AP Stats 1.11 Random SamplingAP Stats 1.11 Random Sampling

APS TAT ISTICS · UNIT1B

Random Sampling

Topic 1.11 Study Guide

In one sentence: a random sample uses chance to pick units, and the four methods (SRS, stratified, cluster, systematic) differ in how the groups are formed and who gets picked.

With or without replacement

Without replacement: a selected unit is set aside, so it can be chosen only once. Most surveys work this way.

With replacement: the unit goes back before the next pick, so it could be chosen again (the fish that is caught, measured, released).

The four methods

Method How it works Best when

Simple random

sample (SRS)

Every possible sample of size n has the same

chance. Number the units, use a random number

generator, ignore repeats.

You have a complete list and no strong

groups

Stratified Split into non-overlapping groups (strata) that are

alike inside. Take an SRS from every stratum,

combine.

Groups differ on the variable.

Guarantees every group is represented

and can be more precise.

Cluster Split into groups (clusters) that are each a mini

version of the population. SRS of clusters, then

survey everyone in chosen clusters.

Saving time and money, clusters look

alike

Systematic Random start in the first k, then every kth unit. Units are already in a line or list Stratified takes SOME from ALL groups. Cluster takes ALL from SOME groups. The exam loves to test this.

Trap. 'Every individual has an equal chance' is not the definition of an SRS. The definition is that every possible sample of size n has an equal chance. A systematic sample gives everyone a chance but is not an SRS, because items 23 and 24 can never both appear.

Stratified vs cluster at a glance

Stratified Cluster

Groups are Homogeneous inside, different from each other Heterogeneous inside, similar to each other Who is selected Some individuals from every group All individuals from some groups Main goal Represent every group, more precision Save time and money

How to describe an SRS (FRQ model)

1. Obtain a list of the whole population and number it 1 to N.

2. Use a random number generator for integers from 1 to N, ignoring repeats, until n unique numbers appear.

3. Sample the units with those numbers.

Stratified: name the strata, then do an SRS of the stated size inside each. Cluster: number the clusters, SRS of clusters, survey every unit in each. Always say the sizes add up (10 teams × 30 = 300). Worked examples

1. A hospital has 600 nurses, 150 physicians, 250 technicians and samples 60, 15, 25 respectively. Stratified (proportional: 10% of each).

2. Provider picks 8 of 120 neighborhoods, surveys every household there. Cluster. 3. Inspector picks one of the first 40 packages, then every 40th. Systematic.

4. Opinions differ sharply between urban and rural residents. Best plan: stratify by urban and rural, because it guarantees both groups appear.

Practice

1. Vet numbers 340 files and picks 30 unique ones with a random number generator. 2. Hotel chain randomly picks 8 of 94 locations and surveys every guest at those 8. 3. Customers split into four shipping regions, SRS of 50 from each.

4. Pick one of the first 25 toys, then test every 25th.

5. A battery is tested and put back before the next pick. Which type of sampling, and can a battery be tested twice?

6. City has 60,000 households in 15 neighborhoods that differ a lot in commute distance. Give one advantage of stratifying by neighborhood over an SRS.

7. Sitbit has 600 teams of 30. Describe a cluster sample of 300 users.

8. Why might a systematic sample be easier than an SRS for 100 of 10,000 packages on a belt? Answer key

1. SRS.

2. Cluster.

3. Stratified.

4. Systematic.

5. With replacement. Yes.

6. Every neighborhood is guaranteed to appear, which reduces variability and gives a more precise estimate.

7. Number teams 1 to 600, randomly pick 10 teams, survey all 30 users on each (10 × 30 = 300).

8. They are already in order. Pick one of the first 100 at random, then inspect every 100th, with no need to number and hunt for scattered packages.