Mathematics 9709/73 — May/June 2017
Cambridge A-Level · Probability & Statistics 2 · worked solutions for every part, with the mark scheme
Topics Sampling and Estimation · Hypothesis Tests · Linear Combinations of Random Variables · The Poisson Distribution · Continuous Random Variables
A residents’ association has 654 members, numbered from 1 to 654. The secretary wishes to send a questionnaire to a random sample of members. In order to choose the members for the sample she uses a table of random numbers. The first line in the table is as follows.
1096 4357 3765 0431 0928 9264
The numbers of the first two members in the sample are 109 and 643. Find the numbers of the next three members in the sample.
Approach
Since the members are numbered from 1 to 654, each member is represented by a 3-digit number from 001 to 654. Group the random digits into blocks of 3, reject any block greater than 654, and reject any repeated number. The first two accepted blocks are 109 and 643.
Working
Group the random digits into 3-digit blocks:
109 | 643 | 573 | 765 | 043 | 109 | 289 | 264
The first two valid numbers are already given:
109, 643
Continue through the blocks:
- 573 is valid
- 765 is greater than 654, so reject it
- 043 is valid (member 43)
- 109 is a repeat, so reject it
- 289 is valid
Therefore the next three members are 573, 43 (or 043) and 289.
Answer
573, 43 (or 043), 289
573, 43 (or 043), 289
Walkthrough
The secretary needs a random sample from members numbered 1 to 654. Since 654 has three digits, every member can be identified by a three-digit number, with leading zeros used for numbers below 100, for example member 43 is 043. We therefore read the random number table in three-digit blocks. We ignore any block above 654 because no member has that number, and we ignore repeats because the same member cannot be chosen twice. The first two blocks that satisfy these rules are 109 and 643. Continuing, the next block is 573, which is valid. Then 765 is too large, so it is skipped. Next 043 corresponds to member 43. Then 109 appears again, so it is skipped as a repeat. Finally 289 is valid, so it is the third new member.
Key Takeaways
This question tests the practical use of a random number table to obtain a simple random sample. The key skills are: matching the numbering system to the digits in the table, rejecting numbers outside the required range, and avoiding duplicates.
Common Mistakes
A common mistake is to read the table as four-digit numbers instead of three-digit blocks. Another is to include 765, which is greater than 654. A further mistake is to include the second 109, forgetting that a member cannot be selected twice. Some students may also write 043 as 43; this is acceptable.
Things to Be Careful About
Remember that member numbers run from 1 to 654, so 000 would also be invalid. Leading zeros are part of the three-digit representation, so 043 means member 43. The mark scheme allows other correct uses of the table as long as the same consistent rule is applied, but the standard answer is 573, 43 (or 043), 289.
The rest of this paper
6 more questions- Q2Sampling and Estimation4M
- Q3Linear Combinations of Random Variables5M
- Q4Sampling and Estimation · Hypothesis Tests8M
- Q5The Poisson Distribution9M
- Q6Continuous Random Variables10M
- Q7Hypothesis Tests11M