One Sample ttest

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One Sample t-test
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Review of Z-test

• Used to compare sample mean to a known
  population, for which you have mu and sigma.
• Enables you to determine whether difference
  between sample and population means is due to
  chance.

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Introducing the t-test

• Used when s is not known and must be estimated
  using sample standard deviation (s).
• The t-statistic is a substitute for z whenever s
  is unknown
• New error term
• SM Estimated standard error
• Provides an estimate of the average distance
  between a sample mean and the population mean
• t-test has its own table of critical values
• There are different types of t-tests (one sample,
  independent samples, and dependent samples)

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One Sample t-test Formula
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One Sample t-test Example

• Research Question Do birds find staring
  aversive?
• State Statistical Hypothesis
• H0 µplain side 30 min
• H1 µplain side 30 min
• Sample Descriptive Statistics
• M35, s9, n16
• Compute standard error of estimate

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One Sample t-test Example

• Set decision Criteria
• two-tailed test (nondirectional)
• Critical values in t-table based on df n-1
• For our sample, n16, therefore df16-1 15
• If a.05, from t-table, tcrit 2.131
• Compute t-test statistic
• Make decision Reject Ho if tobtained gt tcrit
• For our example, reject Ho because tobtained
  2.22 gt tcrit 2.131

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The t-distribution
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t test Critical Values versus z test Critical
Values
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t table
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Example Repeated two-tailed (Directional Test)

• Research Hypothesis Birds avoid eyespots.
• State Statistical Hypothesis
• H0 µplain side lt 30 min
• H1 µplain side gt 30 min
• Sample Descriptive Statistics
• M35, s9, n16
• Compute standard error of estimate

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One Sample t-test Example

• Set decision Criteria
• one-tailed test (directional)
• Critical values in t-table based on df n-1
• For our sample, n16, therefore df16-1 15
• If a.05, from t-table, tcrit 1.753
• Compute t-test statistic
• Make decision Reject Ho if tobtained gt tcrit
• For our example, reject Ho because tobtained
  2.22 gt tcrit 1.753

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Writing-up Test Results in APA Format
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Writing-up Test Results in APA Format

• Birds spent a significantly greater amount of
  time on the plain side (M35) compared to the
  spotted side (M25) of the chamber, t(15) 2.13,
  p lt .05.

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What does plt.05 mean?
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Assumptions of the t-test

• Same as for the z-test
• Independent Observations
• Normality
• For small samples (e.g., if n lt 30), violations
  are a problem and affect the validity of the
  hypothesis test. But if sample size is
  sufficiently large (e.g., if ngt30),moderate
  violations are not a big problem.

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How big is the effect?

• We can use Cohens d to estimate effect size