8.1 Hypothesis test

Definition

Hypothesis: a statement about a population parameter.

• The mean height for an adult male is 5’9”.
• The drug lowers patients’ blood pressure.
• Record the heights from 20 people, the sample mean is 5’7”.
  • Not a hypothesis

• Simple hypothesis
  • The parameter takes a single value (e.g., \(\mu=8\)).
• Composite hypothesis
  • The parameter takes more than one value (\(\mu > 8\)).

Null hypothesis \(H_0\)

The statement that is initially assumed to be true.

• It is the default hypothesis that we stick with unless evidences are strongly against it.
  • It is the “favored” or “protected” hypothesis
• It usually states that nothing interesting is happening.
  • The drug has no effect on blood pressure.
  • The coin is fair.
  • Criminal trials: “Innocent until proven guilty”

Alternative hypothesis \(H_a\)

• It is the hypothesis that is contradictory to \(H_0\).
• It states something interesting is happening.
  • The drug lowers patients’ blood pressure.
  • The coin is biased.
  • The defendant is guilty.
• The burden of proof is placed on the alternative hypothesis.

Hypothesis testing

• Many scientific questions can be boiled down to a yes or no:
  • Is something interesting happening, or not?
  • Is the new drug effective or not?

Hypothesis testing: A method to decide whether we should reject the null hypothesis or not based on data.

The two types of errors in hypothesis testing

\(H_0\): the person is not pregnant.

	Fail to reject \(H_0\)	Reject \(H_0\)
\(H_0\) is true	Correct	Type I error (false alarms)
\(H_0\) is false	Type II error (misses)	Correct

	Fail to reject \(H_0\)	Reject \(H_0\)
\(H_0\) is true	Correct	Type I error (false alarms)
\(H_0\) is false	Type II error (misses)	Correct

\[ \small{ \begin{aligned} \text{P}(\text{Type I error})&=\text{P}(\text{reject $H_0$ when it is true}) \\ &=\alpha \\ \\ \text{P}(\text{Type II error})&=\text{P}(\text{fail to reject $H_0$ when it is false}) \\ &=\beta \\ \end{aligned} } \]

Type I vs. type II errors

• In many cases, a type I error (false alarms) is more serious than a type II error (misses).
• Criminal trial
  • Type I error: Innocent person wrongfully convicted.
  • Type II error: Guilty person going free.
• Drug testing
  • Type I error: Approving a drug that does nothing.
  • Type II error: Not approving an effective drug.

Significance level

• In practice, we often specify the largest value of \(\alpha\) that can be tolerated.
• The resulting \(\alpha\) is called the significance level.
• Common significance levels \(\alpha=0.05, 0.01, 0.001\).