Hypothesis tests

The courtroom of statistics

• A drug company claims their new medicine works better than the old one. How do you decide if the evidence is strong enough?
• Hypothesis testing 假设检验 is the statistical equivalent of a courtroom: you assume the defendant (the null hypothesis 原假设) is innocent until proven guilty beyond reasonable doubt.

Setting up the hypotheses

• A hypothesis test uses sample data to judge a claim.
• The null hypothesis $H_0$ — the default position (usually "no change" or "no effect").
• The alternative 备择假设 $H_1$ — what you suspect is true.
• One-tailed 单尾 if $H_1$ points one way ($\mu > 50$); two-tailed 双尾 if both ($\mu \neq 50$).

A two-tailed 5% test rejects H0 in the shaded tails beyond plus or minus 1.96

Running the test

• Fix a significance level 显著性水平 (often 5%), compute a test statistic 检验统计量, and see if it lands in the rejection region 拒绝域.

• If $|z|$ exceeds the critical value 临界值 (1.96 for 5% two-tailed, 1.645 for 5% one-tailed), reject $H_0$.

A one-tailed test at 5% rejects H₀ only if the test statistic lands in the shaded tail.

Types of error

• Type I error 第一类错误: rejecting $H_0$ when it is true (false positive). Probability = significance level $\alpha$.
• Type II error 第二类错误: accepting $H_0$ when it is false (false negative).

Worked example — z-test

• $H_0: \mu = 50$, $H_1: \mu \neq 50$ (two-tailed, 5% level). $\sigma = 8$, $n = 64$, $\bar{x} = 52$.
• $z = \dfrac{52 - 50}{8/\sqrt{64}} = \dfrac{2}{1} = 2$.
• Critical value for 5% two-tailed: $1.96$. Since $2 > 1.96$, reject $H_0$.
• Set a null and alternative hypothesis; the test statistic falls in the critical region (reject) or the acceptance region. A Type II error is accepting a false $H_0$. Tests may use a binomial or Poisson distribution.

You've got it

Write from a test that does not reject

• H0 says the mean wait is 8 minutes. H1 says it is greater than 8. The test statistic is 1.2 and the critical value is 1.96.