Chapter 3: Producing Data

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Chapter 3 Producing Data
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1. Inferential Statistics

• Population The population is the group of
  people or things that were interested in. This
  group is defined by whatever question is being
  asked.
• Example 1 Do Texas AM students have breakfast
  regularly?
• How many populations are of interest?
• One
• What is the population of interest?
• All current Texas AM students

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1. Inferential Statistics

• Example 2 Is the IQ of women the same as the IQ
  of men?
• How many populations are of interest?
• Two
• What are the populations of interest?
• All women and all men
• Example 3 Which is more effective at lowering
  the heart rate of mice, no drug (control), drug
  A, drug B, or drug C?
• How many populations are of interest?
• Four
• What are the populations of interest?
• All mice taking no drug, all mice taking drug A,
  all mice taking drug B, all mice taking drug C

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1. Inferential Statistics

• Suppose we have no previous information about
  these questions. How could we answer them?
• Census
• Advantages
• We get everyone, we know the truth
• Disadvantages
• Expensive, Difficult to obtain, may be
  impossible.
• Sample A subset of the population selected for
  the study
• Advantages
• Take less time and money. Feasible.
• Disadvantages
• Uncertainty about the truth. We may have error.

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1. Inferential Statistics

• Example 1 Do Texas AM students have breakfast
  regularly?
• Population all current Texas AM students
• Sample 1 all STAT 30X students
• Sample 2 students who come to Blocker building
  on Monday morning
• Sample 3 randomly choose names from the phone
  book and ask them
• .
• One population has many sampling methods

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1. Inferential Statistics

• General Idea of Inferential Statistics
• 1. Take a sample from the whole population.
• 2. Summarize the sample using important
  statistics.
• 3. Use those summaries to make inference about
  the whole population.
• 4. We realize there may be some error involved in
  making inference.

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1. Inferential Statistics

• Example
• Question Can Aspirin reduce the risk of heart
  attack?
• 1. Take Sample Sample of 22,071 male physicians
  between the ages of 40 and 84, randomly assigned
  to one of two groups. One group took an ordinary
  aspirin tablet every other day. The other group
  took a placebo every other day. This group is
  the control group.
• 2. Summary statistic The rate of heart attacks
  in the group taking aspirin was only 55 of the
  rate of heart attacks in the placebo group.
• 3. Inference to population Taking aspirin causes
  lower rate of heart attacks in humans.

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2. Sampling a Single Population

• Basics for sampling
• Sampling should not be biased A sampling method
  is biased if any part of the population cant get
  in.
• Example 1 Only select STAT 30X students ---
  biased
• The selection of an individual in the population
  should not affect the selection of the next
  individual independence.
• Example 1 Survey one student, then ask him to
  introduce his friend to you. --- dependent
• Sampling should be large enough to adequately
  cover the population. A good sample is one that
  is collected in such a way that it is
  representative of the population.
• Example 1 only ask 3 students --- sample size
  too small

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2. Sampling a Single Population

• Sampling Techniques
• Simple Random Sample (SRS) every member of the
  population has an equal chance of being selected.

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2. Sampling a Single Population

• Sampling Techniques
• Simple Random Sample (SRS)
• Assign every individual a number and randomly
  select n numbers using a random number table (or
  computer generated random numbers).
• Table B at the back of the book is random digits.
  
• Example1 Obtain a list of all TAMU students and
  assign every student a number. Using a random
  number table, select 50 of them.
• Example Obtain a list of all SSN for individuals
  in the U.S. who are over 65. Using a random
  number table, select 50 of them.

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2. Sampling a Single Population

• Sampling Techniques SRS Exercise
• Choose a SRS three names from the following
  employees of a small company. Bechhofer
  Brown Ito Kesten
  Kiefer Spitzer Taylor Wald
  WeissUse the numerical labels attached to the
  names above and the following list of random
  digits. Read the list of random digits from left
  to right, starting at the beginning of the
  list.11793 20495 05907 11384 44982 20751 27498
  12009 45287
• The simple random sample is
• a)     1(1)79
• b)    Bechhofer, then Bechhofer again, then
  Taylor
• c)     Bechhofer, Taylor, Weiss

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2. Sampling a Single Population

• Sampling Techniques
• Stratified Random Sample Divide the population
  into several strata. Then take a SRS from each
  stratum.

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2. Sampling a Single Population

• Sampling Techniques
• Stratified Random Sample
• Example1 Obtain a list of all TAMU students and
  divide them into colleges. Then randomly sample
  10 from each college.
• Example Obtain a list of all SSN for individuals
  in the U.S. who are over 65. Divide up the SSNs
  into region of the country (time zones). Then
  randomly sample 30 from each time zone.
• Advantage Each stratum is guaranteed to be
  randomly sampled
• Disadvantage No longer a truly random sample

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Sampling a Single Population

• Sampling Techniques
• Cluster Sample Divide the population into
  several strata or clusters. Then take a SRS of
  clusters.

