Inference for DistributionsInference for DistributionsInference for the Mean of a PopulationSection 7.1Statistical inference in practiceEmphasis turns from statistical reasoning to statistical practice:p Population standard deviation, σ, unknown. Inference on μ and comparisons of μ between populationspopulationsExample: Cola sweetnessDoes storage reduce the sweetness of cola? The loss in sweetness after storage is measured by a random gysample of n = 10 professional tasters.2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3Want to test H0: μ = 0 versus Ha: μ > 0 Use of the one-sample z test requires knowledge of σWe have the estimates=1 196 ofσbut thisWe have the estimate s 1.196 of σ, but this introduces additional random variability Can’t ignore since n is small.The t distributionsAssume a SRS from a N(μ, σ) population. The t statistichas a t distribution with n–1 degrees of freedomg The statistic SE = s/√n is the standard error of √ SE = s/√n estimates In generalt(k) denotes a t distribution withkdegreesIn general, t(k) denotes a t distribution with kdegrees of freedomComparison of t(k) with N(0, 1)A t(k) density curve resembles that of a standard NormalSimilarities: Both are centered at zero, symmetric, mound-shapedDifferences: t(k) has an additional parameter, k = deg. of freedom The sampling distribution of t statistic depends on sample size t(k) has larger spread, but close match for large k If T is t(k) then σT> 1, but σT≈ 1 if k is large Larger spread reflects additional variability due to SE = s/√nCalculating t probabilities and critical valuesSuppose T is t(k). In Excel: For c > 0, tdist(c, k, 1) = P(T ≥ c) For c > 0, tdist(c, k, 2) = 2P(T ≥ c) For 0 < α < 1, tinv(α, k)is the c for which P(T ≥ c) = α/2()One-sample t test Assumptions: SRS of size n from a Normal population Hypotheses: H0: μ = μ0versus a one- or two-sided Ha Test statistic: P-value: P(T ≥ -t) for Ha: μ< μ0P(T ≥ t) for Ha: μ > μ02P(T≥|t|)fH2P(T≥|t|)for Ha: μ≠μ0where T is t(n –1)Example: Cola sweetness (continued)Data: SRS of size n = 10 from a Normal population of professional tasters. pHypotheses: H0: μ = 0 versus Ha: μ > 0Summary statistics: and s = 1.196TtttitiTest statistic:PlP(T2 70) 0 012 i hk19dfP-value:P(T≥2.70) = 0.012, with k= n–1 = 9 d.f.Decision: Reject H0at significance level α = 0.05, and conclude a loss of sweetnessConfidence intervals in testingWhen H0is rejected, a natural follow-up question is: how large is the effect that has been detected?gExample: Cola sweetness (continued) H0is rejected with α = 0.05, indicating evidence of a loss of sweetness How much sweetness is lost?Answer with a confidence intervalOne-sample t confidence interval Assumptions: SRS of size n from a Normal population Target parameter: μ CI formula: For confidence level C, the interval iswhere t* is such that P(T ≥ t*) = (1 – C)/2, with T being t(n1)t(n–1)Example: Cola sweetness (continued)How much sweetness is lost?95% CI: P(T ≥ 2.26) = 0.025, using k= n–1 = 9 d.f.⇒ t* = 2.26, and the interval isC l d l f t b t 016 d188Conclude a loss of sweetness between 0.16 and 1.88 units, on averageRobustnessWith larger samples, one-sample t procedures become robust against violations of the Normality assumptiongypSome guidelines: If n < 15, the Normality assumption is criticalIf15 d l i b f tli dIf n ≥15, proceed only in absence of outliers and strong skewness If n ≥ 40, the procedures are generally robustExample: Cola sweetness (continued)Normality may be hard to verify when n is small. Often Normality is argued from one’s understanding of the phenomenon under studyMatched pairs experimentsThe cola sweetness study is an example of a matched-pairs experiment:pp The raw measurements came in pairs (x1, x2)x1 = sweetness before storagex2= sweetness after storage2 g But we analyzed the differences within pairsx = x1– x2Comments on matched pairsCommon matched pairs settings: Response before and after exposure to a stimulus. Pairs of very similar subjects (i.e., identical twins) applied different treatmentsWhen treatments are randomized, matched pairs is a randomized, comparative experimentrandomized, comparative experimentInference for DistributionsInference for DistributionsComparing Two MeansSection 7.2The two-sample setup Objective: compare two distinct populations through random samples drawn respectively from themppyPopulation 1Population 2Sample 1Sample 2 May represent distinct treatments of a randomized comparative experiment.comparative experiment. Samples are assumed to be drawn independently of each othereach otherNotationPopulation 1 Population 2μ1μ2σ1σ2nce⇓⇓ ⇓⇓pendenn1n2Indeps1s2Basic approach to inferenceObjective: Calculate a confidence interval for μ1– μ2or test H0: μ1= μ2Ubi df0μ1μ2Starting point: Estimate μ1– μ2withUnbiased for μ1–μ2 If both populations are Normal then The z-score of isThe two-sample t statisticTwo-sample t procedures are based on the two-sample t statisticz-score with estimated σ1and σ2If both populations are Normal then t is approximately t(k) with two possible d.f. formulas Satterthwaite’s formula: k = smaller of n1– 1 and n2–1 Easier computation Yields conservative confidence and significance levelsTwo-sample t test Assumptions: independent SRSs drawn from distinct Normal populationspp Hypotheses: H0: μ1= μ2versus a one- or two-sided Ha Test statistic: P-value: P(T ≥ -t) for Ha: μ1< μ2P(T≥t)forH:μ>μP(T≥t) for Ha: μ1> μ22P(T ≥ |t|)for Ha: μ1≠μ2whereTist(k) withkas abovewhere Tis t(k) with kas aboveExample: Directed readingDo directed reading activities improve reading ability? Measure degree of reading power (DRP) in: tt tggp()n1= 21 third-graders under directed readingn=23third-graders under conventional readingtreatmentn2= 23third-graders under conventional readingWant to test H0: μ1= μ2versus Ha: μ1> μ2controlExample: Directed reading (continued)Data: Independent SRS of sizes n1= 21 and n2= 23 from Normal populations students’ DRP measurements ppHypotheses: H0: μ1= μ2versus Ha: μ1> μ2Summary statistics:Test statistic:Example: Directed reading (continued)Test statistic: t = 2.31 P-value*: P(T ≥ 2.31) = 0.016, with k = smaller of n1– 1 and n2– 1 = 21 – 1 = 20 d.f.Decision: Reject H0at significance