The t Tests

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The t Tests

• Independent Samples

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The t Test for Independent Samples

• Observations in each sample are independent (not
  from the same population) each other.
• We want to compare differences between sample
  means.

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Sampling Distribution of the Difference Between
Means

• Imagine two sampling distributions of the mean...
• And then subtracting one from the other
• If you create a sampling distribution of the
  difference between the means
• Given the null hypothesis, we expect the mean of
  the sampling distribution of differences, ?1- ?2,
  to be 0.
• We must estimate the standard deviation of the
  sampling distribution of the difference between
  means.

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Pooled Estimate of the Population Variance

• Using the assumption of homogeneity of variance,
  both s1 and s2 are estimates of the same
  population variance.
• If this is so, rather than make two separate
  estimates, each based on some small sample, it is
  preferable to combine the information from both
  samples and make a single pooled estimate of the
  population variance.

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Pooled Estimate of the Population Variance

• The pooled estimate of the population variance
  becomes the average of both sample variances,
  once adjusted for their degrees of freedom.
• Multiplying each sample variance by its degrees
  of freedom ensures that the contribution of each
  sample variance is proportionate to its degrees
  of freedom.
• You know you have made a mistake in calculating
  the pooled estimate of the variance if it does
  not come out between the two estimates.
• You have also made a mistake if it does not come
  out closer to the estimate from the larger
  sample.
• The degrees of freedom for the pooled estimate of
  the variance equals the sum of the two sample
  sizes minus two, or (n1-1) (n2-1).

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Estimating Standard Error of the Difference
Between Means
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The t Test for Independent Samples An Example

• Stereotype Threat

This test is a measure of your academic ability.
Trying to develop the test itself.
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The t Test for Independent Samples An Example

• State the research question.
• Does stereotype threat hinder the performance of
  those individuals to which it is applied?
• State the statistical hypotheses.

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The t Test for Independent Samples An Example

• Set the decision rule.

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The t Test for Independent Samples An Example

• Calculate the test statistic.

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The t Test for Independent Samples An Example

• Calculate the test statistic.

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The t Test for Independent Samples An Example

• Calculate the test statistic.

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The t Test for Independent Samples An Example

• Decide if your result is significant.
• Reject H0, - 2.37lt - 1.721
• Interpret your results.
• Stereotype threat significantly reduced
  performance of those to whom it was applied.

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Assumptions

• 1) The observations within each sample must be
  independent.
• 2) The two populations from which the samples are
  selected must be normal.
• 3) The two populations from which the samples are
  selected must have equal variances.
• This is also known as homogeneity of variance,
  and there are two methods for testing that we
  have equal variances
• a) informal method simply compare sample
  variances
• b) Levenes test Well see this on the SPSS
  output
• Random Assignment
• To make causal claims
• Random Sampling
• To make generalizations to the target
  population

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Which test?

• Each of the following studies requires a t test
  for one or more population means. Specify
  whether the appropriate t test is for one sample
  or two independent samples.
• College students are randomly assigned to undergo
  either behavioral therapy or Gestalt therapy.
  After 20 therapeutic sessions, each student earns
  a score on a mental health questionnaire.
• One hundred college freshmen are randomly
  assigned to sophomore roommates having either
  similar or dissimilar vocational goals. At the
  end of their freshman year, the GPAs of these 100
  freshmen are to be analyzed on the basis of the
  previous distinction.
• According to the U.S. Department of Health and
  Human Services, the average 16-year-old male can
  do 23 push-ups. A physical education instructor
  finds that in his school district, 30 randomly
  selected 16-year-old males can do an average of
  28 push-ups.

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Handout Example
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Effect Size

• 1) Simply report the actual results of the study.
• (a) Most direct method.
• (b) Can be misleading.
• 2) Calculate Cohens d or ? (preferred).
• (a) Magnitude of effect size is standardized by
  measuring the mean difference between two
  treatments in terms of the standard deviation.
• (b) d (M1-M2)/?sp2
• (c) Evaluate using the following criteria
• i) .20 small effect
• ii) .50 medium effect
• iii) gt .80 large effect

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Effect Size Example

• In the study evaluating stereotype threat, the
  null hypothesis was rejected, with M16.58,
  M29.64, and sp2 9.59.
• Calculate Cohens d, and evaluate the magnitude
  of this measure (small, medium, or large).
• Compare effect size to z table to determine where
  the mean of one group is relative to the other.

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Type 1 Error Type 2 Error
Scientists Decision Reject null hypothesis
Fail to reject null hypothesis
Type 1 Error Correct Decision probability
? Probability 1- ? Correct decision Type 2
Error probability 1 - ? probability ?
Null hypothesis is true Null hypothesis is false
Type 1 Error
Type 2 Error
?
?
Cases in which you reject null hypothesis when it
is really true
Cases in which you fail to reject null hypothesis
when it is false
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Power and sample size estimation

• Power is the probability of correctly rejecting a
  null hypothesis.
• In social sciences we typically use .80.
• What determines the power of a study
• Effect size
• Sample size
• Variance
• a
• One vs. two tailed tests

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If you want to know

• Sample Size
• Need to know
• a
• ß
• ?
• Power
• Need to know
• a
• N per condition
• ?

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Calculating sample size

• Remember stereotype threat example
• ? .99
• Say we want to perform a test with
• power (1-ß) .80
• Two tailed alpha .05

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Solving for n 1-ß .80
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Solving for n 1-ß .80 one tailed
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Solving for n 1-ß .90
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Solving for n 1-ß .90 one tailed
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Calculating power

• Say we did the same study with n of 5 in each
  condition (N 10)
• We want to know how much power we have to find d
  or ? .99.
• Again we are using a two tailed test with a .05.

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Using Piface
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What if I did a one tailed test?
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Spss Homework Hint

• For the two sample t tests you will need to
  create two variables, cond (X) and score (Y)