Questions on sampling and the large data set

Question 1

Define the terms population, census and sample.

Accepted 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.

Question 2

State what is needed before a simple random sample can be taken, and the property every member must have.

Accepted 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.

Question 3

State two ways in which a quota sample differs from a stratified sample.

Accepted Answer

A stratified sample selects at random inside each stratum, while a quota sample lets the interviewer take whoever is available until the quota is filled, so it is not random. B1 for the selection. A stratified sample also needs a sampling frame and the size of each stratum known in advance, and a quota sample needs neither. B1 for the frame and the stratum sizes. The two samples can carry identical proportions and the difference still matters, because only random selection makes a sample unbiased by construction.

Question 4

State the structure of the Pearson Edexcel large data set: how many recording stations it holds and where they are, which months of the year it covers, and which two years.

Accepted Answer

Eight recording stations: five in the UK, at Camborne, Heathrow, Hurn, Leeming and Leuchars, and three overseas, at Beijing, Jacksonville and Perth. B1 for eight stations split five and three. The records run from May to October only, so half of each year is absent. B1 for the six months. The two years are 1987 and 2015. B1 for both years. Perth is in the southern hemisphere, so its May-to-October records cover winter while the UK stations' cover summer, and a comparison that overlooks that is setting two different seasons against each other.

Question 5

A college has 720 first-year and 480 second-year students. Describe how to take a sample of 50, stratified by year group.

Accepted 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. 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.

Question 6

Describe how to take a systematic sample of 60 from a payroll list of 1200 workers.

Accepted 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.

Question 7

A researcher stands outside a gym at 7 am and questions people about their exercise habits. Name this sampling method and explain the bias.

Accepted 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. Ease of collection is the method's only advantage.

Question 8

A hospital employs 450 nurses, 180 doctors and 270 support staff. Describe how to take a sample of 40 stratified by job type, and state one advantage of stratifying here rather than taking a simple random sample of 40.

Accepted Answer

Proportions first. The hospital employs 450 + 180 + 270 = 900 people, so the nurses supply 40 × 450/900 = 20, the doctors 40 × 180/900 = 8 and the support staff 40 × 270/900 = 12. M1 for the fractions, A1 for 20, 8 and 12, which rebuild the sample size of 40. Then number the members of each group and draw a simple random sample of the required size from within each, using random numbers. B1 for numbering and random selection inside every stratum. The advantage is that the sample is certain to carry the hospital's 5 : 2 : 3 split by job type, whereas a simple random sample matches it only on average and one unlucky draw could hold very few doctors, who are the smallest of the three groups. B1 for the advantage tied to this population.

Question 9

A factory line produces 600 items in a day, and a sample of 50 is taken systematically from a numbered list of them after a random start. The machine has 12 heads working in rotation, so every twelfth item on the list comes from the same head. Explain why this systematic sample is unsatisfactory, and state one way to repair it.

Accepted Answer

The interval is 600/50 = 12, which is exactly the machine's cycle. B1 for the interval matching the period of the pattern. Every item selected therefore comes from the same head, so the sample describes one head rather than the machine, and a fault on any of the other eleven would go unseen however large the sample grew. B1 for the group left out. The repair is to take a simple random sample instead, or to use an interval that has no factor in common with the cycle length. B1 for a repair.

Question 10

A market researcher is told to interview 20 men and 20 women outside a supermarket on a Tuesday morning. Name the sampling method, give one advantage of it here, and name a group the sample under-represents.

Accepted Answer

Quota sampling: the number wanted from each group is fixed in advance and the interviewer fills it from whoever is available. B1 for the method named. Its advantage here is that it needs no sampling frame and is quick and inexpensive, which matters because no list of the town's shoppers exists to sample from. B1 for an advantage tied to this survey. It under-represents people in full-time employment, who cannot be outside a supermarket on a Tuesday morning, so the sample leans towards those who are retired, work at home or are not employed. B1 for the group named. The quota fixes the balance of men and women and leaves every other imbalance in the sample untouched.

Question 11

A club has 96 members. Describe how to take a simple random sample of 8 of them using two-digit random numbers, and state what to do with a random number of 97, 98, 99 or 00, and with a number that has already been drawn.

Accepted Answer

Number the members 01 to 96, one number each, from a complete list of the membership. B1 for numbering a complete frame. Read two-digit random numbers from a generator or a table and take the member carrying each number, stopping once eight members have been chosen. B1 for the selection rule. A number of 97, 98, 99 or 00 names nobody, so ignore it and read the next; discarding those four keeps every member's chance of selection equal at 1 in 96. B1 for ignoring them. A repeat is ignored in the same way and the next number taken, because the sample is drawn without replacement and one member cannot fill two of the eight places. B1. Reassigning the four unused numbers to members already on the list would give those members two chances each, which breaks the equal-chance condition the method rests on.

Question 12

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.

Accepted 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 fall too small to measure (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. Treating n/a as zero, or tr as missing, or either as an ordinary reading, shifts the mean, so state how each is coded before calculating.

Question 13

A survey about internet reliability is run entirely through an online form. Explain the sampling problem, and its likely effect on the results.

Accepted 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.

Question 14

A student uses the large data set to compare the mean daily maximum gust at Leuchars in 1987 with the mean daily maximum gust at Leuchars in 2015, and reports that the wind at Leuchars has changed. Give three reasons why the comparison may not support that conclusion, and state one check to make on the 1987 records before starting.

Accepted Answer

First, each period runs only from May to October, so the two means describe half a year each, and nothing follows about winter, when the strongest gusts at a Scottish coastal station would be expected. B1. Second, two single years are two observations rather than a trend: weather varies from one year to the next, and a difference between 1987 and 2015 could be ordinary variation rather than change. B1. Third, days recorded as n/a are missing and must be left out of each mean, so the two means may rest on different numbers of days. B1 for the missing data, B1 for the divisors differing and needing to be stated alongside the means. The check to make before starting is whether the 1987 records carry that variable at all: the 1987 data hold fewer variables than the 2015 data, and no comparison can be made from a column that is not there. B1 for checking the 1987 variables.

Question 15

Describe how to take a sample of 30 daily rainfall records for Heathrow in 2015 from the large data set, stratified by month, stating the sampling frame and how many days you would take from each month. Give one reason why taking the first 30 days of the period instead would be worse.

Accepted Answer

The sampling frame is the list of daily rows for Heathrow in 2015, one row for each day from 1 May to 31 October, which is 31 + 30 + 31 + 31 + 30 + 31 = 184 days. B1 for the frame described as the dated daily records, with its 184 days. Stratifying by month, each month supplies 30 × (days in that month)/184, which is 5.05 for a 31-day month and 4.89 for a 30-day one. M1 for the stratified fractions. Every one of the six rounds to 5, and 6 × 5 = 30, so take 5 days from each month. A1 for five days per month totalling 30. Within each month, number the days and select 5 of them at random using random numbers. B1 for random selection inside each stratum. Taking the first 30 days instead would take 1 to 30 May and none from the other five months, so the sample would describe one month's weather and say nothing about May to October: rainfall varies with the season, and that is the variation the stratification is there to cover. B1 for the reason.