Samples & Surveys - Santa Paula High School

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Chapter 4: Designing Studies Section 4.1 Samples and Surveys The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

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Chapter 4 Designing Studies

 4.1

Samples and Surveys

 4.2

Experiments

 4.3

Using Studies Wisely

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Samples and Surveys Learning Objectives

After this section, you should be able to… 

IDENTIFY the population and sample in a sample survey

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IDENTIFY voluntary response samples and convenience samples

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DESCRIBE how to use a table of random digits to select a simple random sample (SRS)

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DESCRIBE simple random samples, stratified random samples, and cluster samples

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EXPLAIN how undercoverage, nonresponse, and question wording can lead to bias in a sample survey

+ Activity: Hiring Discrimination Directions: - Without looking, remove beads one at a time until 8 beads have been chosen. - Count the number of female pilots selected. Record this value in the table in your notes. Simulation

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Activity: The Federalist Papers 

The Federalist Papers are a series of 85 essays supporting the ratification of the U.S. Constitution. At the time they were published, the identity of the authors was a secret known to just a few people. Over time, however, the authors were identified as Alexander Hamilton, James Madison, and John Jay.

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The authorship of 73 of the essays is fairly certain, leaving 12 in dispute. However, thanks in some part to statistical analysis, most scholars now believe that the 12 disputed essays were written by Madison alone or in collaboration with Hamilton.

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There are several ways to use statistics to help determine the authorship of a disputed text. One example is to estimate the average word length in a disputed text and compare it to the average word lengths of works where the authorship is not in dispute.

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and Sample

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 Population

Definition: The population in a statistical study is the entire group of individuals about which we want information. A sample is the part of the population from which we actually collect information. We use information from a sample to draw conclusions about the entire population.

Population

Sampling and Surveys

The distinction between population and sample is basic to statistics. To make sense of any sample result, you must know what population the sample represents

Collect data from a representative Sample...

Sample

Make an Inference about the Population.

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Populations and Samples

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 Population All students at the school

Sample 100 students surveyed

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Populations and Samples

The quality control manager at a bottling company selects a sample of 10 cans from the production line every hour to see whether the volume of the soda is within acceptable limits Population All cans produced that hour

Sample 10 cans inspected

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Populations and Samples Think of your own example! Be ready to share with the class in 2 minutes!

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to Sample Badly

Definition: Convenience Sample Choosing individuals who are easiest to reach results in a convenience sample.

Convenience samples often produce unrepresentative data…why? Definition: Bias The design of a statistical study shows bias if it systematically favors certain outcomes.

Sampling and Surveys

How can we choose a sample that we can trust to represent the population? There are a number of different methods to select samples.

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 How

+ AP EXAM TIP ALERT When you describe bias, you will be expected to describe the DIRECTION of the bias. Example: Explain how using a convenience sample of students in your statistics class to estimate the percentage of all high school students who have a graphing calculator could result in bias. Answer: This sample would probably include a much higher percentage of students with a graphing calculator than in the population of the entire high school because a graphing calculator is required for the statistics class. This method would overestimate the percentage of students with a graphing calculator.

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to Sample Badly

samples are almost guaranteed to show bias. So are voluntary response samples, in which people decide whether to join the sample in response to an open invitation.

Definition: A voluntary response sample consists of people who choose themselves by responding to a general appeal. Voluntary response show bias that because Voluntarysamples responses means people people with strong opinions (often in once! the same can respond more than Think direction) are most likely to Idol respond. about American or Dancing with the Stars – you can vote multiple times per phone number or email address!

Sampling and Surveys

 Convenience

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 How

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Check your understanding: Pg 211  A farmer brings a juice company several crates of Answer: Convenience sampling. This could

oranges week. A company inspector at lead him each to think that the oranges are looks of better 10 oranges from the top of each crate before quality than they actually are. He might deciding whether to buy all the oranges. overestimate the amount of high quality Answer: oranges inVoluntary the bin! response sampling. Only those who feel particularly strongly about the issue are  The ABC program Nightline once asked whether the likely to respond. In this case, people who are United Nations should continue to have its happy with the what theywere want and headquarters in location the Unitedhave States. Viewers are lesstolikely to respond. This method might invited call one telephone number to respond “Yes” and anotherthe for percentage “No.” There was a chargewho for are underestimate of people callingand either number. More than 168,000 callers happy overestimate the percentage of people responded, and 67% said “No.” who are unhappy.

