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 (or “favored” or “protected”) hypothesis that we stick with unless evidences are strongly against it.
  • It usually states that nothing “interesting” is happening.
    • The drug has no effect on blood pressure.
    • The coin is fair.
    • A criminal trial analogy: “Innocent until proven guilty”
      • \(H_0\): The defendant is innocent.

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 simple yes or no:
    • Is something “interesting” happening, or not?
    • Is the new drug effective (\(H_a\)) or not (\(H_0\))?

Hypothesis testing: A method to decide whether we should reject the null hypothesis (\(H_0\)) 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

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

Type I vs. type II errors

  • In many cases, a type I error (false alarms) is often 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 \(\alpha=0.05, 0.01, 0.001\).