Lecture Unit 3

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 election of 1948 The Predictions

The Candidates Crossley Gallup Roper The ResultsTruman 45 44 38 50Dewey 50 50 53 45

Lecture Unit 3 Objectives1. Given a survey sample, determine whether the

sample is a simple random sample, a stratified sample, a cluster sample, or a systematic sample.

2. Choose a simple random sample, stratified random sample, cluster sample, and systematic random sample in a variety of situations.

3. Explain the affect of sample size when determining whether a sample is representative of the population.

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…

3 Key Ideas That Enable Us to Make the Stretch

Idea 1: Examine a Part of the WholeThe 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.

Sampling methods

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.

CNN on-line surveys:

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.”

Example: hospital employee drug use

• 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

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.

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 oncalculator).

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.

Hospital example (cont.)

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

01 NCSU 02 UNC 03 Duke 04 Wake F 05 BC 06 UM 07 Maryl.08 Clem 09 UVA 10 VaTech 11 GaTech 12 FSU 13 OSU 14 ILL15 IN 16 PUR 17 IOWA 18 MSU 19 Mich 20 PennS 21 NorthW 22 MN 23 WISC

96927 19931 36089 74192 77567 88741 48409 41903The first 3 schools in a random sample selected from the ACC and Big Ten using the above random numbers are:

1 2 3 4 5

0% 0% 0%0%0%

1. UVA, UM, UNC

2. UVA, NCSU, Duke

3. UVA, UM, UVA

4. Clem, Mich, Duke

5. Mich, OSU, Maryl

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 versus sample• 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

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 parameter µ

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

3.2 Simple Random SamplesDesire the sample to be representative of the

population from which the sample is selected

Each individual in the population should have an equal chance to be selected

Is this good enough?

Example Select a sample of high school students as follows:

1. Flip a fair coin

2. If heads, select all female students in the school as the sample

3. If tails, select all male students in the school as the sample

Each student has an equal chance to be in the sample Every sample a single gender, not representative Each individual in the population has an equal chance

to be selected. Is this good enough? NO!!

Simple Random SamplesA simple random sample (SRS) of size n

consists of n units from the population chosen in such a way that every set of n units has an equal chance to be the sample actually selected.

Simple Random Samples (cont.)

• Suppose a large History class of 500 students has 250 male and 250 female students.

• To select a random sample of 250 students from the class, I flip a fair coin one time.

• If the coin shows heads, I select the 250 males as my sample; if the coin shows tails I select the 250 females as my sample.

• What is the chance any individual student from the class is included in the sample?

• This is a random sample. Is it a simple random sample?½

NO!

Not every possible group of 250 students has an equal chance to be selected.

Every sample consists of only 1 gender – hardly representative.

Sampling Frame

• 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 NCSU, we might obtain a list of all registered full-time students from Registration & Records.

– 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 sampleAcademic dept wishes to randomly choose

a 3-member committee from the 28 members of the dept

01 Abbott 08 Goodwin 15 Pillotte 22 Theobald

02 Cicirelli 09 Haglund 16 Raman 23 Vader

03 Crane 10 Johnson 17 Reimann 24 Wang

04 Dunsmore 11 Keegan 18 Rodriguez 25 Wieczoreck

05 Engle 12 Lechtenb’g 19 Rowe 26 Williams

06 Fitzpat’k 13 Martinez 20 Sommers 27 Wilson

07 Garcia14 Nguyen 21 Stone 28 Zink

Solution

• Use a random number table; read 2-digit pairs until you have chosen 3 committee members

• For example, start in row 121:• 71487 09984 29077 14863 61683 47052 62224 51025

Garcia (07) Theobald (22) Johnson (10) Your calculator generates random numbers; you

can also generate random numbers using Excel

Sampling Variability

• Suppose we had started in line 145?• 19687 12633 57857 95806 09931 02150 43163 58636

• Our sample would have been

19 Rowe, 26 Williams, 06 Fitzpatrick

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!!

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

Sex• Male• Female

Age• under 20• 20-30• 31-40• 41-50

Occupation• professional• clerical• blue-collar

Stratified Random Sampling

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

Stratified Random Sampling

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

Stratum Income Population proportion

1 under $15,000 25% 2502 15,000-29,999 40% 4003 30.000-50,000 30% 3004 over $50,000 5% 50

Stratum size

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 selected 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 widelydispersed geographically.

e.g., all Toyota dealerships in NorthCarolina

Mean length of sentencesin our course text

• We would like to assess thereading level of our course textbased 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 all sentences on the selected pages or select a SRS of sentences from each of the selected 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 sentencesin 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 from the

randomly chosen pages.

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. For example: 3, 13, 23, 33, 43, 53, …, 493.• 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?

Yes

1/10

No

Summary: What have we learned? A representative sample can offer us important

insights about populations.– It’s the size of the sample, not its fraction of the

larger population, that determines the precision of the statistics it yields.

There are several ways to draw samples, all based on the power of randomness to make them representative of the population of interest:– Simple Random Sample, Stratified Sample, Cluster

Sample, Systematic Sample, Multistage Sample

Summary: What have we learned? (cont.)

Bias can destroy our ability to gain insights from our sample:

– Nonresponse bias can arise when sampled individuals will not or cannot respond.

– Response bias arises when respondents’ answers might be affected by external influences, such as question wording or interviewer behavior.

Summary: What have we learned? (cont.)Bias can also arise from poor sampling methods:

– Voluntary response samples are almost always biased and should be avoided and distrusted.

– Convenience samples are likely to be flawed for similar reasons.

– Even with a reasonable design, nonrepresentative sample frames create bias.

Undercoverage occurs when individuals from a subgroup of the population are selected less often than they should be.

End of Lecture Unit 3