Reading

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Lecture 11
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Reading

• You should have completed chapter 6.

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Evidence of a correlation

• The margin of error for red depends on the number
  of red marbles in the sample
• 500 red marbles, so -0.05.
• f(LR) 0.65
• P(LR) 0.65 - .05
• So 0.6 to 0.7 of the red population are large.
• 100 non-red marbles, so - 0.10 f(LN) 0.35
• P(LN) 0.35 -0.1
• So 0.25 to 0.45 of the non-red population are
  large. fig. 6.3 whole

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Lack of evidence for a correlation

• Sample size 600 fig.
• 500 red, 100 non-red
• 270 / 500 large, f(LR) 270/500 0.54
• P(LR) 0.54-0.05
• So 0.49 to 0.59 of the red population are large.
• 46/100 large, f(LN) 46/100 0.46
• P(LN) 0.46-0.01
• So 0.36 to 0.56 of the non-red population are
  large.

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Estimating the Strength of Correlation

• The maximum allowed difference is between the top
  of the higher interval and the bottom of the
  lower interval.
• The minimum allowed difference is between the
  bottom of the higher interval and the top of the
  lower interval.
• Fig. 6.3, 0.7-0.250.45 and 0.60-0.450.15
• Estimated strength of the correlation is (0.45,
  0.15)
• Fig 6.4, 0.59-0.360.25 and 0.49-0.56
• Estimated strength of the correlation is (0.23,
  -0.07)
• The negative number indicates that there may be
  no correlation.

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• The standard deviation of the difference is less
  than the standard deviation of the estimates.
• This is because we are getting evidence from the
  whole sample of 600 marbles combined.
• So rather than a certainty of 95, our estimate
  of the strength of correlation is 99 certain.

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Statistical Significance

• A correlation in the sample may be due to a
  correlation in the population, or it may be due
  to an accident of sampling.
• A correlation is statistically significant iff it
  is unlikely to be an accident of sampling.
• The sample frequencies are statistically
  significant iff the corresponding interval
  estimates do not overlap.
• Iff the interval estimates do overlap, the
  differences in sample frequencies are not
  statistically significant.

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Evaluating Statistical HypothesesThe Real World
Population

• The data only tell us about the population
  actually sampled.
• This is different from the population of
  interest.
• E.g. a phone survey of voting intentions doesnt
  include people without phones.
• The population actually sampled must have members
  who are all equally likely to be in the sample
  (random sampling).

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The Sample Data

• When evaluating a hypothesis, it will usually
  only be one part of the sample data that is
  relevant. That part must be identified.

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The Statistical Model

• Evaluating a statistical hypothesis consists in
  comparing the real world to a model.
• Sometimes the model is suggested by the data,
  though not always.
• The possible models we have are proportions,
  distributions and correlations.

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Random Sampling

• How well does the study fit random sampling?
• Random sampling
• a) All members of the population have an equal
  chance of being selected.
• b) There is no correlation between the outcome of
  one selection and another.
• This is an ideal that is only approximated in
  practice.

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Evaluating the Hypothesis

• Assuming random sampling, what does the data tell
  us?
• What estimate for a proportion do we have, with
  what margin of error?
• Is there evidence of a correlation?
• Is it strong evidence?

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Summary

• How well does the data support the evaluation of
  stage 5?
• Is the sample random enough for the hypothesis to
  be supported?
• This depends on how random the sampling procedure
  was, and how strong the evidence is.

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Non-random sampling 1Stratified Sampling

• The sample should contain the same proportion of
  each sub-section as the population.
• Example If the population of America is 20
  hispanic, the sample should be 20 hispanic.
• Advantages More precision.
• Can be administratively easier to focus on
  certain groups.
• Disadvantages Requires identifying appropriate
  strata.
• More complex to analyse results.

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Non-random Sampling 2Cluster Sampling

• If some areas of the population are out of reach,
  sample a cluster of individuals from those within
  reach.
• Rather than sample one child from 30 schools,
  sample 30 children from one school.
• Each cluster should be a small scale
  representation of the sample.
• The individuals within the cluster should be as
  heterogenous as possible. (This is a weakness of
  the HBSC study.)
• Advantage Cheaper / More practical. Saves travel
  time.
• Disadvantages Higher margin of error.
• Clusters may be similar to each other.

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Case StudyHealth Behaviour in School-Aged
Children

• Aim To discover the behaviour and concerns of
  school aged children related to health.
• Hopeful outcomes
• Identify specific groups at risk.
• Understanding the factors that dispose people to
  develop health problems.
• Develop effective intervention strategies.
• (Did they interview the same students at a later
  date?)

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• How do you get a random sample of 11-15 year
  olds?
• They used the school systems.
• This excluded from the study all those not in the
  school system
• Home-schooled
• In detention centres
• Homeless

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• A sample of over 123,000 from 28 countries was
  obtained.
• This consisted of students answering survey
  questions.
• What does this tell us about the proportion of
  children who like school a lot?

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The Real World Population

• The population of interest is children in the 28
  countries between 11 and 15.
• The population sampled is children in school
  during the testing period of 11, 13 and 15 who
  were competent enough in their national language.

