Hypothesis Testing

This chapter systematically studies the basic ideas and steps of hypothesis testing, two types of errors, and hypothesis testing for the mean and variance of a normal population.

Basic Ideas and Steps of Hypothesis Testing

• Null hypothesis $H_0$, alternative hypothesis $H_1$
• Significance level $\alpha$
• Test statistic, P-value
• Rejection region, acceptance region
• Steps: propose hypotheses → select statistic → determine rejection region → calculate sample value → draw conclusion

Two Types of Errors

• Type I error: reject $H_0$ when $H_0$ is true
• Type II error: accept $H_0$ when $H_0$ is false

Hypothesis Testing for a Single Normal Population

• Mean test (variance known/unknown): Z-test/T-test
• Variance test: chi-square test

Hypothesis Testing for Two Normal Populations

• Mean difference test: two-sample T-test
• Variance ratio test: F-test

Exercises

1. Briefly describe the general steps of hypothesis testing.
2. Let $X \sim N(\mu, \sigma^2)$, $\sigma$ known, test $H_0: \mu=\mu_0$. Write the test statistic.
3. Explain the meaning of Type I and Type II errors.
4. If the variances of two normal populations are unknown but equal, which test should be used for the mean difference?
5. Let the variances of two populations be $\sigma_1^2, \sigma_2^2$. To test $H_0: \sigma_1^2=\sigma_2^2$, which test should be used?