STAT 101: Day 2Data Collection: Sampling

1/18/12

• Sample versus Population• Statistical Inference• Sampling Bias• Simple Random Sample• Other Sources of Bias

Section 1.2

Course Website

http://stat.duke.edu/courses/Spring12/sta101.2/

Sample vs Population

• A population includes all individuals or objects of interest

• A sample is all the cases that we have collected data on, usually a subset of the population

• Statistical inference is the process of using data from a sample to gain information about the population

The Big Picture

Population

Sample

Sampling

Statistical Inference

Most Important to You

Which of the following is most important to you?

(a) Athletics(b) Academics(c) Social Life(d) Community Service(e) Other

Most Important to You

• Suppose researchers studying student life at Duke use the results of our clicker question to investigate what Duke students find important

• What is the sample?• What is the population?

• Can the sample data be generalized to make inferences about the population? Why or why not?

Sampling

Population

Sample

Sampling

GOAL: Select a sample that is similar to the population, only smaller

Dewey Defeats Truman?

Dewey Defeats Truman?

• The paper was published before the conclusion of the 1948 presidential election, and was based on the results of a large telephone poll which showed Dewey sweeping Truman

• However, Harry S. Truman won the election

• What went wrong?

Sampling Bias

• Sampling bias occurs when the method of selecting a sample causes the sample to differ from the population in some relevant way

• If sampling bias exists, we cannot trust generalizations from the sample to the population

Sampling

Population Sample

Sample

Can you avoid sampling bias?

• The next slide shows Lincoln’s Gettysburg Address. The entire population, all words in his address, will be shown to you.

• Your task: Select a sample of 10 words that resemble the overall address. Write them down.

• Calculate the average number of letters for the words in your sample

• Place a dot above your sample average on the board

Lincoln’s Gettysburg Address“Four score and seven years ago our fathers brought forth, on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal. Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battle-field of that war. We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this. But, in a larger sense, we can not dedicate—we can not consecrate—we can not hallow—this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they here gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.”

Can you avoid sampling bias?

• Actual average: 4.29 letters

• People are TERRIBLE at selecting a good sample, even when explicitly trying to avoid sampling bias!

• We need a better way…

Random Sampling

• How can we make sure to avoid sampling bias?

• Imagine putting the names of all the units of the population into a hat, and drawing out names at random to be in the sample

Take a RANDOM sample!

Random Sampling

• Before the 2008 election, the Gallup Poll took a random sample of 2,847 Americans. 52% of those sampled supported Obama

• In the actual election, 53% voted for Obama

• Random sampling is a very powerful tool!!!

Selecting a Random Sample• Option 1: Actually draw names out of a hat

• Option 2: Number all units in the population, and generate random numbers

Online: http://www.random.org/integers/

RStudio: To generate n random numbers between 1 and max, use

sample(1:max, n)

> sample(1:100,5) [1] 66 4 51 18 70

Selecting a Random Sample• Option 3: Use RStudio to randomly sample

directly from a vector of population units

population = vector of population unitsn = sample size

sample(population, n)

“Random” Numbers

1. Pick 10 “random” numbers between 1 and 268. Write these numbers down.

(Note: When choosing a real sample, you should use technology to generate random numbers. This is simply for illustrative purposes in class.)

2. Using the next slide, calculate the average number of letters in the words corresponding to your random numbers

3. Place a dot above this average on the board

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Random vs Non-Random Sampling

• Random samples have averages that are centered around the correct number

• Non-random samples may suffer from sampling bias, and averages may not be centered around the correct number

• Only random samples can truly be trusted when making generalizations to the population!

Bowl of Soup Analogy

Think of tasting a bowl of soup…

• Population = entire bowl of soup• Sample = whatever is in your tasting bites

• If you take bites non-randomly from the soup (if you stab with a fork, or prefer noodles to vegetables), you may not get a very accurate representation of the soup

• If you take bites at random, only a few bites can give you a very good idea for the overall taste of the soup

Simple Random Sample

• These methods generate a simple random sample

• In a simple random sample, each unit of the population has the same chance of being selected, regardless of the other units chosen for the sample

• More complicated random sampling schemes exist, but will not be covered in this course

Realities of Sampling

• While a random sample is ideal, often it isn’t feasible. A list of the entire population may not be available, or it may be impossible or too difficult to contact all members of the population.

• Sometimes, your population of interest has to be altered to something more feasible to sample from. Generalization of results are limited to the population that was actually sampled from.

