Lessons

- What is Hypothesis Testing
- Test Statistic, Type I and type II Errors, and Significance Level
- Decision Rule in Hypothesis Testing
- p-Value in Hypothesis Testing
- Selecting the Appropriate Test Statistic
- Hypothesis Testing with t-statistic
- Hypothesis Testing with z-statistic
- Tests Concerning Differences in Means
- Paired Comparision Tests - Mean Differences When Populations are Not Independent
- Hypothesis Tests Concerning Variances
- Chi-square Test – Test for value of a single population variance
- F-test - Test for the Differences Between Two Population Variances
- Non-parametric Tests

# Tests Concerning Differences in Means

Sometimes we want to test if the mean values differ between two populations. We can assume that the two populations are normally distributed and that the samples are drawn independently.

We can combine observations from both samples to get a pooled estimate of the unknown population variance.

The hypothesis can be formed as follows:

- H0: µ1 - µ2 = 0 versus HA: µ1 - µ2 ≠ 0
- H0: µ1 - µ2 ≤ 0 versus HA: µ1 - µ2 > 0
- H0: µ1 - µ2 ≥ 0 versus HA: µ1 - µ2 < 0

### Case 1: Normally distributed populations, population variances unknown, but assumed to be equal

### Case 2: Normally distributed populations, population variances unequal and unknown

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