Chapter 3: Producing Data

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Chapter 3 Producing Data

• Inferential Statistics
• Sampling
• Designing Experiments

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Inferential Statistics

• We start with a question about a group or groups.
• The group(s) we are interested in is(are) called
  the population(s).
• Examples
• What is the average number of car accidents for a
  person over 65 in the United States?
• For the entire world, is the IQ of women the same
  as the IQ of men?
• How many times a day should I feed my goldfish?
• Which is more effective at lowering the
  heartrate of mice, no drug (control), drug A,
  drug B, or drug C?

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Inferential Statistics

• Example 1 What is the average number of car
  accidents for a person over 65 in the United
  States?
• How many populations are of interest?
• One
• What is the population of interest?
• All people in the U.S. over age 65.

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Inferential Statistics

• Example 2 For the entire world, 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

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Inferential Statistics

• Example 3 How many times a day should I feed my
  goldfish?
• How many populations are of interest?
• One
• What is the population of interest?
• All pet goldfish

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Inferential Statistics

• Example 4 Which is more effective at lowering
  the heartrate 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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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
• Advantages
• Less expensive. Feasible.
• Disadvantages
• Uncertainty about the truth. Instead of surety
  we may have error.

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Inferential Statistics

• Suppose we have no previous information about
  these questions. How could we answer them?
• If we take a census, we have everyone and we have
  no need for inference. We know.
• If we take a sample, we make inference from the
  sample to the whole population.
• For these four questions, it is not likely we can
  get a census. We will need to use a sample.
• Obviously, for each population we are interested
  in, we must get a separate sample.

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Inferential Statistics

• General Idea of Inferential Statistics
• We take a sample from the whole population.
• We summarize the sample using important
  statistics.
• We use those summaries to make inference about
  the whole population.
• We realize there may be some error involved in
  making inference.

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Inferential Statistics

• Example (1988, the Steering Committee of the
  Physicians' Health Study Research Group)
• Question Can Aspirin reduce the risk of heart
  attack in humans?
• 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 (headache or not).
  The other group took a placebo every other day.
  This group is the control group.
• 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.
• Inference to population Taking aspirin causes
  lower rate of heart attacks in humans.

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Sampling a Single Population

• Basics for sampling
• Sampling should not be biased no favoring of any
  individual in the population.
• Example of a biased sample
• Select goldfish from a particular store
• The selection of an individual in the population
  should not affect the selection of the next
  individual independence.
• Example of non-independent sample
• Choosing cards from a deck without replacement

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Sampling a Single Population

• Basics for sampling
• Sampling should be large enough to adequately
  cover the population.
• Example of a small sample
• Suppose only 20 physicians were used in the
  aspirin study.
• Sampling should have the smallest variability
  possible.
• We know there is some error want to minimize it.

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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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Sampling a Single Population

• Sampling Techniques
• Simple Random Sample (SRS) every member of the
  population has an equal chance of being selected
• Assign every individual a number and randomly
  select 30 numbers using a random number table (or
  computer generated random numbers).
• 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.
• Table B at the back of the book is random digits.
  

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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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Sampling a Single Population

• Sampling Techniques
• Stratified Random Sample
• Advantage Each stratum is guaranteed to be
  randomly sampled
• 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.

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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 using all the observations in each.

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Sampling a Single Population

• Sampling Techniques
• Cluster Sample
• Advantage May be the only feasible method, given
  resoures.
• 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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Sampling a Single Population

• Sampling Problems
• Voluntary response
• Internet surveys
• Call-in surveys
• Convenience sampling
• Sampling friends
• Sampling at the mall
• Dishonesty
• Asking personal questions
• Not enough time to respond honestly

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Sampling a Single Population

• Undercoverage Some groups in the population are
  left out when the sample is taken
• Nonresponse An individual chosen for the sample
  cant be contacted or does not cooperate
• Response Bias Results that are influenced by
  the behavior of the respondent or interviewer
• For example, the wording of questions can
  influence the answers
• Respondent may not want to give truthful answers
  to sensitive questions

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Sampling More than One Population

• We sample from more than one population when we
  are interested in more than one variable.
• As previously discussed, one variable is chosen
  to be the response variable and the other is
  selected as the explanatory variable.
• Examples
• Comparing decibel levels of 4 different brands of
  speakers
• Determining time to failure of 3 different types
  of lightbulbs
• Comparing GRE scores for students from 5
  different majors

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Sampling More than One Population

• 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
• Number of Samples needed?
• Four

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Sampling More than One Population

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

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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
• Number of Samples needed?
• Five

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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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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 lightbulbs, each in a
  different light socket.
• We wouldnt know if the differences were due to
  the different types of lightbulbs or different
  light sockets.
• Type and Socket confounded.

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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, which will be explained shortly

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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 lightbulb and the type
  of light socket influence the time to failure of
  a lightbulb.
• It is likely that different types of lightbulbs
  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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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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Sampling More than One Population

• Randomized Experiment
• Example Suppose that before we want to test the
  effect of aspirin on the physicians, 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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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.
• Unfortunately or perhaps fortunately, we cannot
  always use a randomized experiment
• Often impossible or unethical, particularly with
  humans.

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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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Sampling More than One Population

• Observational Study
• Advantage The data is much easier 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.
• Case-Control Study A study in which cases
  having a particular condition are compared to
  controls who do not. The purpose is to find out
  whether or not one or more explanatory variables
  are related to a certain disease.
• Although you cant usually determine cause and
  effect, these studies are more efficient and they
  can reduce the potential confounding variables.

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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. The individuals decide that for
  themselves.
• 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?
• What are some possible confounding variables?
• How might we reduce the effect of these
  confounding variables?

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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?
• What are some possible confounding variables?
• How might we control these confounding variables?

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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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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 m.
• We call the standard deviation of a population s
  and the variance s2.
• When we are talking about percentages, we call
  the population proportion p. (or pi).
• 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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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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Inference Overview

• We use sample statistics to make inference about
  population parameters

m
s
s
p
p
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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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Sampling Distribution

• 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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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.
• 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.

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Bias and Variability