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Sampling a Single Population

• Sampling Techniques
• Cluster Sample
• Advantage May be the only feasible method, given
  resources.
• Example Obtain a list of all SSNs for
  individuals in the U.S. who are over 65. Sort
  the SSNs by the last 4 digits making each set of
  100 a cluster. Use a random number table to pick
  the clusters. You may get the 4100s, 5600s and
  8200s for example.

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Sampling a Single Population

• Sampling Techniques
• Multi-Stage Sample Divide the population into
  several strata. Then take a SRS from a random
  subset of all the strata.

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Sampling a Single Population

• Sampling Techniques
• Multi-Stage Sample
• Advantage May be the only feasible method, given
  resources.
• Example Obtain a list of all SSN for individuals
  in the U.S. who are over 65. Divide up the SSNs
  into 50 states. Randomly select 10 states. Then
  randomly sample 40 from each of the selected
  states.

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2. Sampling a Single Population

• Sampling Techniques Exercise
• A small college has 500 male and 600 female
  undergraduates. A simple random sample of 50 of
  the male undergraduates is selected and,
  separately, a simple random sample of 60 of the
  female undergraduates is selected. The two
  samples are combined to give an overall sample of
  110 students. The overall sample is
• a)     a simple random sample
• b)    a stratified random sample
• c)     a cluster sample
• d)    none of the above

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2. Sampling a Single Population

• Sampling Problems
• Voluntary response Internet surveys, Call-in
  surveys
• E.g. Survey about earning. People who take the
  survey can get free T-shirts. Busy people wont
  come, and these people often have high earnings.
  So our sample mean/median may be lower than the
  true mean/median.
• Convenience sampling Sampling friends, Sampling
  at the mall
• Problem They may have similar interests
• Dishonesty Asking personal questions, Not enough
  time to respond honestly

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2. Sampling a Single Population

• Cautions about Sample Surveys
• Undercoverage Some groups in the population are
  left out when the sample is taken
• Ex) sample survey of households will miss not
  only homeless people but prison inmates and
  students in dormitories.
• Nonresponse An individual chosen for the sample
  cant be contacted or does not cooperate
• Ex) phone survey, mail survey

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2. Sampling a Single Population

• Cautions about Sample Surveys
• Response Bias Results that are influenced by
  the behavior of the respondent or interviewer
• The wording of questions can influence the
  answers
• Eg) (Text p254) How do Americans feel about
  government help for the poor?
• Only 13 think we are spending too much on
  assistance to the poor.
• But 44 think we are spending too much on
  welfare.
• It seems that assistance to the poor is nice,
  hopeful word. Welfare is negative word.
• Respondent may not want to give truthful answers
  to sensitive questions

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2. Sampling a Single Population

• Ex. In order to assess the opinion of students at
  the University of Minnesota on campus snow
  removal, a reporter for the student newspaper
  interviews the first 12 students he meets who are
  willing to express their opinions. The method of
  sampling used is
• a)    simple random sampling
• b)    the Gallup Poll
• c)    voluntary response
• d) a census

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3. Sampling More than One Population

• We sample from more than one population when we
  are interested in more than one variable.
• One response variable and one explanatory
  variable. The populations are defined by the
  values the explanatory variable takes on.
• Example 1 Comparing decibel levels of 4
  different brands of speakers
• What is the explanatory variable?
• Brand
• What is the response variable?
• Decibel Level
• Number of Populations?
• Four

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3. Sampling More than One Population

• Example 2 Determining time to failure of 3
  different types of light bulbs
• What is the explanatory variable?
• Type
• What is the response variable?
• Time to Failure
• Number of Populations?
• Three

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3. Sampling More than One Population

• Example 3 Comparing GRE scores for students from
  5 different majors
• What is the explanatory variable?
• Major
• What is the response variable?
• GRE score
• Number of Populations?
• Five

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3. Sampling More than One Population

• Important Considerations
• Each sample should represent the population it
  corresponds to well.
• Samples from more than one population should be
  as close to each other in every respect as
  possible except for the explanatory variable.
  Otherwise we may have confounding variables.
• Two variables are confounded if we cannot
  determine which one caused the differences in the
  response.

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3. Sampling More than One Population

• Important Considerations
• Examples of Confounding
• Suppose we compared the decibel levels of the
  four different speaker brands, each with a
  different measuring instrument
• We wouldnt know if the differences were due to
  the different brands or different instruments.
• Brand and Instrument are then confounded.
• Suppose we compared the time to failure of the
  three different types of light bulbs, each in a
  different light socket.
• We wouldnt know if the differences were due to
  the different types of light bulbs or different
  light sockets.
• Type and Socket confounded.

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3. Sampling More than One Population

• Important Considerations
• Examples of Confounding
• Suppose we obtained GRE scores for each major,
  each from a different university.
• We wouldnt know if the differences were due to
  the different majors or different universities.
• Major and University are then confounded.
• Confounding can be avoided by using good sampling
  techniques

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3. Sampling More than One Population

• Important Considerations
• It is also possible that more than one (possibly
  several) explanatory variable can influence a
  given response variable.
• Example
• Perhaps both the type of light bulb and the type
  of light socket influence the time to failure of
  a light bulb.
• It is likely that different types of light bulbs
  work better for different sockets.
• This concept is known as interaction.
• Interaction The responses for the levels of one
  variable differ over the levels of another
  variable.