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to Sample Well: Random Sampling

The statistician’s remedy 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.

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Random sampling, the use of chance to select a sample, is the central principle of statistical sampling.

Definition: A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected. In practice, people use random numbers generated by a computer or calculator to choose samples. If you don’t have technology handy, you can use a table of random digits.

Sampling and Surveys

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 How

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Choosing an SRS in our classroom # 1 – I need an SRS of size 4. I will put all of your names into a hat and pick out 4 names.

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# 2 – To be equitable, I want a sample with 2 girls and 2 boys. I need two hats now – one with only girls’ names and the other with only boys’ names. Is this still an SRS?

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NO. A sample of 4 students that has for example all boys, all girls, or some other mixture other than 2 and 2 cannot be chosen. In an SRS, every possible sample of size 4 needs to have the same chance of being chosen.

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to Choose an SRS with Table D

How to Choose an SRS Using Table D

Step 1: Label. Give each member of the population a numerical label of the same length. Step 2: Table. Read consecutive groups of digits of the appropriate length from Table D. Your sample contains the individuals whose labels you find.

Sampling and Surveys

Definition: A table of random digits is a long string of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with these properties: • Each entry in the table is equally likely to be any of the 10 digits 0 - 9. • The entries are independent of each other. That is, knowledge of one part of the table gives no information about any other part.

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 How

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Problem: Use Table D at line 130 to choose an SRS of 4 hotels.

01 Aloha Kai 02 Anchor Down 03 Banana Bay 04 Banyan Tree 05 Beach Castle 06 Best Western 07 Cabana

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08 Captiva 09 Casa del Mar 10 Coconuts 11 Diplomat 12 Holiday Inn 13 Lime Tree 14 Outrigger

15 Palm Tree 16 Radisson 17 Ramada 18 Sandpiper 19 Sea Castle 20 Sea Club 21 Sea Grape

22 Sea Shell 23 Silver Beach 24 Sunset Beach 25 Tradewinds 26 Tropical Breeze 27 Tropical Shores 28 Veranda

64817 87174 09517 84534 06489 87201 97245

Sampling and Surveys

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How to Choose an SRS (Page 213)

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 Example:

69 05 16 48 17 87 17 40 95 17 84 53 40 64 89 87 20 Our SRS of 4 hotels for the editors to contact is: 05 Beach Castle, 16 Radisson, 17 Ramada, and 20 Sea Club.

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Another Example: Mall Hours The management company of a local mall plans to survey a random sample of 3 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 3 stores.

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SRS or Not? There are 1400 Students at Santa Paula High School. You ask the 44 students in your Statistics class to take a survey.

NO – Not every possible group of 44 students at SPHS has the same chance of being chosen because most of the students are not in Statistics. This is a convenience sample.

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SRS or Not? There are 100 kids playing soccer in the park. You ask the 10 kids in line at the drinking fountain to answer a question.

NO – Not every possible group of 10 kids has the same chance of being chosen. This is a convenience sample.

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SRS or Not? There are 100 kids playing soccer in the park. You decide to ask 5 girls and 5 boys to answer a question.

NO – Not every possible group of 10 kids has the same chance of being chosen. A group of 10 boys or a group of 8 boys and 2 girls does not have the same chance as the method you decided to use.

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SRS or Not? There are 100 kids playing soccer in the park. You use team rosters to randomly select 10 kids to survey.

YES –every possible group of 10 kids has the same chance of being chosen. This is a Simple Random Sample (SRS).

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Replacement: with or without? Sampling with replacement allows individuals to be selected more than once. Their names go back into the hat after each selection. Sampling without replacement does not allow individuals to be selected more than once. Once they are chosen, their name does not go back into the hat. If you are using table D to select a SRS, you simply indicate that you are “skipping” repeats. If you are using a calculator to select a SRS, generate extra numbers because the calculator samples with replacement.

