PSYC2009 Study Guide - Final Guide: Halfwidth And Fullwidth Forms, Null Hypothesis, Confidence Interval
Comparing Two Means
Confidence Interval for the Difference Between Two Means
• Pr (x̅1 - x̅2 - w < μ1 - μ2 < x̅1 - x̅2 + w) = 1 - α with w(tα/)(SE).
• The variance of the difference between two means is based on a weighted sum of the
variances of each mean, called the pooled variance (average of variance 1 and 2).
• The weights are determined by the size of each sample.
• S2pooled = [(N1 - 1) s21 + (N2 - 1) s22]/[(N1 - 1) + (N2 - 1)]
• The resulting pooled SE is serr = (s2pooled [1/N1 + 1/N2])
• In order for the pooled variance to be a sensible estimate, we must assume that the two
sample variances differ only because of sampling error, i.e. the homogeneity of variance
assumption.
• The df = (N1 - 1) + (N2 - 1), or just df = N1 + N2 - 2,
• Constructing the CI for μ1 – μ2:
1) Calculate the means and SDs for each sample, and the pooled SE, serr.
2) Then decide the confidence level and obtain α.
3) Using Table A.3, find the cell entry in the row corresponding to df = N1 + N2 - 2, and the
column corresponding to a 2-tail area of α.
o Denote this cell entry by tα/.
o For small to moderate sample sizes, the half-width of the CI is w = (tα/)(SE).
o If the df value is larger than any of the values in Table A.3, use the normal
distribution instead and find the appropriate value of zα/ from Table A.2.
o For larger samples such as these, the half-width is w = (zα/)(SE).
4) The resulting CI statement may be completed by plugging the sample means, half-width w,
and α into our formula Pr (x̅1 - x̅2 - w < μ1 - μ2 < x̅1 - x̅2 + w) = 1 - α.
• z.005 = 2.5758, t.005 = 2.6040 (which gives a slightly wider interval).
• If you want to be conservative about rejecting null hypotheses then choose the wider
interval/t.
• You can't compare two CIs for each of the means and see if they overlap because the overlap
doesn't have enough theory behind it to tell us how much of the distribution occurs in this
overlaps.
The Between-Subjects t-test
• The t-statistic compares the difference between the sample means against the null hypothesis
difference, and divides that by the pooled SE: t(df) = [(x̅1 - x̅2) - (μ1 - μ2)]/serr, where df = N1 + N2
- 2.
• The equivalent CI around t(df) is the same as for the one-sample version: Pr(-tα/ < t(df) < tα/) =
1 - α.
• We are able to reject a null hypothesis of no difference using the t test by replacing μ1 - μ2 in
the formula for t(df) with o, and then ask whether the resulting value of t(df) is a plausible one
or not by finding out whether it lies inside the CI around t(df).
Effect Size
• For assessing the size of difference between means.
• For the distribution-based measure, we use Cohen's d for two independent samples: d = [(x̅1 -
x̅2) - (μ1 - μ2)]/spooled.
• Here's how to get d from a reported t-statistic in the psyc research literature: d = t (1/N1 +
1/N2).
Comparing More Than Two Means
Problem of Multiple Confidence Intervals
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