Notes › EENG 3421: Probability and Statistics for Engineers Lecture 26
Mean Hypothesis Tests
219 words 2 min Modified
Table of Contents
Core Setup
- A hypothesis test evaluates a claim about a population parameter using sample information
Definition 26.1 (Null and Alternative Hypotheses).
Null hypothesis $H_0$ is the default claim to be tested Alternative hypothesis $H_1$ is the competing claim
- Test types
- right tailed
- left tailed
- two tailed
- Significance level $\alpha$
- probability threshold used for deciding rejection
Two Common Approaches
- p value approach
- critical value approach
Definition 26.2 (p Value).
The p value is the probability under $H_0$ of observing a sample statistic at least as extreme as the observed value
Proposition 26.3 (p Value for One Tailed Tests).
For a one tailed test, the p value is the area in the relevant tail beyond the observed value of the sample statistic
Proposition 26.4 (p Value for Two Tailed Tests).
For a two tailed test, the p value is twice the tail area beyond the observed value of the sample statistic
Decision Rule Using P Value
Theorem 26.5 (Decision Rule).
Reject $H_0$ if
$$ \text{p value} \le \alpha $$Do not reject $H_0$ if
$$ \text{p value} > \alpha $$Procedure Skeleton
- State $H_0$ and $H_1$
- Select the distribution to use for the test statistic under $H_0$
- Determine rejection and nonrejection regions or compute p value
- Compute test statistic from sample
- Decide reject or fail to reject
References
- Course lecture slides 23–26
Sources
- Course lecture slides 23–26