Sampling methods

In Units 1 and 2 you described data that was already sitting in front of you. Now you step back and ask where data comes from. How a sample is chosen decides whether its results can tell you anything trustworthy about the larger group you actually care about.

Populations, samples and censuses

Every statistical question is about some population: the entire group of individuals you want information about. The population might be all 2,300 students at a high school, every registered voter in a state, or every battery produced at a factory this week.

Measuring every individual in the population is called a census. A census sounds ideal, but it is usually too slow, too expensive or simply impossible. (If testing a battery's lifetime means running it until it dies, a census would destroy the whole week's production.) So instead you collect data from a sample, a subset of the population, and use it to draw conclusions about the whole population.

A number that describes a population, like the true proportion of all voters who support a ballot measure, is a parameter. A number computed from a sample, like the proportion of 500 surveyed voters who support it, is a statistic. You use statistics to estimate parameters.

The central question of this lesson is: how should you pick the sample? The answer, over and over, is use chance.

Simple random samples

Notice the definition is about groups, not just individuals. It is not enough for every individual to have the same chance of being chosen; every possible set of $n$ individuals must be equally likely.

To take an SRS you need a sampling frame, a list of the individuals in the population. Then you can select the sample in a few standard ways:

• Slips of paper (the "hat" method). Write each name on an identical slip, mix the slips thoroughly in a container, and draw $n$ slips without looking.
• Technology. Number the individuals $1$ to $N$. Use a random number generator to produce integers from $1$ to $N$, ignoring repeats, until you have $n$ different numbers. The individuals with those numbers form the sample.
• A table of random digits. Give each individual a label with the same number of digits (for $N = 850$, use $001$ to $850$). Read groups of three digits from the table, skipping $000$, numbers above $850$, and repeats.

On the AP exam, a complete description of an SRS names the labeling, the random mechanism, what to do with repeats, and how many individuals to select. Sampling in this course is almost always without replacement: once someone is chosen, they can't be chosen again.

Other random sampling methods

An SRS is the baseline, but it isn't always the most practical or the most precise design. Three other random methods show up constantly in AP Statistics.

Stratified random sampling

First divide the population into groups called strata (singular stratum) of individuals that are similar in some way that might affect the response, such as grade level or income bracket. Then take a separate SRS from each stratum and combine them into one sample.

Stratifying works best when individuals within each stratum are similar to one another and the strata differ from each other. It guarantees every stratum is represented, and it usually produces a more precise estimate (less sample-to-sample variability) than an SRS of the same size.

A common approach is proportional allocation: each stratum's share of the sample matches its share of the population.

$$\text{sample size from a stratum} = \frac{\text{stratum size}}{\text{population size}} \times n$$

Cluster sampling

Divide the population into groups called clusters, often based on location, such as classrooms, city blocks or apartment buildings. Randomly select some of the clusters, then collect data from every individual in the chosen clusters.

Cluster sampling is used because it is cheaper and more convenient. Surveying every student in 8 randomly chosen homerooms is far easier than tracking down 200 students scattered across the building. Ideally each cluster looks like a small version of the population, so clusters should be mixed (heterogeneous) inside.

Systematic random sampling

Choose a starting point at random from the first $k$ individuals in an ordered list or line, then select every $k$th individual after that. If you want a sample of $n$ from a population of $N$, use $k = \dfrac{N}{n}$ (rounded down if needed).

Systematic sampling is easy to carry out in the field, for example selecting every 20th customer leaving a store after picking a random start from 1 to 20. It can go wrong if the list has a repeating pattern.

Practice

Practice 1

A grocery chain wants to know how customers rate its checkout speed. At one store, a manager randomly picks a number from 1 to 15, surveys that customer in line, and then surveys every 15th customer after that for the rest of the day. What type of sample is this?

ASimple random sampleBStratified random sampleCSystematic random sampleDCluster sample

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Practice 2

A school has 45 homerooms. To learn how students feel about the cafeteria, the student council randomly selects 8 homerooms and surveys every student in those homerooms. What type of sample is this?

ACluster sampleBStratified random sampleCSimple random sampleDSystematic random sample

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Practice 3

A company has 2,400 employees: 1,500 hourly workers, 700 salaried staff and 200 managers. It will select a stratified random sample of 120 employees, using proportional allocation by job type. How many managers should be in the sample?

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

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Practice 4

A county clerk has an ordered list of 1,800 property owners and wants a systematic random sample of 60 of them. After choosing a random starting point, every $k$th owner on the list will be selected. What value of $k$ should the clerk use?

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

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Practice 5

A club has 30 juniors and 20 seniors. The advisor randomly selects 6 juniors and, separately, 4 seniors to attend a meeting. Every club member has a $\dfrac{1}{5}$ chance of being chosen. Why is this not a simple random sample of 10 members?

ABecause the sample is too small to be an SRSBBecause juniors and seniors do not have the same chance of being chosenCBecause the advisor did not use a random number generatorDBecause not every group of 10 members is equally likely; for example, a group of 10 juniors can never be chosen

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Practice 6

A polling firm estimating support for a new city tax stratifies voters by neighborhood income level (low, middle, high) instead of taking an SRS of the same size. Under which condition does stratifying give the biggest improvement in precision?

AWhen voters within each income level hold a wide mix of opinions, but the income levels have similar overall supportBWhen voters within each income level hold similar opinions, but support differs a lot between income levelsCWhen every income level has exactly the same number of votersDWhen the firm surveys every voter in one randomly chosen income level

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Practice 7

A city health department wants to estimate the proportion of households with a working smoke detector. It numbers all 600 residential blocks, randomly selects 25 blocks, and inspects every household on those blocks. What is the main advantage of this design compared with an SRS of the same number of households from across the city?

AIt removes all bias from the estimate.BIt guarantees a more precise estimate than an SRS.CIt is cheaper and faster, because inspectors travel to only 25 locations.DIt guarantees that every neighborhood income level is represented.

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