• A Level Maths

Demo

• A Level
• Sampling

1. Explain the terms population, census, and sample.

A population includes all/every person/item/thing, and a census measures/asks every member of a population
A sample is a subset of a census and measures/asks only some members of a population.

2. Suggest an advantage for using a census, and one for using a sample.

A census is guaranteed to be accurate. A sample is easier to collect (quicker/cheaper).

3. A factory wants to find out the average number of seeds its apples contain. Suggest why using a census to collect this information is inappropriate.

Apples are destroyed in order to find the number of seeds they contain and cannot be sold afterwards.

4. What are the similarities and differences between quota sampling and stratified random sampling?

Both are representative of the population since the population is divided into groups/strata, but quota sampling is not random.

5. A teacher wants to know how well their class of 25 students did on a test. After marking two tests, both of which scored 100%, the teacher concludes that the class did extremely well in the test.

1. Is the teacher correct?
2. How could the teacher be more confident in their conclusion?

1. The teacher could be correct, but the sample size is too small to know for sure.
2. The teacher could mark more tests.

6. Each student at a school is given a number from 1 to 1000 by how old they are, with the oldest student given the number 1. A school selects a sample of the students by choosing every student whose number ends in 1. Explain whether this is a systematic sample.

No - a systematic sample selects the first person at random.

7. A year 9 student is carrying out a survey in front of a supermarket to find out how much shoppers spend by asking the first 5 people who leave the shop.

1. What is the name of the type of sampling used by the pupil?
2. Suggest an advantage and a disadvantage of this type of sampling.
3. Suggest two ways the student could improve their data collection method while still using the same sampling method.

1. Opportunity sampling
2. It is very easy to do, but it is not a random sample and can be biased.
3. Come back at a different time of day to ask more people, visit a different shop and ask people there.

8. The organiser of a school disco wants to find out what types of music the students like. It is thought that boys and girls have different preferences for music. There are 120 boys and 80 girls at the school, and the organiser wants a sample of 14 students.

1. What type of sampling method should the organiser use?
2. Explain how the organiser should use this method to obtain a sample.

1. Stratified sampling
2. 40% of the students are girls, so 40% of the sample (6) should be girls. The organiser should randomly select (e.g. by numbering the students and generating a random number) 6 girls and 8 boys.

9. The first 5 students exiting the dining hall were asked to rate their lunch on the scale of 1 to 10. Their ratings are: 7 3 9 4 5

1. Find the average rating of the school lunch that day.
2. What is the name of the sampling method used?
3. Suggest two improvements to improve the reliability of the data.

1. $5.6$
2. Opportunity sampling
3. Ask more people, ask people at different times

10. A school wants to know how long their 200 year 12 students spend doing homework on average, by sampling 20 students.

1. Describe how the school could use random sampling to take a sample.
2. Describe how the school could use systematic sampling to take a sample

1. Each student can be assigned a number from 1 to 200. A random number generator can be used to find 20 students for random sampling.
2. The students could be listed by some random way (e.g. by giving them a random number) - NOT alphabetically. A random student from the first 20 students can be selected, and then every 20 students after that can be selected for systematic sampling.

11. The head of sixth form wants to find out whether students are satisfied with their teachers. There are 125 year 12 students and 145 year 13 students. In year 12, there are 70 boys and 55 girls, and in year 13, there are 70 boys and 75 girls. Describe a suitable method to randomly sample 30 students.

There are a total of 270 students. We need $\frac{125}{270}\times 30 = 14$ year 12s and $16$ year 13s. Of the year 12s, we need $\frac{70}{125}\times 14 = 8$ boys and $6$ girls, and in year 13, we need $\frac{70}{145}\times 16 = 8$ boys and $8$ girls. Students from each year group should be split into boys and girls, and given a number. A random number generator should be used to select the appropriate number of students.

12. On Monday, 50 students were late to school. A teacher standing at the entrance used opportunity sampling to find the first 5 students who arrived late, and found that, on average, students are only 3 minutes late.

1. Suggest whether this is likely to be a good estimate.
2. Suggest a more appropriate sampling method to obtain a sample size of 5.

1. The first five students are the least late students, so the average late student is likely to be significantly later.
2. The teacher could write down how late all the students are, and use random sampling or systematic sampling to find 5 students.