Psychology 340

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Statistics for the Social Sciences

• Psychology 340
• Spring 2010

Using t-tests Basic introduction and 1-sample
t-tests
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Outline (for content up to Exam 2)

• Review t-tests (note next exam is March 4, right
  before spring break)
• One sample, related samples, independent samples
• Effect sizes with t-tests
• Confidence intervals in t-test type designs

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• Statistical analysis follows design

• The one-sample z-test can be used when

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• Statistical analysis follows design

• The one-sample t-test can be used when

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Testing Hypotheses

• Hypothesis testing a five step program

• Step 1 State your hypotheses
• Step 2 Set your decision criteria
• Step 3 Collect your data
• Step 4 Compute your test statistics
• Compute your estimated standard error
• Compute your t-statistic
• Compute your degrees of freedom
• Step 5 Make a decision about your null hypothesis

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Performing your statistical test

• What are we doing when we test the hypotheses?
• Consider a variation of our memory experiment
  example

• Conclusions
• the memory treatment sample are the same as
  those in the population of memory patients.
• they arent the same as those in the population
  of memory patients

H0
HA
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Performing your statistical test

• What are we doing when we test the hypotheses?

Real world (truth)
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Performing your statistical test

• What are we doing when we test the hypotheses?
• Computing a test statistic Generic test

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Performing your statistical test
Test statistic
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Performing your statistical test
Test statistic
Diff. Expected by chance
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Performing your statistical test
Test statistic
Diff. Expected by chance
Estimated standard error
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Performing your statistical test
Test statistic
Diff. Expected by chance
Estimated standard error
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One sample t-test

• The t-statistic distribution (a transformation of
  the distribution of sample means transformed)
• Varies in shape according to the degrees of
  freedom

• New table the t-table

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One sample t-test

• The t-statistic distribution (a transformation of
  the distribution of sample means transformed)

• To reject the H0, you want a computed test
  statistics that is large
• The alpha level gives us the decision criterion

• New table the t-table

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One sample t-test

• New table the t-table

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One sample t-test

• What is the tcrit for a two-tailed hypothesis
  test with a sample size of n 6 and an a-level
  of 0.05?

df n - 1 5
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One sample t-test

• What is the tcrit for a one-tailed hypothesis
  test with a sample size of n 6 and an a-level
  of 0.05?

Distribution of the t-statistic
a 0.05
n 6
One-tailed
df n - 1 5
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One sample t-test

• An example One sample t-test

• We give a n 16 memory patients a memory
  improvement treatment.

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

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One sample t-test

• An example One sample t-test

• Step 1 State your hypotheses

• We give a n 16 memory patients a memory
  improvement treatment.

µTreatment gt µpop 60

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

µTreatment lt µpop 60
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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

• Step 2 Set your decision criteria

a 0.05

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

• Step 2 Set your decision criteria

a 0.05
One -tailed

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

a 0.05
One -tailed

• Step 3 Collect your data

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

a 0.05
One -tailed

• Step 4 Compute your test statistics

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

-2.5
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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

a 0.05
One -tailed
t -2.5

• Step 4 Compute your test statistics

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

a 0.05
One -tailed

• Step 5 Make a decision about your null hypothesis

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

tcrit -1.753
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One sample t-test

• An example One sample t-test

H0 µTreatment gt µpop 60
HA µTreatment lt µpop 60

• We give a n 16 memory patients a memory
  improvement treatment.

a 0.05
One -tailed

• Step 5 Make a decision about your null hypothesis

• How do they compare to the general population
  of memory patients who have a distribution of
  memory errors that is Normal, µ 60?

- Reject H0
-1.753 tcrit
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Assumptions of t-statistics

• Values in the sample must be independent
  observations
• Each observation is independent of all of the
  other observations (see pg 253 of your textbook
  for examples of non-independence)
• The population that is sampled must be normally
  distributed
• It turns out that t-tests are pretty robust
  against violations of this assumption, especially
  with large samples

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Effect Sizes Power for 1-sample t Test
Recall for the 1-sample z-test
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Using SPSS 1 sample t

• Entering the data
• Observations go into one column

e.g., 2, 6, 5, 9

• Performing the analysis
• Analyze -gt Compare means -gt one sample t-test
• Identify which column to do your test on
• Enter the Test value this is the population
  mean in the H0
• Be careful, the default is 0, this may not be
  what you want

• Reading the output
• Mean of the sample, the computed-t, degrees of
  freedom, p-value (Sig.) (also reminds you of your
  test value)

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Next time

• Related samples t-tests
• Independent samples t-tests