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3. Sampling More than One Population

• Good Sampling Techniques
• Randomized Experiment
• Observational Studies

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3. Sampling More than One Population

• Randomized Experiment
• The key to a randomized experiment the treatment
  (explanatory variable) is randomly assigned to
  the experimental units or subjects.

Random Assignment
Compare
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3. Sampling More than One Population

• Randomized Experiment
• Example Suppose we wish to do a study on the
  effect of aspirin on mice, comparing heart rates.
• We obtain a random sample of 100 mice.
• We randomly assign 50 mice to receive a placebo.
• We randomly assign 50 mice to receive aspirin.
• After 20 days of administering the placebo and
  aspirin, we measure the heart rates and obtain
  summary statistics for comparison.

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3. Sampling More than One Population

• Randomized Experiment
• The single greatest advantage of a randomized
  experiment is that we can infer causation.
• Through randomization to groups, we have
  controlled all other factors and eliminated the
  possibility of a confounding variable.
• We cannot always use a randomized experiment
• Often impossible or unethical, particularly with
  humans.

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3. Sampling More than One Population

• Observational Study
• We are forced to select samples from different
  pre-existing populations

Simple Random Sample
Compare
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3. Sampling More than One Population

• Observational Study
• Example 1 Suppose we are interested in comparing
  GRE scores for students in five different majors
• We cannot do a randomized experiment because we
  cannot randomly assign individuals to a specific
  major.
• Thus, we observe students from 5 different
  pre-existing populations the five majors.
• We obtain a random sample of size 15 from each of
  the five majors.
• We calculate statistics and compare the 5 groups.
• Can we say being in a specific major causes
  someone to get a higher GRE score?

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3. Sampling More than One Population

• Observational Study
• Example 2 Suppose we are interested finding out
  which age group talks the most on the telephone
  0-10 years, 10-20 years, 20-30 years, or 30-40
  years
• We cannot do a randomized experiment because we
  cannot randomly assign individuals to an age
  group.
• Thus, we observe (through polling or wire
  tapping) individuals from 4 different
  pre-existing populations the four age groups.
• We obtain a random sample of size 25 from each of
  the four age groups.
• We calculate statistics and compare the 4 groups.
• Can we say being in a specific age group causes
  someone to talk more on the telephone?

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3. Sampling More than One Population

• Observational Study
• Advantage The data is much more easy to obtain.
• Disadvantages
• We cannot say the explanatory variable caused the
  response
• There may be lurking or confounding variables
• Observational studies should be more to describe
  the past, not predict the future.

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4. Inference Overview

• Recall that inference is using statistics from a
  sample to talk about a population.
• We need some background in how we talk about
  populations and how we talk about samples.

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4. Inference Overview

• Describing a Population
• It is common practice to use Greek letters when
  talking about a population.
• We call the mean of a population ? .
• We call the standard deviation of a population ?
  and the variance ? 2.
• When we are talking about percentages, we call
  the population proportion ?.
• It is important to know that for a given
  population there is only one true mean and one
  true standard deviation and variance or one true
  proportion.
• There is a special name for these values
  parameters.

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4. Inference Overview

• Describing a Sample
• It is common practice to use Roman letters when
  talking about a sample.
• We call the mean of a sample .
• We call the standard deviation of a sample s and
  the variance s2.
• When we are talking about percentages, we call
  the sample proportion p.
• There are many different possible samples that
  could be taken from a given population. For each
  sample there may be a different mean, standard
  deviation, variance, or proportion.
• There is a special name for these values
  statistics.

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4. Inference Overview

• We use sample statistics to make inference about
  population parameters

m
s
s
p
p
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4. Inference Overview

• Sampling Variability
• There are many different samples that you can
  take from the population.
• Statistics can be computed on each sample.
• Since different members of the population are in
  each sample, the value of a statistic varies from
  sample to sample.

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4. Inference Overview

• The sampling distribution of a statistic is the
  distribution of values taken by the statistic in
  all possible samples of the same size from the
  same population.
• We can then examine the shape, center, and spread
  of the sampling distribution.

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4. Inference Overview Bias and Variability

• Bias concerns the center of the sampling
  distribution. A statistic used to a parameter is
  unbiased if the mean of the sampling distribution
  is equal to the true value of the parameter being
  estimated.
• To reduce bias, use random sampling. The values
  of a statistic computed from an SRS neither
  consistently overestimates nor consistently
  underestimates the value of the population
  parameter.

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4. Inference Overview Bias and Variability

• Variability is described by the spread of the
  sampling distribution.
• To reduce the variability of a statistic from an
  SRS, use a larger sample. You can make the
  variability as small as you want by taking a
  large enough sample. The variability of a
  statistic from a random sample does not depend on
  the size of the population, as long as the
  population is large enough.

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4. Inference Overview