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 gathering two samples from the same
  population.
• And then subtracting the mean of one from the
  mean of 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 (i.e., the standard error 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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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)