From STA238 Probability Statistics and Data Analysis II

Hypothesis testing

Someone hands you a value for the parameter. Assume it is true, then see where your estimator landed relative to it. If that position is freakishly unlikely, the claim is wrong.

The three steps

1. State the hypotheses.

$H_0$ : the default. Always an equality, $\mu = \mu_0$ .
$H_1$ : what you conclude if you reject. $\mu > \mu_0$ , $\mu < \mu_0$ , or $\mu \neq \mu_0$ .

2. Compute the test statistic. The standardization from confidence intervals, with $\mu_0$ plugged in for the unknown $\mu$ :

$t = \frac{\bar x - \mu_0}{s / \sqrt n}$

3. Get the $p$ -value and compare to $\alpha$ . Reject $H_0$ when $p < \alpha$ .

$H_1$	$p$ -value
$\mu > \mu_0$	$P(T \ge t)$
$\mu < \mu_0$	$P(T \le t)$
$\mu \neq \mu_0$	$2 \cdot P(T \ge \lvert t \rvert)$

The $p$ -value is a CDF evaluation on whichever distribution supplies $c$ in confidence intervals. Same table.

Two routes to the same answer

$p$ -value: reject when $p < \alpha$ .
Confidence interval: reject when $\mu_0$ falls outside the $(1-\alpha)$ interval.

Equivalent.

What you write

Reject: “At the $\alpha = 0.05$ level we reject $H_0$ ; there is evidence that $\mu > 80$ .”

Fail to reject: “At the $\alpha = 0.05$ level we fail to reject $H_0$ ; there is not enough evidence that $\mu > 80$ .”

Never “accept $H_0$ .” Failing to reject is not evidence for $H_0$ .

Strength of evidence

$p$	Interpretation
$> 0.10$	No evidence against $H_0$
$0.05$ to $0.10$	Weak
$0.01$ to $0.05$	Moderate
$0.001$ to $0.01$	Strong
$< 0.001$	Very strong

1 Someone hands you a value for the parameter. Assume it is true, then see where your estimator

2 landed relative to it. If that position is freakishly unlikely, the claim is wrong.

4 ## The three steps

6 **1. State the hypotheses.**

8 - $H_0$: the default. Always an equality, $\mu = \mu_0$.

9 - $H_1$: what you conclude if you reject. $\mu > \mu_0$, $\mu < \mu_0$, or $\mu \neq \mu_0$.

11 **2. Compute the test statistic.** The standardization from

12 [confidence intervals](03%20Confidence%20intervals.md), with $\mu_0$ plugged in for the

13 unknown $\mu$:

15 $$t = \frac{\bar x - \mu_0}{s / \sqrt n}$$

17 **3. Get the $p$-value and compare to $\alpha$.** Reject $H_0$ when $p < \alpha$.

19 | $H_1$ | $p$-value |

20 |---|---|

21 | $\mu > \mu_0$ | $P(T \ge t)$ |

22 | $\mu < \mu_0$ | $P(T \le t)$ |

23 | $\mu \neq \mu_0$ | $2 \cdot P(T \ge \lvert t \rvert)$ |

25 The $p$-value is a CDF evaluation on whichever distribution supplies $c$ in

26 [confidence intervals](03%20Confidence%20intervals.md). Same table.

28 ## Two routes to the same answer

30 - **$p$-value:** reject when $p < \alpha$.

31 - **Confidence interval:** reject when $\mu_0$ falls outside the $(1-\alpha)$ interval.

33 Equivalent.

35 ## What you write

37 Reject: "At the $\alpha = 0.05$ level we reject $H_0$; there is evidence that $\mu > 80$."

39 Fail to reject: "At the $\alpha = 0.05$ level we fail to reject $H_0$; there is not enough

40 evidence that $\mu > 80$."

42 Never "accept $H_0$." Failing to reject is not evidence for $H_0$.

44 ## Strength of evidence

46 | $p$ | Interpretation |

47 |---|---|

48 | $> 0.10$ | No evidence against $H_0$ |

49 | $0.05$ to $0.10$ | Weak |

50 | $0.01$ to $0.05$ | Moderate |

51 | $0.001$ to $0.01$ | Strong |

52 | $< 0.001$ | Very strong |