t(ea) for Two: Test between the Means of Different Groups

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t(ea) for Two Test between the Means of
Different Groups

• When you want to know if there is a difference
  between the two groups in the mean
• Use t-test.
• Why cant we just use the difference in score?
• Because we have to take the variability into
  account.
• T difference between group means
• sampling variability

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One-Sample T Test

• Evaluates whether the mean on a test variable is
  significantly different from a constant (test
  value).
• Test value typically represents a neutral point.
  (e.g. midpoint on the test variable, the average
  value of the test variable based on past research)

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Example of One-sample T-test

• Is the starting salary of company A (17,016.09)
  the same as the average of the starting salary of
  the national average (20,000)?
• Null Hypothesis
• Starting salary of company A National average
  
• Alternative Hypothesis
• Starting salary of company A National average

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• SPSS demo (employee data)
• Review
• Standard deviation Measure of dispersion or
  spread of scores in a distribution of scores.
• Standard error of the mean Standard deviation of
  sampling distribution. How much the mean would be
  expected to vary if the differences were due only
  to error variance.
• Significance test Statistical test to determine
  how likely it is that the observed
  characteristics of the samples have occurred by
  chance alone in the population from which the
  samples were selected.

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z and t

• Z score standardized scores
• Z distribution normal curve with mean value z0
• 95 of the people in the given sample (or
  population) have
• z-scores between 1.96 and 1.96.
• T distribution is adjustment of z distribution
  for sample size (smaller sampling distribution
  has flatter shape with small samples).
• T difference between group means
• sampling variability

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Confidence Interval

• A range of values of a sample statistic that is
  likely (at a given level of probability, i.e.
  confidence level) to contain a population
  parameter.
• The interval that will include that population
  parameter a certain percentage ( confidence
  level) of the time.

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Confidence Interval for difference and Hypothesis
Test

• When the value 0 is not included in the interval,
  that means 0 (no difference) is not a plausible
  population value.
• It appears unlikely that the true difference
  between Company As salary average and the
  national salary average is 0.
• Therefore, Company As salary average is
  significantly different from the national salary
  average.

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Independent-Sample T test

• Evaluates the difference between the means of two
  independent groups.
• Also called Between Groups T test
• Ho ?1 ?2
• H1 ?1 ?2

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Paired-Sample T test

• Evaluates whether the mean of the difference
  between the paired variables is significantly
  different than zero.
• Applicable to 1) repeated measures and 2) matched
  subjects.
• Also called Within Subject T test Repeated
  Measures T test.
• Ho ?d 0
• H1 ?d 0

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SPSS Demo

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Analysis of Variance (ANOVA)

• An inferential statistical procedure used to test
  the null hypothesis that the means of two or more
  populations are equal to each other.
• The test statistic for ANOVA is the F-test (named
  for R. A. Fisher, the creator of the statistic).

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T test vs. ANOVA

• T-test
• Compare two groups
• Test the null hypothesis that two populations has
  the same average.
•  
• ANOVA
• Compare more than two groups
• Test the null hypothesis that two populations
  among several numbers of populations has the same
  average.

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ANOVA example

• Example Curricula A, B, C.
• You want to know what the average score on the
  test of computer operations would have been
• if the entire population of the 4th graders in
  the school system had been taught using
  Curriculum A
• What the population average would have been had
  they been taught using Curriculum B
• What the population average would have been had
  they been taught using Curriculum C.
• Null Hypothesis The population averages would
  have been identical regardless of the curriculum
  used.
• Alternative Hypothesis The population averages
  differ for at least one pair of the population.

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ANOVA F-ratio

• The variation in the averages of these samples,
  from one sample to the next, will be compared to
  the variation among individual observations
  within each of the samples.
• Statistic termed an F-ratio will be computed. It
  will summarize the variation among sample
  averages, compared to the variation among
  individual observations within samples.
• This F-statistic will be compared to tabulated
  critical values that correspond to selected alpha
  levels.
• If the computed value of the F-statistic is
  larger than the critical value, the null
  hypothesis of equal population averages will be
  rejected in favor of the alternative that the
  population averages differ.

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Interpreting Significance

• plt.05
• The probability of observing an F-statistic at
  least this large, given that the null hypothesis
  was true, is less than .05.

