Monday, October 29

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Monday, October 29
Independent samples t-test for the difference
between two means.
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H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
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(No Transcript)
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How do we know if the difference between these
means, of 53.75 - 51.16 2.59, is reliably
different from zero?
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We could find confidence intervals around each
mean...

95CI 52.07 ? ? boys ? 55.43
95CI 49.64 ? ? girls ? 52.68
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H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
But we can directly test this hypothesis...
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H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
To test this hypothesis, you need to know the
sampling distribution of the difference between
means.
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??X1-X2
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H0 ?1 - ?2 0 H1 ?1 - ?2 ? 0
To test this hypothesis, you need to know the
sampling distribution of the difference between
means.
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??X1-X2
which can be used as the error term
in the test statistic.
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The sampling distribution of the difference
between means.
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This reflects the fact that two independent
variances contribute to the variance in the
difference between the means.
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The sampling distribution of the difference
between means.
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This reflects the fact that two independent
variances contribute to the variance in the
difference between the means.
Your intuition should tell you that the variance
in the differences between two means is larger
than the variance in either of the means
separately.
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The sampling distribution of the difference
between means, at n ?, would be
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The sampling distribution of the difference
between means.
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??X1-X2 ? ?21 ? ?22

n1 n2
Since we dont know ?, we must estimate it with
the sample statistic s.
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The sampling distribution of the difference
between means.
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??X1-X2 ? ?21 ? ?22

n1 n2
Rather than using s21 to estimate ?21 and s22 to
estimate ?22 , we pool the two sample estimates
to create a more stable estimate of ?21 and ?22
by assuming that the variances in the two
samples are equal, that is, ?21 ?22 .
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sp2 sp2
sX1-X2

N1 N2
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sp2 sp2
sX1-X2

N1 N2
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sp2 sp2
sX1-X2

N1 N2
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Because we are making estimates that vary by
degrees of freedom, we use the t-distribution to
test the hypothesis.
at (n1 - 1) (n2 - 1) degrees of freedom
(or N-2)
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Assumptions

• X1 and X2 are normally distributed.
• Homogeneity of variance.
• Samples are randomly drawn from their respective
  populations.
• Samples are independent.