Section 4.1 Notes - COMPLETE
1
Population vs. Sample
We draw samples from a population because we are interested in inferring something about the population based on the sample. We sample when a census is impractical.
In order to draw a sample, we identify the population of interest and then choose a sample that is representative (hopefully) of the population of interest.
Section 4.1 Notes - COMPLETE
2
POPULATION = the group we WANT information about
SAMPLE = the group we GET information about
SAMPLING FRAME = the group of individuals from which we will draw our sample (ideally the entire population - but not always)
Why do we sample?
Parameters* of interest Corresponding sample Statistics**
*Parameter: a value describing the entire population**Statistic: a value describing the sample
Section 4.1 Notes - COMPLETE
3
Example: Identify the population and sample in each of the following settings.
(a) The student government at a high school surveys 100 of the students at the school to get their opinions about a change to the bell schedule.
(b) The quality control manager at a bottling company selects a sample of 10 cans from the production line every hour to see if the volume of the soda is within acceptable limits.
An archaeological dig turns up large numbers of pottery shards, broken stone implements, and other artifacts. Students working on the project classify each artifact and assign it a number. The counts in different categories are important for understanding the site, so the project director chooses 2% of the artifacts at random and checks the students' work. What are the population and sample here?
Section 4.1 Notes - COMPLETE
4
Sample Surveys
A “sample survey” is a study that uses an organized plan to choose a sample that represents some specific population.
1. Define a population we want to describe.
2. Say exactly what we want to measure (the variable(s)).
3. Decide how to choose a sample from the population. This is known as the sampling design.
When sampling goes badly...
Sampling Errors:Mistakes made in the process of taking a sample that could lead to inaccurate information about the population
Non-Sampling Errors:Errors that arise in the data collection process as a result of factors OTHER THAN taking a sample
Section 4.1 Notes - COMPLETE
5
Voluntary Response Sampling:• Consists of people who choose themselves by responding to a
general appeal. • Biased because people with strong opinions (often in the same
direction) are most likely to respond.
Example:
Convenience Sampling:• Consists of individuals who are easiest to reach• Tend to have similar opinions and typically are not
representative of the population
Example:
These are SAMPLING ERRORS
Bad Sampling Designs
Undercoverage - when some groups in the population are left out of the process of choosing the sample.
Example:
This is an example of a SAMPLING ERROR
Section 4.1 Notes - COMPLETE
6
BIAS
A sampling method is considered biased when it, by its nature, tends to over- or underemphasize some characteristics of the population of interest.
Wording of Questions- confusing or leading questions can change respondents answers leading to a strong bias - even the order of the questions asked is important.
Response Bias- a systematic pattern of incorrect responses - this can occur because of the type of question asked and the perceived "correct" answer or because of the person asking the question.
Example:
These are both examples of NON-SAMPLING ERRORS
http://www.youtube.com/watch?v=G0ZZJXw4MTA
Nonresponse - When an individual chosen for the sample can't be contacted or refuses to participate.
Example:
Also, a NON-SAMPLING ERROR
Section 4.1 Notes - COMPLETE
7
When sampling goes well...
The statistician’s remedy to sampling error is to allow impersonal chance to choose the sample. A sample chosen by chance rules out both favoritism by the sampler and self-selection by respondents.
Random sampling, the use of chance to select a sample, is the central principle of statistical sampling.
SimpleRandomSamples
Asimplerandomsample(SRS)ofsizenconsistsofnindividualsfromthepopulationchoseninsuchawaythateverysetofnindividualshasanequalchancetobethesampleactuallyselected.
UltimatelyitiseasiertoshowsomethingisNOTaSRSratherthanitis-howdoyoupossiblyshowthatEVERYsetofnindividualscanbethesample.
TheeasiestwaytoshowthatasamplingmethodDOESNOTSATISFYTHEDEFINITIONOFSRSistoshowthatthereissomegroupofnindividualsthathasNOPOSSIBLEWAYofbeingchosen-thusnotallgroupsarepossibleandcan'tbeequallylikelytobethesampleselected.
Section 4.1 Notes - COMPLETE
8
In order to gather an SRS, we can use a table of Random digits or our calculators to generate random digits.
using Table (what you will always do in this class):1. The table contains a long string of digits 0, 1, 2, 3, 4, 5, 6, 7, 8,
& 92. Each entry in the table is equally likely to be any of the 10
digits 0 to 93. The entries are independent of each other. That is, knowledge
of one part of the table gives no information about any other part
using the Calculator:1. Select MATH, arrow over to PRB, select #5 randInt2. The inputs need to be your starting value (typically 0 or 1), the
ending value, and how many random values you want3. Keep in mind, if there are repeats you may need to generate
more random digits
Section 4.1 Notes - COMPLETE
9
Using either method, you need to:
1. LABEL - Give each member of the population a numerical label of the same length.• i.e. if your population is 900 individuals you will number them
000 - 899
2. TABLE/CALCULATOR - use successive groups of digits of the length you used as labels from the table of random digits to find the sample• i.e. if your population is 900 individuals you will read 3 digits at
a time
3. Be sure to address the issue of numbers outside of your population when using a table and the issue of repeated values for both methods.• i.e. if your population is 900 individuals what will you do with
numbers greater than 899?
