Inferences about central values (.) ]µnormal., 5 # Inferences about. using data: C", C#,..., C8 (collected as a random sample) Point estimate How good is the estimate?.s œc 1 œ C" C# âc8 8 Confidence interval ( 5 # known) A 1001 α % confidence interval (CI) for. is 5 CD 5 αî#, CDαÎ# œ6,? 8 8
A CI is constructed so that under repeated sampling, the long-run proportion of such intervals that contain. will be 1 α: P ]D 5 1 ]D 5 αî#. αî# 8 8 œ α What is P 5 CD CD 5.? αî# 8 αî# 8 What does the estimate say about contending models? Statistical hypothesis tests ( 5 # known) First model:. œ.! (known constant) Null hypothesis H (often the skeptic's hypothesis)!
Second model:..! Alternative hypothesis H (often the research a hypothesis) Distribution of ] under each model: Decision to abandon first model (null) for the second model (alternative) based on a rejection region. Reject H! in favor of H if a C exceeds a critical value: reject H if C C! + do not reject H if CC! +
α: probability of rejecting the null model, given the null model is the true model (set small enough to convince skeptics) α œ 0.05: everyday science α œ 0.01: important stuff (medical, etc.) Hypotheses: H!:. Ÿ.! ( œ, but get all values free!) H a :..! Test statistic: rephrase in terms of D (instead of C) Dœ C.! 5Î8 Rejection region (decision rule) reject H if! D D α do not reject H if! DD α 5 Cα œ Dα 8.!
One-sided in the other direction (left-tailed test) Hypotheses: H!:..! H a :..! Rejection region (decision rule) reject H if! DŸ D α do not reject H if! D D α
Two-sided hypothesis test H :. œ.!! H a :. Á.! Rejection region: reject H if C is outside of C, C! 1 α α # # Test statistic: Dœ C.! 5Î8 Rejection region: reject H if D! D α Î# do not reject H if D D α! Î#
T -values for hypothesis tests The :-value for a hypothesis test is the probability that, under repeated sampling, the test statistic would be as extreme as the observed value of the test statistic, given that the null hypothesis is true. Right-sided: H :. Ÿ.!! H a :..! :œt^ D reject H if D D Í reject H if :Ÿ! α! α Left-sided: H!:..! H a :..! :œt^ÿd reject H if DŸ D Í reject H if :Ÿ! α! α Two-sided: H :. œ.!! H a :. Á.! :œ2t^ D reject H if D D Í reject H if : Ÿ α! αî#!
Note: the :-value is not the probability of the null hypothesis!! ( probability of a hypothesis is meaningless in frequentist statistics) Relationship between CIs and two-sided hypothesis tests A 1001 α % CI for. is the set of all values of.! for which the null hypothesis H!:. œ.! would not be rejected in a test against the alternative hypothesis H a :. Á.!
Inferences for. with 5 # unknown ]µ normal., 5 # ; ]", ]#,..., ] 8 a random sample; then ]. Xœ 8 WÎ 8 has a Student's t distribution with 1 degrees of freedom (df). Student's t distribution approaches a standard normal distribution as 8p. Table 2 (Appendix, p. 1093) lists selected percentiles of the Student's t distribution for various df values. The 1001 αth percentile of the Student's t distribution is denoted as > + (with some particular df specified) Confidence interval ( 5 # unknown) A 1001 α % CI for. is (df œ8"ñ = C> = αî#, C> αî# œ6,? 8 8
Hypothesis tests ( 5 # unknown) Case 1. H :. Ÿ. vs. H :.. (right-tailed test)!! a! Case 2. H :.. vs. H :. Ÿ. (left-tailed test)!! a! Case 3. H :. œ. vs. H :. Á. (two-tailed test)!! a! α specified; df œ81 Test statistic: >œ C.! =Î8 :-value: Case 1. Case 2. Case 3. :œpx > :œpx Ÿ> :œ 2 PX > Rejection region: Case 1. Reject H! if > > α (:Ÿα) Case 2. Reject H! if >Ÿ > α (:Ÿα) Case 3. Reject H if > > (: Ÿ α)! αî#