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### Sampling

Sample Surveys

Producing Valid Data

“If you don’t believe in random sampling, the next time you have a blood test tell the doctor to take it all.”

The Predictions

The Candidates Crossley Gallup Roper The Results

Truman 45 44 38 50

Dewey 50 50 53 45

Beyond the Data at Hand to the World at Large

- We have learned ways to display, describe, and summarize data, but have been limited to examining the particular batch of data we have.
- We’d like (and often need) to stretch beyond the data at hand to the world at large.
- Let’s investigate three major ideas that will allow us to make this stretch…

Idea 1: Examine a Part of the Whole

- The first idea is to draw a sample.
- We’d like to know about an entire population of individuals, but examining all of them is usually impractical, if not impossible.
- We settle for examining a smaller group of individuals—a sample—selected from the population.

Examples

1. Think about sampling something you are cooking—you taste (examine) a small part of what you’re cooking to get an idea about the dish as a whole.

2. Opinion polls are examples of sample surveys, designed to ask questions of a small group of people in the hope of learning something about the entire population.

Convenience sampling: Just ask whoever is around.

- Example: “Man on the street” survey (cheap, convenient, often quite opinionated or emotional => now very popular with TV “journalism”)
- Which men, and on which street?
- Ask about gun control or legalizing marijuana “on the street” in Berkeley or in some small town in Idaho and you would probably get totally different answers.
- Even within an area, answers would probably differ if you did the survey outside a high school or a country western bar.
- Bias: Opinions limited to individuals present.

Voluntary Response Sampling:

- Individuals choose to be involved. These samples are very susceptible to being biased because different people are motivated to respond or not. Often called “public opinion polls.” These are not considered valid or scientific.
- Bias: Sample design systematically favors a particular outcome.

Ann Landers summarizing responses of readers

70% of (10,000) parents wrote in to say that having kids was not worth it—if they had to do it over again, they wouldn’t.

Bias: Most letters to newspapers are written by disgruntled people. A random sample showed that 91% of parents WOULD have kids again.

Bias: People have to care enough about an issue to bother replying. This sample is probably a combination of people who hate “wasting the taxpayers money” and “animal lovers.”

Response Bias

- Work hard to avoid influencing responses.
- Response bias refers to anything in the survey design that influences the responses.
- For example, the wording of a question can influence the responses:

Example: hospital employee drug use

Administrators at a hospital are concerned about the possibility of drug abuse by people who work there. They decide to check on the extent of the problem by having a random sample of the employees undergo a drug test. The administrators randomly select a department (say, radiology) and test all the people who work in that department – doctors, nurses, technicians, clerks, custodians, etc.

- Why might this result in a biased sample?
- Dept. might not represent full range of employee types, experiences, stress levels, or the hospital’s drug supply

Example (cont.)

Name the kind of bias that might be present if the administration decides that instead of subjecting people to random testing they’ll just…

- a. interview employees about possible drug abuse.
- Response bias: people will feel threatened, won’t answer truthfully
- b. ask people to volunteer to be tested.
- Voluntary response bias; only those who are “clean” would volunteer

Bias

- Bias is the bane of sampling—the one thing above all to avoid.
- There is usually no way to fix a biased sample and no way to salvage useful information from it.
- The best way to avoid bias is to select individuals for the sample at random.
- The value of deliberately introducing randomness is one of the great insights of Statistics – Idea 2

Idea 2: Randomize

- Randomization can protect you against factors that you know are in the data.
- It can also help protect against factors you are not even aware of.
- Randomizing protects us from the influences of all the features of our population, even ones that we may not have thought about.
- Randomizing makes sure that on the average the sample looks like the rest of the population

Idea 2: Randomize (cont.)

Individuals are randomly selected. No one group should be over-represented.

Sampling randomly gets rid of bias.

Random samples rely on the absolute objectivity of random numbers. There are tables and books of random digits available for random sampling.

Statistical software cangenerate random digits (e.g., Excel “=random()”,

ran# button on

calculator).

Idea 2: Randomize (cont.)

- Not only does randomizing protect us from bias, it actually makes it possible for us to draw inferences about the population when we see only a sample.

Listed in the table are the names of the 20 pharmacists on the hospital staff. Use the random numbers listed below to select three of them to be in the sample.

04905 83852 29350 91397 19994 65142 05087 11232

Hospital example (cont.)Idea 3: It’s the Sample Size!!

- How large a random sample do we need for the sample to be reasonably representative of the population?
- It’s the size of the sample, not the size of the population, that makes the difference in sampling.
- Exception: If the population is small enough and the sample is more than 10% of the whole population, the population size can matter.
- The fraction of the population that you’ve sampled doesn’t matter. It’s the sample size itself that’s important.

