S1: Sampling
Infer properties of populations from samples; understand limitations of sampling
📋 Key Concepts
Definition: A sample is a smaller group selected from a larger population. We use samples to make inferences (educated guesses) about the whole population.
Key Terms
| Term | Definition |
|---|---|
| Population | The entire group being studied |
| Sample | A smaller group taken from the population |
| Inference | Using sample data to draw conclusions about the population |
| Bias | When the sample doesn't fairly represent the population |
| Representative | A sample that accurately reflects the population |
Important: A larger sample size generally gives more accurate results, but the method of sampling is equally important.
📝 Why Sample?
Reasons for sampling:
- Too expensive to survey everyone
- Too time-consuming to collect all data
- Some tests are destructive (e.g. testing light bulbs until they fail)
- Practical limitations
Example 1
A factory produces 10,000 light bulbs per day. Explain why the quality control manager uses a sample rather than testing every bulb.
Solution:
Testing every bulb would be:
- Time-consuming - 10,000 bulbs would take too long
- Destructive - testing requires burning out each bulb
- Expensive - uses resources and staff time
A sample of 100 bulbs can give a good estimate of quality.
📝 Sampling Methods
Random Sampling: Every member of the population has an equal chance of being selected. This reduces bias.
Types of Random Sampling
Simple Random Sample: Every possible sample of size n has an equal chance of being chosen.
Method: Assign each member a number, then use random number generator or lottery to select.
Method: Assign each member a number, then use random number generator or lottery to select.
Stratified Sampling: The population is divided into groups (strata), then a random sample is taken from each group proportional to its size.
Example 2
A school has 200 Year 7 students and 300 Year 8 students. A sample of 50 students is needed using stratified sampling.
Solution:
Total students = 200 + 300 = 500
Sample size = 50
Year 7: (200 ÷ 500) × 50 = 20 students
Year 8: (300 ÷ 500) × 50 = 30 students
Select 20 students randomly from Year 7 and 30 students randomly from Year 8.
Systematic Sampling: Select every kth item from a list after a random starting point.
Formula: k = population size ÷ sample size
Formula: k = population size ÷ sample size
Example 3
A company wants to select 40 employees from a list of 200 for a survey using systematic sampling.
Solution:
k = 200 ÷ 40 = 5
Choose a random starting number between 1 and 5 (e.g. 3)
Select employees: 3, 8, 13, 18, 23, 28, ...
Select every 5th employee starting from position 3.
📝 Bias in Sampling
Bias occurs when the sample does not fairly represent the population, leading to misleading conclusions.
Common Sources of Bias
- Selection bias: Some groups are more likely to be chosen than others
- Response bias: People may not answer truthfully
- Non-response bias: People who don't respond may have different views
- Leading questions: Questions that suggest a particular answer
Example 4
A researcher stands outside a gym and asks people about their exercise habits. Explain why this sample might be biased.
Solution:
This sample is biased because:
- People at a gym are more likely to exercise regularly
- People who don't exercise aren't represented
- The sample doesn't represent the whole population
A better approach would be to survey a random sample from the general population.
📝 Limitations of Sampling
Key Limitations:
- Sample may not be representative of the population
- Results are estimates, not exact values
- Sampling error - natural variation between samples
- Larger samples reduce but don't eliminate error
Example 5
A sample of 50 people is asked about their favourite sport. Can we conclude that the results apply to the whole country?
Solution:
No, because:
- Sample size is too small for a whole country
- Sample may not be representative (location, age, interests)
- There is sampling error - another sample might give different results
Need a larger, more representative sample for valid conclusions.
❓ Practice Questions
Q1: Define what is meant by a "population" in statistics.
Q2: A school has 120 boys and 180 girls. A stratified sample of 50 students is needed. How many boys and girls should be selected?
Q3: A researcher wants to survey 25 people from a list of 200 using systematic sampling. What value of k should be used?
Q4: Explain why asking people leaving a cinema about their views on films might produce a biased sample.
Q5: Give two advantages of using a sample rather than surveying the whole population.