• In practice, think hard about potential sources of sampling bias, and try your best to avoid them

Non-Random Samples

Suppose you want to estimate the average number of hours that Duke students spend studying each week. Which of the following is the best method of sampling?

(a) Go to the library and ask all the students there how much they study

(b) Email all Duke students asking how much they study, and use all the data you get

(c) Give a clicker question in STAT 101 and force every student to respond

(d) Stand outside the Bryan Center and ask everyone going in how much they study

Bad Methods of Sampling

• Sampling units based on something obviously related to the variable(s) you are studying

– Sampling only students in the library when asking how much they study, or sampling only students taking a statistics class

– “Today’s Poll” on fitnessmagazine.com asked “Have you ever hired a personal trainer?”. 27% of respondents said “yes” – can we infer that 27% of all humans have hired a personal trainer?

Bad Methods of Sampling

• Letting your sample be comprised of whoever chooses to participate (volunteer bias)

– Emailing or mailing the entire population, and then making conclusions about the population based on whoever chooses to respond

–Example: An airline emails all of it’s customers asking them to rate their satisfaction with their recent travel

Road Safety• The Federal Office of Road Safety in Australia conducted a study on the effects of alcohol and marijuana on performance

• Participants were volunteers who responded to advertisements for the study on rock radio stations

• Volunteers were given a random combination of the two drugs, then their performance was observed

• What is the sample? What is the population?• Is there sampling bias?• Will the results be informative and/or do you think the study is worth conducting?

DATA

Data Collection and Bias

PopulationSample

Sampling Bias?

Other forms of bias?

Other Forms of Bias

• Even with a random sample, data can still be biased, especially when collected on humans

• Other forms of bias to watch out for in data collection:

– Question wording– Context– Inaccurate responses– Many other possibilities – examine the specifics of each study!

Question Wording

• “Do you think the US should allow public speeches against democracy?”

• “Do you think the US should not forbid public speeches against democracy?”

Source: Rugg, D. (1941). “Experiments in wording questions,” Public Opinion Quarterly, 5, 91-92.

21% said speeches should be allowed

39% said speeches should be not be forbidden

Question Wording• A random sample was asked: “Should there be a tax cut, or should money be used to fund new government programs?”

• A different random sample was asked: “Should there be a tax cut, or should money be spent on programs for education, the environment, health care, crime-fighting, and military defense?”

Tax Cut: 60% Programs: 40%

Tax Cut: 22% Programs: 78%

Context• Ann Landers column asked readers

“If you had it to do over again, would you have children?

• The first request for data contained a letter from a young couple which listed worries about parenting and various reasons not to have kids=> 30% said “yes”

• The second request for data was in response to this number, in which Ann wrote how she was “stunned, disturbed, and just plain flummoxed”Þ95% said “yes”

Having Children• If we were to run the question all by itself in the newspaper with a request for responses, could we trust the results?

(a) Yes(b) No

Having Children

Newsday conducted a random sample of all US adults, and asked them the same question, without any additional leading materialÞ91% said “yes”

Do you think the true proportion of parents who are happy they had children is close to 91%?

(a) Yes(b) No

Inaccurate Responses

• In a study on US students, 93% of the sample said they were in the top half of the sample regarding driving skill

Svenson, O. (February 1981). "Are we all less risky and more skillful than our fellow drivers?".  Acta Psychologica 47 (2): 143–148.

• From random sample of all US college students, 22.7% reported using illicit drugs. Do you think this number is accurate?Substance Abuse and Mental Health Services Administration (2010). “Results from the 2009 National Survey on Drug Use and Health: Volume 1.” Summary of National Findings (Office of Applied Studies, NSDUH Series H-38A, HHS Publication No. SMA 10-4856Findings). Rockville, MD, heeps://nsduhweb.rti.org/

Summary• Data is collected on a sample, and we would like to use

the data to make inferences to the larger population

• Sampling bias can occur when the sample does not resemble the population

• Sampling bias can be avoided by random sampling

• Bias exists when the sample data do not accurately reflect the true population data, and bias can occur in many ways

• When making conclusions based on data, STOP AND THINK ABOUT HOW THE DATA WERE COLLECTED!

Summary

Always think critically about how the data were collected, and recognize that not all forms of data

collection lead to valid inferences

To Do• Complete the class survey on Sakai (due Monday,

1/23)

• Email me if you still need a textbook

• Email me with your gmail adress if you still need an RStudio account

• Buy a clicker (grading starts 1/30)(go to this google doc if you want to buy one used

from a previous student)