MYP Standard Mathematics 9.1 Sampling Techniques Notes

Why We Sample And What Makes A Sample Useful

1. We take samples because measuring everyone can be too slow, expensive, or impractical.
2. The goal is to use the sample to make an inference (a conclusion) about the population.
3. Whether that inference is reasonable depends on how the sample was chosen.

Simple Random Sampling Gives Each Member An Equal Chance

1. In a simple random sample, every member of the population has an equal chance of being selected, and the selection is genuinely random.
2. This reduces selection bias because you are not choosing people based on convenience.
3. Common ways to select a simple random sample include pulling names from a hat (when the group is small) or using a calculator/computer random number generator (RNG).

How To Take A Simple Random Sample (Method)

1. Define the population clearly.
2. Create a sampling frame (a complete list).
3. Number the members from 1 to $N$.
4. Use an RNG to generate $n$ distinct numbers.
5. Select the individuals with those numbers.

Systematic Sampling Uses A Random Start And A Fixed Interval

1. A systematic sample is formed from an ordered sampling frame by:
   1. choosing a random starting point, then
   2. selecting items at regular intervals.
2. If the sampling frame has $N$ items and you want a sample of size $n$, a common interval is $$k \approx \frac{N}{n}$$
3. Choose a random start from 1 to $k$, then take every $k$th item.

Stratified Sampling Preserves Proportions Of Key Groups

1. Use a stratified sample when the population contains important subgroups (called strata) and you want them fairly represented.
2. Here, "fairly represented" means the proportions in the sample match the proportions in the population.
3. For example, if 60% of the population is in one group, about 60% of the sample should come from that group.

How To Build A Stratified Sample

1. Choose the strata (for example, class, age band, or car type).
2. Find each group size in the population.
3. Decide the overall sample size $n$.
4. Compute each group's sample size using $$ni = \frac{Ni}{N}\times n$$
5. From each group, take a simple random sample of size $n_i$.

Quota Sampling Is Practical When The Sampling Frame Is Unknown

1. Sometimes a sampling frame is unknown or does not exist (for example, surveying "people in the downtown area").
2. In these cases, fully random methods may be impossible.
3. A quota sample is used by collecting responses until you have enough data in each chosen category (for example, 15 local men and 15 local women).
4. You might stand somewhere and ask people who are willing to talk until each quota is reached.

Response Rate Affects How Much You Can Trust A Survey

1. Even if you choose a careful sampling method, not everyone selected will respond.
2. This matters because non-response can distort results.
3. If you invite $n$ people and receive $r$ responses, then $$\text{response rate} = \frac{r}{n}\times 100\%$$
4. A low response rate can create non-response bias if the people who do not respond are systematically different from those who do.

Sample Size And Reliability: Bigger Is Usually Better (But Not Always)

1. In general, a larger sample size makes generalizations more reliable because it reduces sampling variability (the natural differences you would get if you repeated the sampling process).
2. However, bias does not disappear just because the sample is large.
3. A large biased sample can give a very precise answer to the wrong question.