Sampling Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Sampling questions. See exactly how to solve problems on population, definitions, sample, census.

populationdefinitionssamplecensussampling framesystematic
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
In a study of the heights of all 12001200 students at a college, state what is meant by the \textbf{population} in this context.

Worked solution

  1. Recall what a population means

    Population=every member of the group being studied\text{Population} = \text{every member of the group being studied}

    In statistics the population is the whole group we are interested in, not just the people we measure. It is the complete set we want to draw conclusions about.

  2. Identify the group in this study

    Group studied=students at the college\text{Group studied} = \text{students at the college}

    The study is about the heights of the college students, so the group of interest is those students. That tells us what the population must be.

  3. State the size and conclusion

    N=1200N = 1200

    There are 1200 students, and every one of them is part of the group being studied. So the population is all 1200 students.

Answer
The population is all 12001200 students at the college.
Question 2
2 markseasy
A researcher measures the heights of 5050 of the 12001200 students. Which statement best describes this set of 5050 students?

Worked solution

  1. Recall what a sample is

    Sample=a selected subset of the population\text{Sample} = \text{a selected subset of the population}

    A sample is a smaller group chosen from the population. We study the sample to learn about the whole population.

  2. Compare with the population

    50<120050 < 1200

    Here 50 is only part of the 1200 students, so it is not the whole population. That makes it a subset.

  3. Conclude

    These 50 students form a sample\text{These } 50 \text{ students form a sample}

    Because it is a selected part of the population, the 50 students are a sample. This is the correct description.

Answer
A sample: a selected subset of the population.
Question 3
2 markseasy
What is meant by a \textbf{census}?

Worked solution

  1. Recall the meaning of census

    Census=observe every member of the population\text{Census} = \text{observe every member of the population}

    A census collects data from every single member of the population. Nobody is left out.

  2. Contrast with a sample

    Sample⊊Population\text{Sample} \subsetneq \text{Population}

    A sample only looks at part of the population, but a census looks at all of it. This is the key difference.

  3. State the definition

    Census: measure the whole population\text{Census: measure the whole population}

    So a census is a survey of the complete population. That is the correct statement.

Answer
A census observes or measures every member of the population.
Question 4
2 markseasy
A \textbf{sampling frame} is used when selecting a sample. What is a sampling frame?

Worked solution

  1. Recall the idea of a frame

    Frame=a list of the population\text{Frame} = \text{a list of the population}

    To choose a sample fairly we usually need a list of everyone we could pick. That list is the sampling frame.

  2. Give an example

    e.g. a register of all students\text{e.g. a register of all students}

    A school register or an electoral roll are common sampling frames. We select our sample from such a list.

  3. State the definition

    Frame lists all sampling units\text{Frame lists all sampling units}

    So a sampling frame is the list of all members of the population we can sample from. That is the correct answer.

Answer
A list of all the members of the population from which the sample is taken.
Question 5
2 markseasy
A population of 200200 members is to be sampled systematically to give a sample of size 2525. Find the sampling interval kk.

Worked solution

  1. Write down the known values

    N=200,n=25N = 200,\quad n = 25

    N is the size of the whole population and n is the size of the sample we want. We need the gap between chosen members.

  2. Recall the interval formula

    k=Nnk = \frac{N}{n}

    In systematic sampling we pick every k-th member. The interval k is the population size divided by the sample size.

  3. Substitute and evaluate

    k=20025=8k = \frac{200}{25} = 8

    Dividing 200 by 25 gives 8. So we choose every 8th member of the list.

Answer
k=8k = 8

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