Hypothesis testing: A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.
State null H0 and alternative hypotheses H1.
Select a level of significance, α.
Identify the test statistic. Formulate a decision rule. Critical-value Approach P-value Approach Confidence interval Approach
Take a sample, arrive at decision. Do Not Reject, H0 Reject H0 and accept H1
Type I error: Rejecting the null hypothesis, H0, when it is true. (False positive, 假阳性)
Type II error: Not to reject the null hypothesis when it is false. (False negative, 假阴性)
Two-tailed test: Θ0 = {µ0} H0 : µ = µ0, Ha : µ = µ0.
Upper-tailed test: Θ0 = (−∞, µ0] H0 : µ ≤ µ0, Ha : µ > µ0.
Lower-tailed test: Θ0 = [µ0, ∞) H0 : µ ≥ µ0, Ha : µ < µ0.
Variance is known
Z-test Under H0, the distribution of z is known, i.e., z ∼ N(0, 1)
Population proportion Bernoulli（p）:
reject H0 : p ≤ p0 z ≥ zα
Significance level = P(Reject H0|H0) = P(|z| ≥ c|H0)
p-value: the probability of observing a sample value as extreme as, or more extreme than, the value observed, given that H0 is true.
reject H0 : µ = µ0, if p-value ≤ α
T test Under H0, the distribution of t is known, i.e., t ∼ tn−1, where n − 1 is the degree of freedom. Other same as Z test
When variance is unknown
The power of a hypothesis test is 1 minus the probability of a Type II error.
Two sample test: H0 : µ1 − µ2 = D0, Ha : µ1 − µ2 = D0. variance 1,2 are known,
Two population proportions: H0 : p1 = p2 Ha : p1 = p2
Variance unknown:
Variance unknown:
use
use
Matched samples: new list: µd=µ1 − µ2
Non-parametric: sign test: H0 : p = 0.5, Ha : p = 0.5
Normal approximation:
t~tdf