Example

- i) In the city of Chicago, Illinois, 1,000 likely voters are randomly selected and asked who they are going to vote for in the Chicago mayoral race.
- ii) In the state of Illinois, 1,000 likely voters are randomly selected and asked who they are going to vote for in the Illinois governor's race.
- iii) In the United States, 1,000 likely voters are randomly selected and asked who they are going to vote for in the presidential election.
- Which survey has more accuracy?
- All the surveys have the same accuracy

Idea 3: It’s the Sample Size!!

- Chicken soup
- Blood samples

Does a Census Make Sense?

- Why bother worrying about the sample size?
- Wouldn’t it be better to just include everyone and “sample” the entire population?
- Such a special sample is called a census.

Does a Census Make Sense? (cont.)

- There are problems with taking a census:
- Practicality: It can be difficult to complete a census—there always seem to be some individuals who are hard to locate or hard to measure.
- Timeliness: populations rarely stand still. Even if you could take a census, the population changes while you work, so it’s never possible to get a perfect measure.
- Expense: taking a census may be more complex than sampling.
- Accuracy: a census may not be as accurate as a good sample due to data entry error, inaccurate (made-up?) data, tedium.

Population: The entire group of individuals in which we are interested but can’t usually assess directly.

Example: All humans, all working-age people in California, all crickets

A parameter is a number describing a characteristic of the population.

Sample: The part of the population we actually examine and for which we do have data.

How well the sample represents the population depends on the sample design.

A statistic is a number describing a characteristic of a sample.

Population versus samplePopulation

Sample

Sample Statistics Estimate Parameters

- Values of population parameters are unknown; in addition, they are unknowable.
- Example: The distribution of heights of adult females (at least 18 yrs of age) in the United States is approximately symmetric and mound-shaped with mean µ. µ is a population parameter whose value is unknown and unknowable
- The heights of 1500 females are obtained from a sample of government records. The sample mean x of the 1500 heights is calculated to be 64.5 inches.
- The sample mean x is a sample statistic that we use to estimate the unknown population parameterµ

We typically use Greek letters to denote parameters and Latin letters to denote statistics.

Various claims are often made for surveys. Why are each of the following claims not correct?

- It is always better to take a census than a sample
- Timeliness, expense, complexity, accuracy
- Stopping students on their way out of the cafeteria is a good way to sample if we want to know the quality of the food in the cafeteria.
- Bias; they chose to eat at the cafeteria
- We drew a sample of 100 from the 3,000 students at a small college. To get the same level of precision for a town of 30,000 residents, we'll need a sample of 1,000 residents.
- It’s the sample size, not the size of the population or the fraction of the population that we sample, that is important.

Survey claims (cont.)

- An internet poll taken at the web site www.statsisfun.org garnered 12,357 responses. The majority said they enjoy doing statistics homework. With a sample size that large, we can be pretty sure that most Statistics students feel this way, too.
- Voluntary response bias; size of sample does not remove the bias.
- The true percentage of all Statistics students who enjoy the homework is called a “population statistic.”
- The true percentage is a population parameter

Simple Random Sample

- A simple random sample (SRS) of size n consists of n units from the population chosen in such a way so that every set of n units has an equal chance to be the sample actually selected.

Simple Random Samples (cont.)

- To select a sample at random, we first need to define where the sample will come from.
- The sampling frame is a list of individuals from which the sample is drawn.
- E.g., To select a random sample of students from a college, we might obtain a list of all registered full-time students.
- When defining sampling frame, must deal with details defining the population; are part-time students included? How about current study-abroad students?
- Once we have our sampling frame, the easiest way to choose an SRS is with random numbers.

Warning!

- If some members of the population are not included in the sampling frame, they cannot be part of the sample!! (e. g., using a telephone book as the sampling frame)
- Population: Wal Mart shoppers
- Sampling frame?

Example: simple random sample

- Academic dept wishes to randomly choose a 3-member committee from the 28 members of the dept

00 Abbott 07 Goodwin 14 Pillotte 21 Theobald

01 Cicirelli 08 Haglund 15 Raman 22 Vader

02 Crane 09 Johnson 16 Reimann 23 Wang

03 Dunsmore 10 Keegan 17 Rodriguez 24 Wieczoreck

04 Engle 11 Lechtenb’g 18 Rowe 25 Williams

05 Fitzpat’k 12 Martinez 19 Sommers 26 Wilson

06 Garcia 13 Nguyen 20 Stone 27 Zink

Solution

- Use a random number table; read 2-digit pairs until you have chosen 3 committee members
- For example, start in row 21:

76509 47069 86378 41797 11910 49672 88575

Rodriguez (17) Lechtenberg (11) Engle (04)

- Your calculator generates random numbers; you can also generate random numbers using Excel

Sampling Variability

- Suppose we had started in line 22?

