1. Tests for Two Means – Normal Populations
Suppose we have two independent random samples from two
different Normal populations and we are interested in comparing the
means of these two populations.
The CLT is not necessary and hence the samples can be small.
We differentiate between three cases, variances of the populations
are known, variances are unknown, and variances are unknown but
equal….
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2. Example
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3. Hypothesis Tests with Paired Data
A matched pairs study is a study in which, subjects are matched in
pairs and the outcomes are compared within each matched pair. One
situation calling for match pairs is when observations are taken on
the same subjects, under different conditions.
A match pairs analysis is needed when there are two measurements
or observations on each individual and we want to examine the
difference.
For each individual (pair), we find the difference d between the
measurements from that pair. Then we treat the d’s as one sample.
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4. If the sample size is large, the CLT applies and we perform the test
for no difference between the treatments effect using the standard
normal distribution.
If the sample size is small, the CLT do not apply, we need to assume
the data is normal and test for no difference between the treatments
effect using the t distribution.
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5. Example
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6. Tests for Two Binomial Proportions
Suppose X1, …, Xn are iid Bernoulli(p1) independent of Y1, …, Ym that are iid Bernoulli(p2).
Further, suppose that n and m are large.
The CLT applies to both sample proportions.
The test is conducted using standard normal distribution, using
pooled variance estimate….
The CI is constructed using the standard normal distribution,
without pooling variance estimates…
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7. Example
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8. Test on Pairs of Variances
Suppose are iid independent of that
are iid
We are interested in testing versus a one sided or a two
sided alternative…
Then…
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9. Example
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