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Logic of ANOVA

• If 2 or more populations have identical averages,
  the averages of random samples selected from
  those populations ought to be fairly similar as
  well.
• Sample statistics vary from one sample to the
  next, however, large differences among the sample
  averages would cause us to question the
  hypothesis that the samples were selected from
  populations with identical averages.

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Logic of ANOVA cont.

• How much should the sample averages differ before
  we conclude that the null hypothesis of equal
  population averages should be rejected.
• In ANOVA, the answer to this question is obtained
  by comparing the variation among the sample
  averages to the variation among observations
  within each of the samples.
• Only if variation among sample averages is
  substantially larger than the variation within
  the samples, do we conclude that the populations
  must have had different averages.

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Three types of ANOVA

• One-way ANOVA
• Within-subjects ANOVA (Repeated measures,
  randomized complete block)
• Factorial ANOVA (Two-way ANOVA)

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Sources of Variation

• Three sources of variation
• 1) Total, 2) Between groups, 3) Within groups
• Sum of Squares (SS) Reflects variation. Depend
  on sample size.
• Degrees of freedom (df) Number of population
  averages being compared.
• Mean Square (MS) SS adjusted by df. MS can be
  compared with each other. (SS/df)
• F statistic used to determine whether the
  population averages are significantly different.
  If the computed F static is larger than the
  critical value that corresponds to a selected
  alpha level, the null hypothesis is rejected.

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Computing F-ratio

• SS Total Total variation in the data
• df total Total sample size (N) -1
• MS total SS total/ df total
• SS between Variation among the groups compared.
• df between Number of groups -1
• MS between SS between/df between
• SS within Variation among the scores who are in
  the same group.
• df within Total sample size - number of groups
  -1
• MS within SS within/df within
•  
• F ratio MS between / MS within

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Formula for One-way ANOVA
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Alpha inflation

• Conducting multiple ANOVAs, will incur a large
  risk that at least one of them would be
  statistically significant just by chance.
• The risk of committee Type I error is very large
  for the entire set of ANOVAs.
• Example 2 tests .05 Alpha
• Probability of not having Type I error .95
• .95x.95 .9025
• Probability of at least one Type I error is
• 1-9025 .0975. Close to 10 .
• Use more stringent criteria. e.g. .001

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Relation between t-test and F-test

• When two groups are compared both t-test and
  F-test will lead to the same answer.
• t2 F.
• So by squaring t youll get F
• (or square root of t is F)

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Follow-up test

• Conducted to see specifically which means are
  different from which other means.
• Instead of repeating t-test for each combination
  (which can lead to an alpha inflation) there are
  some modified versions of t-test that adjusts for
  the alpha inflation.
• Most recommended Tukey HSD test
• Other popular tests Bonferroni test , Scheffe
  test

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Within-Subject (Repeated Measures) ANOVA

• SS tr Sum of Squares Treatment
• SS block Sum of Squares Block
• SS error SS total - SS block - SS tr
• MS tr SS tr/k-1
• MSE SS error/(n-1)(k-1)
• F MS tr/MSE

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Within-Subject (Repeated Measures) ANOVA

• Examine differences on a dependent variable that
  has been measured at more than two time points
  for one or more independent categorical
  variables.

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Within-Subject (Repeated Measures) ANOVA
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Factorial ANOVA

• T-test and One way ANOVA
• 1 independent variable (e.g. Gender), 1 dependent
  variable (e.g. Test score)
• Two-way ANOVA (Factorial ANOVA)
• 2 (or more) independent variables (e.g. Gender
  and Academic Standing), 1 dependent variable
  (e.g. Test score)

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(End of Analytic Method I)
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Main Effects and Interaction Effects

• Main Effects
• The effects for each independent variable on the
  dependent variable.
• Differences between the group means for each
  independent variable on the dependent variable.
• Interaction Effect
• When the relationship between the dependent
  variable and one independent variable differs
  according to the level of a second independent
  variable.
• When the effect of one independent variable on
  the dependent variable differs at various levels
  of second independent variable.

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T-distribution

• A family of theoretical probability distributions
  used in hypothesis testing.
• As with normal distributions (or
  z-distributions), t distributions are unimodal,
  symmetrical and bell shaped.
• Important for interpreting data gather on small
  samples when the population variance is unknown.
• The larger the sample, the more closely the t
  approximates the normal distribution. For sample
  greater than 120, they are practically
  equivalent.