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Sampling Methods: The River  Suppose

we wanted to estimate the yield of our corn field. The field is square and divided into 16 equally sized plots (4 rows x 4 columns). A river runs along the eastern edge of the field. We want to take a sample of 4 plots.

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a random number generator or table D, pick a simple random sample (SRS) of 4 plots

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Sampling Methods

The basic idea of sampling is straightforward: take an SRS from the population and use your sample results to gain information about the population. Sometimes there are statistical advantages to using more complex sampling methods.

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One common alternative to an SRS involves sampling important groups (called strata) within the population separately. These “sub-samples” are combined to form one stratified random sample.

Definition: To select a stratified random sample, first classify the population into groups of similar individuals, called strata. Then choose a separate SRS in each stratum and combine these SRSs to form the full sample.

Sampling and Surveys

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 Other

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Sampling Methods

a stratified random sample can sometimes give more precise information about a population than an SRS, both sampling methods are hard to use when populations are large and spread out over a wide area.

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that situation, we’d prefer a method that selects groups of individuals that are “near” one another.

Definition: To take a cluster sample, first divide the population into smaller groups. Ideally, these clusters should mirror the characteristics of the population. Then choose an SRS of the clusters. All individuals in the chosen clusters are included in the sample.

Sampling and Surveys

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 Other

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Sampling at a School Assembly (218)

Describe how you would use the following sampling methods to select 80 students to complete a survey.

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(a) Simple Random Sample

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(b) Stratified Random Sample

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(c) Cluster Sample

Sampling and Surveys

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 Example:

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Example: Hotel on the Beach

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for Sampling

The purpose of a sample is to give us information about a larger population.

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The process of drawing conclusions about a population on the basis of sample data is called inference.

Why should we rely on random sampling? 1)To eliminate bias in selecting samples from the list of available individuals. 2)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.

Sampling and Surveys

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 Inference

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Surveys: What Can Go Wrong?

Most sample surveys are affected by errors in addition to sampling variability.

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Good sampling technique includes the art of reducing all sources of error.

Definition Undercoverage occurs when some groups in the population are left out of the process of choosing the sample. Nonresponse occurs when an individual chosen for the sample can’t be contacted or refuses to participate. A systematic pattern of incorrect responses in a sample survey leads to response bias. The wording of questions is the most important influence on the answers given to a sample survey.

Sampling and Surveys

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 Sample

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Sampling Errors & Nonsampling Errors A sampling error has to do with choosing the sample. -

Voluntary response - Undercoverage

A nonsampling error happens after the sample has been chosen. Nonresponse - Response bias Wording of questions -

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Check your understanding (p224) #1a – sampling error #1b – nonsampling error #1c – Sampling error #2 – By making it sound like disposable diapers are not a problem in the landfill, this question will result in fewer people suggesting that we should ban them. The author of the question highlights several other items that take up more space in the landfill, which makes it look like disposable diapers are really not a problem.

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Activity: See no evil, hear no evil? Follow the directions on Page 206

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Turn in your results to your teacher.

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Teacher: Right-click (control-click) on the graphs to edit the counts.

Sampling and Surveys

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Section 4.1 Samples and Surveys Summary In this section, we learned that… 

A sample survey selects a sample from the population of all individuals about which we desire information.

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Random sampling uses chance to select a sample.

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The basic random sampling method is a simple random sample (SRS).

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To choose a stratified random sample, divide the population into strata, then choose a separate SRS from each stratum.

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To choose a cluster sample, divide the population into groups, or clusters. Randomly select some of the clusters for your sample.

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Section 4.1 Samples and Surveys Summary, con’t In this section, we learned that… 

Failure to use random sampling often results in bias, or systematic errors in the way the sample represents the population.

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Voluntary response samples and convenience samples are particularly prone to large bias.

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Sampling errors come from the act of choosing a sample. Random sampling error and undercoverage are common types of error.

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The most serious errors are nonsampling errors. Common types of sampling error include nonresponse, response bias, and wording of questions.

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Looking Ahead… In the next Section… We’ll learn how to produce data by designing experiments. We’ll learn about  Observational Studies vs. Experiments  The Language of Experiments  Randomized Comparative Experiments  Principles of Experimental Design  Inference for Experiments  Blocking  Matched Pairs Design