Hypothesis Tests: Two Independent Samples

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Hypothesis Tests Two Independent Samples
  
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Violent Videos Again

• Bushman (1998) Violent videos and aggressive
  behavior
• Doing the study Bushmans way--almost
• Bushman had two independent groups
• Violent video versus educational video
• We want to compare mean number of aggressive
  associations between groups

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The Data
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Analysis

• These are Bushmans data, though he had more
  dimensions.
• We still have means of 7.10 and 5.65, but both of
  these are sample means.
• We want to test differences between sample means.
• Not between a sample and a population mean

Cont.
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Analysis--cont.

• How are sample means distributed if H0 is true?
• Need sampling distribution of differences between
  means
• Same idea as before, except statistic is (X1 -
  X2)

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Sampling Distribution of Mean Differences

• Mean of sampling distribution m1 - m2
• Standard deviation of sampling distribution
  (standard error of mean differences)

Cont.
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Sampling Distribution--cont.

• Distribution approaches normal as n increases.
• Later we will modify this to pool variances.

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Analysis--cont.

• Same basic formula as before, but with
  accommodation to 2 groups.
• Note parallels with earlier t

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Our Data
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Degrees of Freedom

• Each group has 100 subjects.
• Each group has n - 1 100 - 1 99 df
• Total df n1 - 1 n2 - 1 n1 n2 - 2 100
  100 - 2 198 df
• t.025(198) 1.97 (approx.)

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Conclusions

• Since 2.66 gt 1.97, reject H0.
• Conclude that those who watch violent videos
  produce more aggressive associates (M 7.10)
  than those who watch nonviolent videos (M
  5.65), t(198) 2.66, p lt .05 (one-tailed).

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IQ PillIndependent Samples
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IQ PillIndependent Samples
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Computing t
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Conclusions

• Since -.88 lt 1.97, fail to reject H0.
• Conclude that those who took the IQ pill did not
  score significantly higher (M 102.4) on an IQ
  test than those who did not take the IQ pill (M
  99.4), t(8) -.88, p gt .10.

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t value for related samples
Larger t value for same data
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Assumptions

• Two major assumptions
• Both groups are sampled from populations with the
  same variance
• homogeneity of variance
• Both groups are sampled from normal populations
• Assumption of normality
• Frequently violated with little harm.

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Pooling Variances

• If we assume both population variances are equal,
  then average of sample variances would be better
  estimate.

Cont.
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Pooling Variances--cont.

• Substitute sp2 in place of separate variances in
  formula for t.
• Will not change result if sample sizes equal
• Do not pool if one variance more than 4 times the
  other.
• Discuss

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Heterogeneous Variances

• Refers to case of unequal population variances.
• We dont pool the sample variances.
• We adjust df and look t up in tables for adjusted
  df.
• Minimum df smaller n - 1.
• Most software calculates optimal df.

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Effect Size for Two Groups

• Extension of what we already know.
• We can often simply express the effect as the
  difference between means (1.45)
• We can scale the difference by the size of the
  standard deviation.
• Gives d
• Which standard deviation?

Cont.
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Effect Size, cont.

• Use either standard deviation or their pooled
  average.
• We will pool because neither has any claim to
  priority.
• Pooled st. dev. v14.8 3.85

Cont.
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Effect Size, cont.

• This difference is approximately .38 standard
  deviations.
• This is a medium effect.
• Elaborate

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Confidence Limits
Cont.
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Confidence Limits--cont.

• p .95 that interval formed in this way includes
  the true value of ?1 - ?2.
• Not probability that ?1 - ?2 falls in the
  interval, but probability that interval includes
  ?1 - ?2.
• Comment that reference is to interval formed in
  this way, not this interval.

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