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Chapter 3: Producing Data

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Title: Statistics 303 Author: hintzee Last modified by: Dept. of Statistics Created Date: 9/5/2002 4:44:12 PM Document presentation format: On-screen Show – PowerPoint PPT presentation

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Title: Chapter 3: Producing Data


1
Chapter 3 Producing Data
  • Inferential Statistics
  • Sampling
  • Designing Experiments

2
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?

3
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.

4
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

5
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

6
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

7
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.

8
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.

9
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.

10
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.

11
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

12
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.

13
Sampling a Single Population
  • Sampling Techniques
  • Simple Random Sample (SRS) every member of the
    population has an equal chance of being selected.

14
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.

15
Sampling a Single Population
  • Sampling Techniques
  • Stratified Random Sample Divide the population
    into several strata. Then take a SRS from each
    stratum.

16
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.

17
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.

18
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.

19
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.

20
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.

21
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

22
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

23
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

24
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

25
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

26
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

27
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.

28
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.

29
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

30
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.

31
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
32
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.

33
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.

34
Sampling More than One Population
  • Observational Study
  • We are forced to select samples from different
    pre-existing populations

Simple Random Sample
Compare
35
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.

36
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?

37
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?

38
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.

39
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.

40
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.

41
Inference Overview
  • We use sample statistics to make inference about
    population parameters

m
s
s
p
p
42
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.

43
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.

44
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.

45
Bias and Variability
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