Hypothesis Testing

Summary

Hypothesis testing uses sample evidence to decide between a null hypothesis and an alternative hypothesis. A test statistic measures how far the sample result is from the null value; if the statistic falls in a predetermined rejection region, the null is rejected. The structure of the alternative determines whether the test is one-tailed or two-tailed.^[1]

Prerequisites

• Point Estimation
• Confidence Intervals
• Central Limit Theorem
• Normal Distribution

Definition / Notation

A test of $H_0:\theta ={\theta }_0$ against $H_a:\theta \ne {\theta }_0$ is two-tailed; tests against $H_a:\theta >{\theta }_0$ or $H_a:\theta <{\theta }_0$ are one-tailed.

Parameters / Assumptions

• A fully specified parametric model under $H_0$ .
• Independent observations, unless the test is designed for dependence.
• The sampling distribution of the test statistic under $H_0$ is known or approximated.
• The significance level $\alpha$ is fixed before seeing the data.

Essential Result

For $H_0:\mu ={\mu }_0$ with known $\sigma$ , the test statistic is

• Two-tailed test ( $H_a:\mu \ne {\mu }_0$ ): reject if $|Z|>z_{\alpha /2}$ .
• One-tailed upper test ( $H_a:\mu >{\mu }_0$ ): reject if $Z>z_{\alpha }$ .
• One-tailed lower test ( $H_a:\mu <{\mu }_0$ ): reject if $Z<-z_{\alpha }$ .

There is a duality between two-sided tests and confidence intervals: a two-sided $100(1-\alpha )\mathrm{\%}$ confidence interval contains exactly the values of ${\mu }_0$ that would not be rejected by a two-sided $\alpha$ -level test.^[2]

Worked Example

Test $H_0:\mu =100$ against $H_a:\mu \ne 100$ with $n=36$ , $\overline{x}=104$ , $\sigma =12$ , and $\alpha =0.05$ . The test statistic is

Since $|2|>1.96$ , reject $H_0$ at the 5% level.

Common Mistakes

• Confusing statistical significance with practical importance.
• Choosing a one-tailed alternative after seeing the data direction.
• Believing that failing to reject $H_0$ proves $H_0$ is true.
• Applying a test whose distribution assumptions are violated.

Connections

• p-values
• Type I and Type II Errors
• Confidence Intervals
• Maximum Likelihood

References