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The Sample Data

• The relevant piece of data is that 24 of
  respondents reported liking school a lot (p.169).

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The Statistical Model

• The model suggested is that the proportion of
  children who like school a lot is 24.

Dont like school a lot
Like school a lot
0.24
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Random Sampling

• How well does the study fit random sampling?
• Random sampling
• a) All members of the population have an equal
  chance of being selected.
• Not satisfied. Cluster sampling Only certain
  schools were selected.
• b) There is no correlation between the outcome of
  one selection and another.
• Not fully satisifed. There is less variation
  within classes than between them e.g. students
  grouped by abilities.

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Evaluating the Hypothesis

• Assuming random sampling, what does the data tell
  us?
• For a sample of 120 000, the margin of error is
  about 1
• So the data suggests that 23-25 of children like
  school a lot.

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Children 11-15
0.23-0.25 like school a lot
Estimate
Dont like school a lot
Like school a lot
0.24
n 120,000
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Summary

• How well does the data support the evaluation of
  stage 5?
• Is the sample random enough for the hypothesis to
  be supported?
• Cluster sampling rather than random sampling was
  used.
• We have to rely to some extent on the care taken
  by the researchers. Given that it is a
  large-scale project by academics, it is
  reasonable to assume it was close to random
  sampling.
• Thus, we have good evidence that the proportion
  of children who like school a lot is 23-25.

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Problems with survey sampling 1Non-random
sampling

• The Kinsey report (1948, 1953) Sample of
  convenience.
• 10 of the sample were homosexual.
• 50 of married males had extra-marital sex.
• Self-Selection.
• 25 of the sample were in prison, and 5 were
  male prostitutes.
• Advantage Much larger sample size than would
  otherwise be possible.

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• 1936 election The Literary Digest predicted Alf
  Landon, the Republican candidate would win by a
  landslide.
• But their sample was chosen from phone books and
  car registration details. The poor of the 1930s
  had neither.
• George Gallup constructed his sample more
  carefully, and predicted that Roosevelt would win.

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Problems with survey sampling 2False responses
(liars)

• Sensitive subjects. Sex, drugs, bullying.
• Telling people what they want to hear.
• The Shy-Tory Factor
• Polls in the British 1992 election put Labour and
  Tories at 38 and 39.
• But the Tories won by 7.6.
• An inquiry by the Market Research Society put the
  difference down to embarassed Tories.

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Example Why do Fox News polls favour Republicans?

• Fox News shows Obama with a 7 point lead.
  http//www.foxnews.com/polls/
• Gallup shows him with a 10 point lead.
  www.gallup.org
• 1. Its Fox News you talk to him
• 2. Telling people what they want to hear.

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Major expected sources of error in the current
polls

• 1. Racism. As it is not socially acceptable in
  most of America to refuse to vote for a black
  candidate, voters who refuse to vote for a black
  candidate will not say so. The Bradley-effect.
  (False responses.)
• 2. Samples are selected by phone numbers. Voters
  who only have cell-phones may be
  under-represented. If there is a correlation
  between voters with only cell-phones and voting,
  the poll may be inaccurate.

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Exercise 6.13

• Is there a correlation between having a college
  education and not drinking?

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1. The Real World Population

• American adults.

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2. The Sample Data

• Among those with a college education, 75
  classified themselves as either light or moderate
  drinkers.
• 49 with a high school education gave these
  responses.

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College education High School Education
Non-drinkers or heavy drinkers
Light or moderate drinkers
0.75
0.49
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3. The Statistical Model

• The model suggested is that there is a positive
  correlation between having a college education
  and being a light or moderate drinker.

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4. Random Sampling

• How well does the study fit random sampling?
• In-home interviews.
• Random sampling
• a) All members of the population have an equal
  chance of being selected.
• Were not told how the homes are selected.
• The homeless are excluded.
• b) There is no correlation between the outcome of
  one selection and another.
• Were not told, but probably satisfied.

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5. Evaluating the Hypothesis

• Assuming random sampling, what does the data tell
  us?
• For a sample of 500 the margin of error is 4.
• Complication Were not told what proportion of
  the population had a college education.
• For n250, margin of error is-0.6

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Non-drinkers or heavy drinkers
0.81 0.69
0.55 0.42
Light or moderate drinkers
Non-drinkers or heavy drinkers
0.75
Light or moderate drinkers
0.49
n500 total
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Strength of Correlation

• 0.81-0.42 0.39
• 0.69-0.55 0.14
• So the estimated strength of correlation is
  0.39, 0.14.
• As this is based on the whole sample of 500, we
  can be 99 of this conclusion.

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6. Summary

• How well does the data support the evaluation of
  stage 5?
• Is the sample random enough for the hypothesis to
  be supported?
• We have to decide based on the report and the
  context in which we find the report. Its
  reasonable to assume that this was a carefully
  conducted study, in which case we have good
  evidence for the conclusion that there is a
  moderate correlation between having a college
  education and light or moderate drinking.