Inference and Inferential Statistics

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Inference and Inferential Statistics

• Methods of Educational Research
• EDU 660

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Inference

• Draw conclusions from the data
• Allow researchers to generalize to a population
  of individuals based on information obtained from
  a sample of those individuals
• Assesses whether the results obtained from a
  sample are the same as those that would have been
  calculated for the entire population

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Probabilistic nature of inference

• How likely is it?
• Are the results that we have seen due to chance
  or some real difference?
• Mean score for 2 different groups
• Example
• X1 23.5 X2 31.6
• Is this a real difference between these
• scores?

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Normal distribution

• A bell shaped curve reflecting the distribution
• of many variables of interest to educators

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Normal distribution

• Characteristics
• 50 of the scores fall above the mean and 50
  fall below the mean
• The mean, median, and mode are the same values
• Most participants score near the mean the
  further a score is from the mean the fewer the
  number of participants who attained that score
• Specific numbers or percentages of scores fall
  between
• ?1 SD, 68
• ?2 SD, 95
• ?3 SD, 99

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Null and Alternative hypotheses

• The null hypothesis represents a statistical tool
  important to inferential tests of significance
• The alternative hypothesis usually represents the
  research hypothesis related to the study

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Null and Alternative hypotheses

• Comparisons between groups
• Null no difference between the means scores of
  the groups
• Alternative there are differences between the
  mean scores of the groups
• Relationships between variables
• Null no relationship exists between the
  variables being studied
• Alternative a relationship exists between the
  variables being studied

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Test of Significance

• Statistical analyses to help decide whether to
  accept or reject the null hypothesis
• Alpha a level
• An established probability or significance level
  which serves as the criterion to determine
  whether to accept or reject the null hypothesis
• Common levels in education
• a .01 1 probability level
• a .05 5 probability level
• a .10 10 probability level

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Type I and Type II Errors

• Correct decisions
• The null hypothesis is true and it is accepted
• The null hypothesis is false and it is rejected
• Incorrect decisions
• Type I error - the null hypothesis is true and it
  is rejected
• Type II error the null hypothesis is false and
  it is accepted

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Type I and Type II Errors
As a becomes smaller there is a smaller chance of
a Type 1 error but a greater chance of a Type 2
error.
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One-Tailed and Two-Tailed Tests

• One-tailed an anticipated outcome in a specific
  direction
• Treatment group mean is significantly
  higher/lower than the control group mean
• Two-tailed anticipated outcome not directional
• Treatment and control groups are equal
• Ample justification needed for using one-tailed
  tests

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One-Tailed and Two-Tailed Tests
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Test of Significance

• Specific tests are used in specific situations
  based on the number of samples and the
  statistics of interest
• One sample tests of the mean, variance,
  proportions, correlations, etc.
• Two sample tests of means, variances,
  proportions, correlations, etc.

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Test of Significance

• Types of inferential statistics
• Parametric tests more powerful tests that
  require certain assumptions to be met
• t - tests
• ANOVA
• Non-parametric tests less powerful
• Chi-Square

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Form a Null Hypothesis

• H0 There is no significant difference in the
  mean scores for the 2 groups
• Acceptance of the null hypothesis
• The difference between groups is too small to
  attribute it to anything but chance
• Rejection of the null hypothesis
• The difference between groups is so large it can
  be attributed to something other than chance
  (e.g., experimental treatment)

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The t Test

• Used to test whether 2 means are significantly
  different at a selected probability
• The t test determines whether the observed
  difference is sufficiently larger than a
  difference that would be expected by chance

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Types of t Tests

• t test for independent samples
• The members of one sample are not related to
  those of the other sample in any systematic way -
  come from the same population
• Examples
• 1. Examine the difference between the mean
  scores for an experimental and control group
• 2. Examine the mean scores for men and women in
  sample

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Types of t Tests

• t test for NonIndependent samples
• Used to compare groups that are formed to examine
  a samples performance on a single measure or
  multiple measures
• Example examining the difference between
  pre-test and post-test mean scores for a single
  class of students

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Analysis of Variance - ANOVA

• ANOVA is used to test whether there is
• a significant difference between 2 or
• more means at a specified significance
• Level (usually 5)
• Example Is there a significant difference in
  the mean scores on a test (µ1, µ2, µ3) of 3
  classes of college students?

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ANOVA

• Omnibus Null Hypothesis
• H0 µ1 µ2 µ3
• Note repeated use of numerous t tests for more
  than 2 means will result in an increased
  probability of type I errors
• p 1 - (1 a)c where c is the number of t
  tests

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Analysis of Variance - ANOVA

• If an ANOVA determines that there is a
  significant difference among a group of means,
  what then?
• Multiple comparison methods are used to determine
  what means are different Scheffe test

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Steps in Statistical Testing

• State the null and alternative hypotheses
• Set alpha level - 0.05, 0.01 etc
• Identify the appropriate test of significance
• Identify the test statistic
• Compute the test statistic and probability level
• Is the probability level less than the specified
  probability?
• Accept or reject hypothesis