1. Tests and confidence intervals for a difference between two population means

2. Introduction
For inference on the difference between the means of two populations, we need samples from both populations. The basic assumptions are:
is a random sample from a distribution with mean and variance
The and samples are independent

3. Allowing different sample sizes
Different samples sizes are allowed because it may be more difficult or expensive to sample one population than another.
Also, the sample sizes may initially be the same, but the actual sizes may differ (because of dropouts, death, etc.).

4. Proposition The expected value of is and the standard deviation is
This result follows due to independence of the samples.
Thus is unbiased for It is the natural estimator of the difference between the means.

5. Test procedures for normal populations with known variances
The statistic
has a standard normal distribution (if the original populations are normal).
To test , replace by the null value to form the test statistic.

6. Rejection regions
Null hypothesis Alternative hypothesis Rejection region or

7. Large-sample tests
When both samples are sufficiently large, ( ), normality and known variances aren’t needed. The following variable is approximately normally distributed: The test statistic is then .

8. Large-sample confidence intervals
Under these conditions, a level confidence interval takes the form
Upper and lower bounds are calculated by retaining the appropriate sign and replacing by

9. The two-sample t-test and confidence interval
When at least one of the sample sizes is small and the population variances are unknown, we need assumptions about the distribution for inference.
We again consider the case when the populations have a normal distribution.

10. Theorem
When the population distributions are both normal, the variable
has approximately a t distribution with df v estimated from the data by
(round v down to the nearest integer).

11. Confidence interval and rejection regions
For , the two-sample confidence interval again takes the form
The two-sample test has rejection regions:
Alternative hypothesis Rejection region