The management company of a local mall plans to survey a random sample of 5 stores to determine the hours they would like to stay open during the holiday season. Use Table D at line 101 to select an SRS of size 5 stores.
Aeropostale Forever 21 Old NavyAll American Burger GameStop Pac SunArby’s Gymboree Panda ExpressBarnes & Noble Haggar Payless ShoesCarter’s for Kids Just Sports Star JewelersDestination Tan Mrs. Fields Vitamin WorldFamous Footwear Nike Factory Store Zales
Section 4.1 Notes - COMPLETE
10
To gather data on a 1200-acre pine forest in Louisiana, the U.S. Forest Service laid a grid of 1410 equally spaced circular plots over a map of the forest. A ground survey visited a sample of 10% of these plots.
(a) How would you label the plots?
(b) What is the total sample size for this study?
(c) Use your calculator to choose the first 10 plots.
Stratified Random Sample
1.Dividethepopulation(orsamplingframe)intostrata(likesub-populations)whicharehomogenousgroupsofindividualsthataresimilarinsomewaythatisimportanttotheresponse
2.ChooseaSRSfromeachstrataproportionalinsizetothestrata'ssizewithinthepopulation.Forexample:Iffemalesrepresent30%ofthepopulationandyouarestratifyingbasedongender,then30%ofyourZinalsampleshouldbefemales.
Section 4.1 Notes - COMPLETE
11
EXAMPLE: The manager of a beach-front hotel wants to survey guests in the hotel to estimate overall customer satisfaction. The hotel has two towers, an older one to the south and a newer one to the north. Each tower has 10 floors of standard rooms (40 rooms per floor) and 2 floors of suites (20 suites per floor). Half of the rooms in each tower face the beach, while the other half of the rooms face the street. This means there are (2 towers)(10 floors)(40 rooms) + (2 towers)(2 floors)(20 suites) = 880 total rooms.
Describe a method for gathering a stratified random sample - there are multiple options!
A club has 30 students:
Abel Fisher Huber Miranda ReinmannCarson Ghosh Jimenez Moskowitz SantosChen Griswold Jones Neyman ShawDavid Hein Kim O'Brien ThompsonDeming Hernandez Klotz Pearl UttsElashoff Holland Liu Potter Varga
and 10 faculty members:
Andrew Fernandez Kim Moore WestBesicovitch Gupta Lightman Vicario Yang
The club can send 4 students and 2 faculty members to a convention. It decides to choose those who will go by random selection. Using the table of random digits, starting at line 18, to choose a stratified random sample of 4 students and 2 faculty members.
Section 4.1 Notes - COMPLETE
12
Cluster Sampling
ClustersamplingseemssimilartostratiZiedrandomsampling,butisinfactVERYdifferentinseveralways:
1.Splitthepopulationintogroups,orclusters,thatfairlyrepresenttheENTIREpopulation-NOTnecessarilyhomogenousgroups!
2.AssignlabelstoENTIREcluster-NOTindividuals
3.RandomlyselectCLUSTERS
4.SampleALLindividualswithinthecluster(i.e.acensus)
Systematic Sampling(not in the textbook...)
1. Inordertogatherasystematicsample,Zirstlabelallindividualsinthepopulation.
2. Thenselectastartingpoint-atRANDOM
3. Lastly,selecteverykthindividualfromthisstartingpoint.(Findkbytakingthepopulationsizedividedbythedesiredsamplesize).
Besurethatthereisnoreasontothinkthattheorderofthelistcanbeassociatedwiththeresponse.
Section 4.1 Notes - COMPLETE
13
Multistage Sampling
Multistage samples are sampling designs that combine several different methods.
Examples:
Inference for Sampling
The purpose of a sample is to give us information about a larger population.
The process of drawing conclusions about a population on the basis of sample data is called inference.
Why should we rely on random sampling?
To eliminate bias in selecting samples from the list of available individuals.
The laws of probability allow trustworthy inference about the population.
Results from random samples come with a margin of error that sets bounds on the size of the likely error.
Larger random samples give better information about the population than smaller samples.
Section 4.1 Notes - COMPLETE
14
Homework: p. 226 #s 1 - 11 odd, 17 - 27 odd, 28, 29, 31, 33, 35
QUIZ (FOR A GRADE) ON SECTION 4.1 THE CLASS RIGHT AFTER WE FINISH THE NOTES
