Hypothesis testing and statistical inference

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Hypothesis testing and statistical inference

• Research Process and Design
• Spring 2006
• Class 10 (Week 11)

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Todays objectives

• To review elements of last weeks class
• To understand logic of inferential statistics
• To explore hypothesis testing
• To get feedback on first draft of methods section

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Logic of Inferential Statistics

• Hope to say something with confidence about the
  population based on a sample
• Use probabilities to state the degree of
  confidence

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Probability samples and standard errors

• Random selection
• Human influence is removed from the selection
  process
• E.g., dice, random number generator
• Probability samples use random selection to draw
  a subset of the sampling frame
• Sampling error arises because of this
• Standard errors allow us to quantify this error

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Laws of Sampling Theory

• Whenever a random sample is taken from a
  population there will be sampling error.
• If sample is truly random, then characteristics
  of sample will be an unbiased estimate of
  population characteristics.
• As sample size increases, the range (the size) of
  sampling error decreases.

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Central Limit Theorem

• The sampling distribution, or the distribution of
  the sampling error for any sample drawn from a
  given population, approximates a normal curve.
• Standard error - standard deviation of the sample
  estimates of means that would be formed if an
  infinite number of samples.

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Standard error

• Relies on the concept of repeated samples from a
  population
• Due to chance, the means of these samples will
  vary around the population mean
• We can measure this variance and determine how
  much the typical sample will deviate from the
  population mean (i.e., the standard deviation or
  SD)
• This SD is the standard error (SE)
• http//www.ruf.rice.edu/lane/stat_sim/sampling_di
  st/index.html

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Standard errors

• Standard error of the mean
• s is the SD from our sample n is sample size
• We can see that as n increases, SE decreases
• Different formulas for different statistics
  (proportions, comparing two means, etc.), but
  they have a similar form

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Confidence Intervals

• The range within which the parameter in question
  could be expected to be included a specified
  percentage of the time if procedure were to be
  repeated.
• C Z statistic associated with the confidence
  level 1.96 corresponds to the .95,
• 2.33 corresponds to the 98 level,
• and 2.58 corresponds to the 99 confidence level

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Standard errors

• Confidence intervals (CI) use SE and tell us the
  precision of our estimates
• 95 CI for a mean
• Very specific definition if we calculated
  similar CIs on 100 similar samples, 95 of them
  would bracket the population parameter
• Does not mean there is a 95 probability that
  population parameter falls in your CI either it
  does or it doesnt
• http//www.ruf.rice.edu/lane/stat_sim/conf_interv
  al/

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Standard errors

• Margin of error in polls is a confidence
  interval, usually a 95 CI

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Central Limit Theorem

• The sampling distribution, or the distribution of
  the sampling error for any sample drawn from a
  given population, approximates a normal curve.
• Standard error - standard deviation of the sample
  estimates of means that would be formed if an
  infinite number of samples is known as the
  standard error.

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Confidence Intervals and Hypothesis Testing

• The range within which the parameter in question
  could be expected to be included a specified
  percentage of the time if procedure were to be
  repeated.
• C Z statistic associated with the confidence
  level 1.96 corresponds to the .95,
• 2.33 corresponds to the 98 level,
• and 2.58 corresponds to the 99 confidence level

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Normal Curve

• Symmetric around mean
• Skewness
• Positively skewed
• Negatively skewed
• Kurtosis
• Leptokurtic
• Platykurtic

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Hypotheses

• How do we know that differences between groups
  are not due to sampling error? How big does a
  correlation need to be to know that difference is
  real and not a result of sampling error?
• Hypothesis testing?
• How do we develop hypotheses?
• Null hypothesis
• Research (or alternative) hypothesis

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Statistical HypothesesNull Hypothesis (H0)

• Statistical statement of no difference in
  population
• No difference between performance of a group and
  accepted benchmark
• e.g., passing a test

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Statistical HypothesesNull Hypothesis (H0)

• No difference between two or more groups that are
  being compared
• e.g., experimental and comparison groups
• No relationship between or among predictor and
  criterion variables
• e.g., relationship between attitude and
  achievement does not exist it is not different
  from zero

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Statistical Hypotheses--Alternative hypothesis
(H1)

• Statistical statement that includes all results
  not explicitly stated in the null hypothesis
• Reflects the existence of differences or
  relationships
• The performances of students in the experimental
  group exceeds the performances of those in the
  comparison group
• A relationship between attitude and achievement
  does exist
• Usually reflects the research hypothesis

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Choice of probability level

• Rejection region or significance level
• plt.10,plt.05, plt.01, plt.001
• Determined a priori
• Significance
• Interpreting results
• One-tailed test
• Two-tailed test

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Error in hypothesis testing
State of Nature
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Feedback on the method section

• Do you have all of the information you need to
  fully assess the method? What do you need?
• Is the study feasible as described in the method?
• How is the study (1)Maximizing experimental
  variation, (2)Minimizing Error Variation, and (3)
  Controlling Extraneous (Confounding) Variation
• Dont hesitate to ask the other group for help in
  areas where you are having trouble.

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In two weeks

• Analysis of variance (ANOVA)
• Reading for next week
• Jaeger Chapters 12-14
• Assignment due Draft of proposal. Bring 3 copies
  to class.
• NO CLASS NEXT WEEK. Work on proposals.