✅ Answers
- The population is the entire group being studied or about which conclusions are to be drawn.
- Boys: (120/300) × 50 = 20; Girls: (180/300) × 50 = 30
- k = 200 ÷ 25 = 8 (select every 8th person)
- Biased because: only people who choose to watch films are surveyed; those leaving cinemas are likely to enjoy films more than general population; excludes people who don't go to cinemas.
- Any two: cheaper, faster, more practical, less resource-intensive, possible when population is very large or infinite.
🎯 Exam Tips
- Always consider whether a sample is representative and free from bias
- Know how to calculate numbers for stratified sampling
- Understand the difference between the population and the sample
- Be able to identify sources of bias in sampling methods
- Remember: larger samples generally give more reliable results
🧠 Problem-Solving Strategies
Problem-Solving
For sampling problems: (1) Identify the population and the sample clearly, (2) For stratified sampling, calculate each group's proportion of the total, then multiply by sample size, (3) For systematic sampling, find k = population size ÷ sample size, then choose random start from 1 to k, (4) Always check your stratified sample numbers add up to the total sample size, (5) Round carefully — you may need to adjust one group to make totals work.
Multi-Step Problem
A school has 150 Year 9, 180 Year 10, and 120 Year 11 students. A stratified sample of 45 students is needed. Calculate how many from each year group.
Solution: Total = 150 + 180 + 120 = 450. Year 9: (150/450) × 45 = 15. Year 10: (180/450) × 45 = 18. Year 11: (120/450) × 45 = 12. Check: 15 + 18 + 12 = 45 ✓. Each group is proportional to its size in the population.
⚠️ Common Errors
Watch Out!
1. Wrong: Calculating stratified sample numbers that don't add up to the total sample size (due to rounding) Correct: Always check the sum matches the sample size — adjust the largest group if needed to account for rounding
2. Wrong: Confusing the population with the sample — e.g. saying "the sample of 1000 people shows 60% prefer tea, so exactly 60% of the whole country prefers tea" Correct: The sample gives an estimate — we can infer the population is likely around 60%, but it's not exact due to sampling error
3. Wrong: Choosing a "random" sample by asking your friends (convenience sampling) and calling it random Correct: Random sampling means every member has an equal chance — convenience samples are biased and not random
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A town has 8000 adults: 3200 are aged 18–30, 2800 are aged 31–50, and 2000 are aged 51+. A researcher wants a stratified sample of 200 adults. (a) Calculate how many from each age group. (b) Describe how to select the people using simple random sampling within each group. (c) Explain one advantage of stratified sampling over simple random sampling for this survey about voting intentions.
(a) Total = 8000. 18–30: (3200/8000) × 200 = 80. 31–50: (2800/8000) × 200 = 70. 51+: (2000/8000) × 200 = 50. Check: 80 + 70 + 50 = 200 ✓
(b) Assign each person in the age group a number. Use a random number generator to select the required number of people. For 18–30, generate 80 unique random numbers from 1 to 3200. Repeat similarly for other groups.
(c) Stratified sampling ensures each age group is represented proportionally. Simple random sampling might accidentally select very few from the 51+ group, meaning their views on voting are underrepresented. Since age affects voting intentions, proportional representation gives more reliable results.
Mark scheme: M1 for correct proportions, A1 for 80, 70, 50, M1 for describing random number selection, A1 for clear method, M1 for identifying representation advantage, A1 for linking to voting context
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A researcher surveys people about their income by standing outside a luxury car dealership on a Monday morning. 50 people are surveyed.
(a) Is this a random sample? Explain.
(b) What type of bias is likely in this sample?
(c) How could the researcher improve the sampling method to get a more representative result?
Answers: (a) No — people at a luxury car dealership are more likely to have higher incomes. Not everyone in the population has an equal chance of being selected. (b) Selection bias — the sample over-represents wealthy individuals and under-represents lower income groups. (c) Use stratified or simple random sampling from the general population, e.g. selecting from the electoral roll or using random phone numbers.
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