about workload.

   (a)

   The sample is stratified by year group, in proportion to the size of each year group. Find the number of DP1 students in the sample.

   [1 mark]

   • A20
   • B33
   • C18
   • D22

   (b)

   Instead, the 600 names are listed alphabetically and a systematic sample of 40 is taken. Find the sampling interval $k$ (every $k$th student is chosen after a random start).

   [1 mark]

   • A15
   • B40
   • C$\frac{1}{15}$
   • D560

   (c)

   Instead, the head of year gives the questionnaire to the first 40 DP students to enter the library one morning. Name this sampling technique and explain why it is likely to be biased in this context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In a reaction-time experiment, 15 volunteers each pressed a button as soon as a light came on. Their times, in milliseconds (ms), had a lower quartile of 220 ms, a median of 238 ms and an upper quartile of 260 ms. The shortest time was 205 ms and the longest was 330 ms.

   (a)

   Find the interquartile range of the times.

   [1 mark]

   • A125 ms
   • B20 ms
   • C40 ms
   • D18 ms

   (b)

   Find the value above which a time is classified as an outlier.

   [1 mark]

   • A300 ms
   • B320 ms
   • C298 ms
   • D380 ms

   (c)

   The 330 ms time belongs to a volunteer who was looking at their phone when the light came on. Determine whether 330 ms is an outlier, and state, with a reason, whether it should be removed from the data.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A researcher collects data from 50 households. For each household she records the number of people living there and the electricity used in one week, measured in kilowatt-hours (kWh).

   (a)

   Classify each of the two variables as discrete or continuous, and justify your classifications.

   [3 marks]

   (b)

   One electricity value was recorded as $-35$ kWh and three households have no electricity value recorded. The mean of the other 46 values is 210 kWh. The researcher's assistant includes the $-35$, records the three missing values as 0 and divides the total by 50.
   Find the mean the assistant obtains, and comment on the assistant's method.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A city council wants to estimate the mean daily commuting time of its 48 000 adult residents. 18 000 of them live in the North district and 30 000 live in the South district. The council plans a stratified sample of 400 residents, stratified by district.

   (a)

   (i) Show that the sampling fraction is $\frac{1}{120}$, and hence find the number of residents to be sampled from each district.
   (ii) A councillor suggests a quota sample instead: an interviewer stands at the main railway station one weekday morning and questions the first 150 North and the first 250 South residents who pass. Give two reasons why this sample is likely to give a biased estimate of the mean commuting time.
   (iii) State one practical difficulty in taking a simple random sample of 400 residents from the whole city.

   [6 marks]

   (b)

   The stratified sample gives a mean commuting time of 42 minutes for the North residents and 30 minutes for the South residents.
   (i) Find an estimate of the mean commuting time for all adult residents of the city.
   (ii) A simple random sample of 400 residents happens to contain 100 North and 300 South residents. Assuming the same mean for each district, find the mean commuting time this sample would give.
   (iii) Hence explain why the stratified sample is more suitable for this investigation.

   [6 marks]

   Total for question 4: 12 marks

End of questions