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Sampling and the large data set questions
You can rarely measure everyone, so you measure some and reason about the rest. This lesson compares the sampling methods in terms of bias, representativeness, practicality and the information each one needs before you can use it, and sets out what the exam's own large data set actually holds.
7 original questions · 23 marks · the sampling and the large data set notes · Statistics
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
Define the terms population, census and sample.
Worked answer
The population is every member of the group under study. A census measures all of them. A sample measures a chosen subset and uses it to say something about the whole. B1 B1 B1 for the three definitions. A census is complete but slow and expensive, and for destructive testing it is impossible. A sample trades completeness for practicality.State what is needed before a simple random sample can be taken, and the property every member must have.
Worked answer
You need a sampling frame, which is a numbered list of the whole population. Every member must then have an equal chance of selection, and every possible sample of that size must be equally likely. B1 for the sampling frame, B1 for the equal chance of selection. Random numbers or a lottery draw deliver that; picking whoever is nearest never does.A college has 720 first-year and 480 second-year students. Describe how to take a sample of 50, stratified by year group.
Worked answer
Do the proportions first. First years supply 50 × 720/1200 = 30 students and second years 50 × 480/1200 = 20, and the two rebuild the sample size of 50. Then, within each year group, number the students and take a simple random sample of the required size using random numbers. M1 for the proportions, A1 for 30 and 20, B1 for numbering within each year group, B1 for the random selection from each. Both halves earn marks, and candidates who do the arithmetic and stop lose the second half every time. Stratifying is worth the trouble here because it guarantees the sample matches the college's 60:40 split, whereas a simple random sample matches it only on average and one unlucky draw could badly over-represent a year.Describe how to take a systematic sample of 60 from a payroll list of 1200 workers.
Worked answer
The interval is 1200/60 = 20. Choose a random starting point between 1 and 20, then take every 20th name from there. B1 for the interval, B1 for the random start, B1 for the rule. The random start is what keeps the method fair; always beginning at the first name would not be. State the interval, the random start and the rule, because all three are needed before the sample can be reproduced.A researcher stands outside a gym at 7 am and questions people about their exercise habits. Name this sampling method and explain the bias.
Worked answer
Opportunity sampling, also called convenience sampling, since it takes whoever happens to be available. The people arriving at a gym at 7 am exercise far more than the population the survey is meant to describe, so every estimate drawn from them overstates activity levels in the same direction. B1 for naming the method, B1 for the group over-represented, B1 for the direction of the bias. Naming the over-represented group is what earns the explanation mark. Ease of collection is the method's only advantage.A student calculates the mean daily rainfall at Heathrow over May to October 2015 from the large data set, and concludes that rainfall across the UK is increasing. Give two reasons why the conclusion is unsafe, and state how values recorded as tr and n/a should be treated.
Worked answer
First, Heathrow is one of only five UK stations in the data set, and one site in west London cannot speak for the whole country. Camborne and Leuchars have very different climates. Second, the claim is about a change over time, yet only 2015 has been used. Without the matching 1987 figures there is nothing to compare, and two summers 28 years apart would in any case be too little to establish a trend. A third reason worth having: each period runs only from May to October, so half of every year is missing and nothing follows about winter rainfall. As for the awkward entries, tr means a trace, a measurable fall below 0.05 mm, so it should be entered as zero or as a very small value rather than skipped. Entries of n/a are missing data and must be left out of the calculation altogether, which reduces the number of days the mean is divided by. B1 for the first reason, B1 for supporting it with the other stations, B1 for the second reason, B1 for supporting it with the missing earlier data, B1 for the treatment of tr and n/a. Silently treating either as zero, or as an ordinary reading, shifts the mean and is exactly the slip these questions are set to catch.A survey about internet reliability is run entirely through an online form. Explain the sampling problem, and its likely effect on the results.
Worked answer
The sampling frame excludes the very people the survey most needs, because anyone with poor or no internet access cannot reach the form. The responses will therefore overstate how reliable the internet is. No increase in sample size repairs this, since the missing group is missing systematically rather than by chance. B1 for the frame excluding those with poor access, B1 for the responses overstating reliability, B1 for a larger sample not helping.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise sampling and the large data set one question at a time
The player marks nothing for you. It shows one question, waits, then shows the worked answer so you can mark yourself, and brings a question back sooner when it went badly.