Chapter 12 Sample Surveys math2200. How to study a population A population is the entire group of...

Chapter 12 Sample Surveys

math2200

How to study a population

• A population is the entire group of individuals we want information about.– Impractical to examine the entire population

• A sample is a smaller group of the population we actually examine in order to gather information.

What is a survey?

• A survey is the process of collecting data from a sample in an attempt to draw conclusions about the entire population.– The conclusions can only be accurate if the

selected sample properly represents the population.

Analogy: Soup tasting.

Opinion polls

• Example of Sample surveys.

• Designed to ask questions of a small group of people in the hope of learning something about the entire population.

Bias

• Sampling methods that tend to over- or under- emphasize some characteristics of the population are said to be biased.

• AVOID BIAS!!!

Landon vs. Roosevelt in 1936

Actually, Roosevelt polled 62.5% of the major-party vote and won 523 out of possible 531 electoral votes.

Reason: Biased Sampling.

The poll results pointed to a Landon victory over Roosevelt, indicating that he would get some 57% of the vote.

Randomization

To avoid biased samples, we should select individuals for the sample at random!

In 1936, using a different sample of 50,000, Gallup predicted that Roosevelt would get 56% of the vote to Landon’s 44%.

The Benefit of Randomization

• Protects against both known and unknown factors.

• Makes it possible to draw inference about the population based on the sample.

Sample Size

• The sample size but not the fraction of the population matters.– Exception: If the population is small enough

and the sample is more than 10% of the whole population, the population size can matter.

• A larger sample size leads a more precise result in general.

Does a Census Make Sense?

• A sample include everyone in the entire population is called a census.

• There are problems with taking a census:– It is hard to track down every individual in the

population.– Population changes.– Taking a census is way more complicated

than sampling.

Populations and Parameters

• A parameter that is part of a model for a population is called a population parameter.

• The statistics that estimate population parameters are called sample statistics.

Notation

Greek letters to denote parameters and Latin letters to denote statistics. Hat on the estimation.

Simple random samples

• If the statistics we compute from a sample reflect the population parameter accurately , the sample is called representative.

• If every possible sample of the size we plan to draw has an equal chance to be selected, a sample drawn in this way is called a Simple Random Sample.

• The sampling frame is a list of individuals from which the sample is drawn.

• Different samples lead to different sample statistics . We call these sample-to-sample differences sampling variability.

Simple Random Sample: Example

• Select 5 students from 19 enrolled in our class

– Sampling frame: 19 enrolled students– Sample: 5 students

• Sample size: 5

1. Number the students from 1 to

2. Use your TI-83 to obtain five random integers between 1 and 80 (no repetition)

Stratified Sampling

• Stratified Sampling– Simple random sampling is applied within

each stratum within the population before results are combined.

• Stratified random sampling can reduce bias.

• Stratifying can also reduce the variability of our results.

Example

• Survey how students feel about funding for the football team. population: 60% men and 40% women. Sample size = 100

• Simple random sampling : 20 men 80 women ? 80 men 20 women?

• Stratified random sampling: 60 men and 40 women. And simple random sampling within men or women.

Cluster Sampling

• Split the population into similar parts or clusters, then select one or a few clusters at random and perform a census within each of them.

Multistage Sampling

• Sampling schemes that combine several methods are called multistage samples.

• Most surveys conducted by professional polling organizations use some combination of stratified and cluster sampling as well as simple random sampling.

Example

To assess the reading level of a book based on the words used

• Randomly select one chapter from one part (stratified sampling)

• Randomly select several pages from each of those chosen chapters (cluster sampling)

• Randomly select a few sentences from each of those chosen pages (simple random sampling)

Systematic sampling

• Example– survey every 10th person on an alphabetical

list of students– Start from a randomly selected individual

Who’s Who?

1. The population of interest?

2. The sampling frame?

3. The target sample, for example, a sample determined by simple random sampling?

4. The actual respondents?

The Valid Survey

BEFORE setting out the survey:– What do I want to know?– Am I asking the right respondents?– Am I asking the right questions?– What would I do with the answers if I had

them; would they address the things I want to know?

• Know what you want to know.• Use the right frame.• Time your instrument.• Ask specific rather than general questions.• Ask for quantitative results when possible.• Be careful in phrasing questions.• Even subtle differences in phrasing can make a

difference.• Give a pilot survey to smaller

The Valid Survey (cont.)

What Can Go Wrong?

• Sample Badly with Volunteers

• Sample Badly, but Conveniently

• Sample from a Bad Sampling Frame

• Watch out for nonrespondents

• Work hard to avoid influencing responses

How to Think About Biases

• Look for biases in any survey you encounter—there’s no way to recover from a biased sample of a survey that asks biased questions.

• Spend your time and resources reducing biases.

• If you possibly can, pretest your survey.• Always report your sampling methods in

detail.

What have we learned?

• A representative sample can offer us important insights about populations.

• There are several ways to draw samples.• Bias can destroy our ability to gain insights from

our sample• Bias can also arise from poor sampling methods• Look for biases in the survey and be sure to

report our methods how the survey was performed.