19689 90332 04315 21358 97248 11188 39062

- Our sample would have been

19 Summers, 03 Dunsmore, 04 Engle

Sampling Variability

- Samples drawn at random generally differ from one another.
- Each draw of random numbers selects different people for our sample.
- These differences lead to different values for the variables we measure.
- We call these sample-to-sample differences sampling variability.
- Variability is OK; bias is bad!!

- under 20
- 20-30
- 31-40
- 41-50

- Sex
- Male
- Female

- Occupation
- professional
- clerical
- blue-collar

- This sampling procedure separates the population into mutually exclusive sets (strata), and then selects simple random samples from each stratum.

Stratified Random Sampling

- With this procedure we can acquire information about
- the whole population
- each stratum
- the relationships among strata.

Stratum Income Population proportion

Stratum size

1 under $15,000 25% 250

2 15,000-29,999 40% 400

3 30.000-50,000 30% 300

4 over $50,000 5% 50

Stratified Random Sampling- There are several ways to build the stratified sample. For example, keep the proportion of each stratum in the population.

A sample of size 1,000 is to be drawn

Total 1,000

Cluster Sampling

- Sometimes stratifying isn’t practical and simple random sampling is difficult.
- Splitting the population into similar parts or clusters can make sampling more practical.
- Then we could select one or a few clusters at random and select an SRS or perform a census within each cluster.
- This sampling design is called cluster sampling.
- If each cluster fairly represents the full population, cluster sampling will give us an unbiased sample.

Cluster Sampling Useful When…

- it is difficult and costly to

develop a complete list of the

population members (making

it difficult to develop a simple

random sampling procedure.)

e.g., all items sold in a grocery store

the population members are widely

dispersed geographically.

e.g., all Toyota dealerships in North

Carolina

Mean length of sentencesin a statistics text

- We would like to assess the

reading level of textbook

based on the length of the sentences.

- Simple random sampling would be awkward:
- number each sentence in the book?
- Better way:
- choose a few pages at random (the pages are the clusters, and it's reasonable to assume that each page is representative of the entire text).
- count the length of the sentences on those pages

Cluster sampling - not the same as stratified sampling!!

- We stratify to ensure that our sample represents different groups in the population, and sample randomly within each stratum.
- Clusters are more or less alike, each heterogeneous and resembling the overall population.
- We select clusters to make sampling more practical or affordable.
- We select an SRS or conduct a census on each selected cluster.

Strata are homogenous (e.g., male, female) but

differ from one another

Multistage Sampling

- Sometimes we use a variety of sampling methods together.
- Sampling schemes that combine several methods are called multistage samples.

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

Mean length of sentences in our course text, cont.

- In attempting to assess the reading level of our course text:
- we might worry that it starts out easy and gets harder as the concepts become more difficult
- we want to avoid samples that select too heavily from early or from late chapters
- Suppose our course text has 5 sections, with several chapters in each section.

Mean length of sentences in our course text, cont.

- We could:
- i) randomly select 1 chapter from each section
- ii) randomly select a few pages from each of the selected chapters
- iii) if altogether this makes too many sentences, we could randomly select a few sentences from each page.
- So what is our sampling strategy?
- i) we stratify by section of the book
- ii) we randomly choose a chapter to represent each stratum (section)
- iii) within each chapter we randomly choose pages as clusters
- iv) finally, we choose an SRS of sentences within each cluster

Systematic Sampling

- Sometimes we draw a sample by selecting individuals systematically.
- For example, you might survey every 10th person on an alphabetical list of students.
- To make it random, you must still start the systematic selection from a randomly selected individual.
- When there is no reason to believe that the order of the list could be associated in any way with the responses sought, systematic sampling can give a representative sample.
- Systematic sampling can be much less expensive than true random sampling.
- When you use a systematic sample, you need to justify the assumption that the systematic method is not associated with any of the measured variables.

Systematic Sampling-example

- You want to select a sample of 50 students from a college dormitory that houses 500 students.
- On a list of all students living in the dorm, number the students from 001 to 500.
- Generate a random number between 001 and 010, and start with that student.
- Every 10th student in the list becomes part of your sample.
- Questions:
- 1) does each student have an equal chance to be in the sample?
- 2) what is the chance that a student is included in the sample?
- 3) is this an SRS?

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