Significance test

Make Hypotheses

• Null hypothesis $H_0$:
  • The null hypothesis is the statement being tested
  • ”=” always appears in the null hypothesis.
• Alternative hypothesis $H_a$:
  • $H_a$ is the complement of $H_0$

The alternative hypothesis is set up based on the property one wants to test. Then the null hypothesis is the complement of alternative hypothesis. Then the null hypothesis is tested, and one hopes to reject the null hypothesis, based on the sample data, so that the alternative hypothesis (one’s inference) is proven true by the data.

	$H_0$ True	$H_a$ True
Reject $H_0$	Type I error	Correct
Fail to reject $H_0$	Correct	Type II error

Test Statistics

• Significance level $\alpha$: The probability of rejecting $H_0$ when $H_0$ is true.

T-test

t distribution \(t = \frac{\bar{Y}-\mu}{s/\sqrt{n}}\)

• Single side:
• Two side

fill in image for reject/accept region

The idea is that if $\mu = \mu_0$ is true, then $t = \frac{\bar{Y} - \mu}{s/\sqrt{n}}$ would fall close to the center with high probability. If $t$ falls far from the center (rejection area), it is less likely to be true that $\mu = \mu_0$.

For example, if $H_0:\mu \geq \mu_0$, then if the sample gives $t«0$, and we believe that $H_0$ is true. Then the probability that this ($t«0$) happens is very small ($\alpha$). Thus, if we reject $H_0$ now, the probability we make mistake is also small. We therefore feel confident to reject $H_0$.