10 exam-style questions on sampling methods, bias, types of data and statistical diagrams. Tap Show answer for full working.
Define a simple random sample. Give one advantage and one disadvantage of using simple random sampling.
Every possible sample of the required size has an equal chance of being selected. Advantage: unbiased / representative in theory. Disadvantage: needs a full sampling frame; can be time-consuming or impractical for large populations.
Distinguish between discrete and continuous data. Classify each of the following: number of siblings, height, shoe size, time taken to run 100 m.
Discrete: countable (usually integer). Continuous: measurable on a continuous scale. Number of siblings — discrete; height — continuous; shoe size — discrete (or ordered categorical); time — continuous.
What is meant by a census? Give one reason why a census might be preferred to a sample, and one reason why a sample might be preferred.
A census collects data from every member of the population. Census preferred for accuracy / complete information. Sample preferred for lower cost, speed, or when testing destroys the item.
A school has 600 students: 200 in Year 12 and 400 in Year 13. A stratified sample of 30 students is required. How many should be selected from each year?
Year 12: \( \dfrac{200}{600}\times30=10 \). Year 13: \( \dfrac{400}{600}\times30=20 \).
Explain what is meant by systematic sampling. Describe how you would select a systematic sample of 20 students from a list of 400.
Select every \( k \)-th member of a list after a random start. Here \( k=400/20=20 \). Choose a random start between 1 and 20, then take every 20th student.
A researcher stands outside a supermarket on a Monday morning and asks the first 50 people who leave about their shopping habits. Identify two sources of bias in this sampling method.
1. Only shoppers at that supermarket / time / day are included (not representative of all shoppers or the wider population). 2. Voluntary response / interviewer selection may favour certain demographics; people in a hurry may refuse.
Describe cluster sampling and give one advantage and one disadvantage compared with stratified sampling.
Population divided into clusters; a random sample of clusters is taken and all (or a sample of) members of selected clusters are included. Advantage: cheaper / easier when clusters are geographically concentrated. Disadvantage: higher sampling error if clusters are heterogeneous; less precise than stratified for the same size.
A company wishes to estimate the average time employees spend commuting. Discuss the relative merits of using a simple random sample, a stratified sample by department, and a convenience sample of employees in the canteen.
SRS: unbiased if frame complete, but may miss small departments. Stratified by department: ensures representation of each department, more precise if commute times differ by department. Convenience (canteen): quick and cheap but biased (only those who eat in canteen / at that time; excludes remote workers etc.).
Explain the difference between a sampling frame and the population. Give an example where the sampling frame does not match the population and the bias that may result.
Population: the whole group of interest. Sampling frame: the list from which the sample is actually drawn. Example: population = all UK adults; frame = electoral register → excludes those not registered, under-represents young / mobile people.
A histogram is drawn for continuous data with unequal class widths. Explain how the heights of the bars should be determined and why frequency density is used. Sketch (describe) the correct heights for classes 0–10 (freq 20), 10–30 (freq 30), 30–40 (freq 25).
Height = frequency density = frequency / class width so that area is proportional to frequency. Densities: \( 20/10=2 \), \( 30/20=1.5 \), \( 25/10=2.5 \). Bars of heights 2, 1.5 and 2.5 respectively.