A3. Statistical Inference Hypothesis Testing for General Population Parameters

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1 Appendix / A3. Statistical Inference / General Parameters- A3. Statistical Inference Hypothesis Testing for General Population Parameters POPULATION H 0 : θ = θ 0 θ is a generic parameter of interest (e.g., μ, π, σ in the one sample case; μ μ, π π, σ / σ in the two sample case) of a random variable X. SAMPLE θ 0 is a conjectured value of the parameter θ in the null hypothesis. In the two sample case for means and proportions, this value is often chosen to be zero if, as in a clinical trial, we are attempting to detect any statistically difference between the two groups (at some predetermined significance level α). For the ratio of variances between two groups, this value is usually one, to test for equivariance. Once a suitable random sample (or two or more, depending on the application) has been selected, the observed data can be used to compute a point estimate ˆ θ that approximates the parameter θ above. For example, for single sample estimates, we take ˆμ = x, ˆ π = p, ˆ σ = s ; for two samples, take ˆμ ˆμ = x x, ˆ π ˆ π = p p, ˆ σ / ˆ σ = s / s. This sample-based statistic is then used to test the null hypothesis in a procedure known as statistical inference. The fundamental question: At some pre-determined significance level α, does the sample estimator ˆ θ provide sufficient experimental evidence to reject the null hypothesis that the parameter value is equal to θ 0, i.e., is there a statistically difference between the two? If not, then this can be interpreted as having evidence in support of the null hypothesis, and we can tentatively accept it, bearing further empirical evidence; see THE BIG PICTURE. In order to arrive at the correct decision rule for the mean(s) and proportion(s) [subtleties exist in the case of the variance(s)], we need to calculate the following object(s): margin of error Confidence Interval endpoints = ˆ θ ± critical value standard error (If θ 0 is inside, then accept null hypothesis. If θ 0 is outside, then reject null hypothesis.) Acceptance Region endpoints = θ 0 ± critical value standard error (If ˆ θ is inside, then accept null hypothesis. If ˆ θ is outside, then reject null hypothesis.) ˆ θ θ0 Test Statistic =, which is used to calculate the p-value of the experiment. standard error (If p-value > α, then accept null hypothesis. If p-value < α, then reject null hypothesis.) The appropriate critical values and standard errors can be computed from the following tables, assuming that the variable X is normally distributed. (Details can be found in previous notes.)

2 One Sample POPULATION PARAMETER H 0 : θ = θ 0 SAMPLE STATISTIC Point Estimate ˆ θ = f(x,, x n ) CRITICAL VALUE (-sided) Appendix / A3. Statistical Inference / General Parameters- MARGIN OF ERROR = product of these two factors: STANDARD ERROR (estimate) Mean H 0 : μ = μ 0 ˆμ = x = x i n Proportion H 0 : π = π 0 ˆ π = p = X n, where X = # Successes n 30: t n, α / or z α / n < 30: t n, α / only n 30: z α / ~ N(0, ) n < 30: Use X ~ Bin(n,π). (not explicitly covered) Any n: s / n n 30: For Confidence Interval: ˆ π ( ˆ π) n For Acceptance Region, p-value: π ( π ) n 0 0 Two Independent Samples Two Paired Samples 3 H 0 : θ θ = 0 Point Estimate ˆ θ ˆ θ Means H 0 : μ μ = 0 x x CRITICAL VALUE (-sided) STANDARD ERROR (estimate) n, n 30: tn+ n, α / or z α / n, n 30: s / n + s / n n, n < 30: Is σ = σ? n, n < 30: Informal: /4 < s /s < 4? s / n + / n Yes t n+ n, α / No Satterwaithe s Test n, n 30: z α / Proportions H 0 : π π = 0 ˆ π ˆ π n, n < 30: (or use Chi-squared Test) Fisher s Exact Test (not explicitly covered) where s = (n ) s + (n ) s n + n n, n 30: For Confidence Interval: ˆ π( ˆ π ˆ ˆ ) n + π( π) n For Acceptance Region, p-value: ˆ π ( ˆ π ) n + n where ˆ π = (X + X ) / (n + n ) k samples (k ) H 0 : θ = θ = = θ k Independent Dependent (not covered) Means H 0 : μ = μ = = μ k F-test (ANOVA) Repeated Measures, Blocks Proportions H 0 : π = π = = π k Chi-squared Test Other techniques For -sided hypothesis tests, replace α / by α. For Mean(s): If normality is established, use the true standard error if known either σ / n or σ / n σ / + n with the Z-distribution. If normality is not established, then use a transformation, or a nonparametric Wilcoxon Test on the median(s). 3 For Paired Means: Apply the appropriate one sample test to the pairwise differences D = X Y. For Paired Proportions: Apply McNemar s Test, a matched version of the Chi-squared Test.

3 Appendix / A3. Statistical Inference / General Parameters-3 HOW TO USE THESE TABLES The preceding page consists of three tables that are divided into general statistical inference formulas for hypothesis tests of means and proportions, for one sample, two samples, and k samples, respectively. The first two tables for -sided Z- and t- tests can be used to calculate the margin of error = critical value standard error for acceptance/rejection regions and confidence intervals. Column indicates the general form of the null hypothesis H 0 for the relevant parameter value, Column shows the form of the sample-based parameter estimate (a.k.a., statistic), Column 3 shows the appropriate distribution and corresponding critical value, and shows the corresponding standard error estimate (if the exact standard error is unknown). Pay close attention to the footnotes in the tables, and refer back to previous notes for details and examples! Two-sided alternative H 0 : θ = θ 0 vs. H A : θ θ 0 Confidence Limits: Column ± (Column 3)() Is Column outside? Acceptance Region: Column ± (Column 3)() Is Column outside? Column Column P(Z > Z-score ), or equivalently, P(Z < Z-score ), for large samples P(T df > T-score ), or equivalently, P(T df < T-score ), for small samples Example: α =.05 0 p.00 extremely p.005 strongly p.0 moderately p.05 borderline p.0 not Reject H 0 Accept H 0

4 Appendix / A3. Statistical Inference / General Parameters-4 One-sided test*, Right-tailed alternative H 0 : θ θ 0 vs. H A : θ > θ 0 Confidence Interval: Column (Column 3)() Is Column outside? Acceptance Region: Column + (Column 3)() Is Column outside? Column Column P(Z > Z-score), for large samples P(T df > T-score), for small samples One-sided test*, Left-tailed alternative H 0 : θ θ 0 vs. H A : θ < θ 0 Confidence Interval: Column + (Column 3)() Is Column outside? Acceptance Region: Column (Column 3)() Is Column outside? Column Column P(Z < Z-score), for large samples P(T df < T-score), for small samples * The formulas in the tables are written for -sided tests only, and must be modified for -sided tests, by changing α / to α. Also, recall that the p-value is always determined by the direction of the corresponding alternative hypothesis (either < or > in a -sided test, both in a -sided test).

5 Appendix / A3. Statistical Inference / General Parameters-5 THE BIG PICTURE STATISTICS AND THE SCIENTIFIC METHOD If, over time, a particular null hypothesis is continually accepted (as in a statistical meta-analysis of numerous studies, for example), then it may eventually become formally recognized as an established scientific fact. When sufficiently many such interrelated facts are collected and the connections between them understood in a coherently structured way, the resulting organized body of truths is often referred to as a scientific theory such as the Theory of Relativity, the Theory of Plate Tectonics, or the Theory of Natural Selection. It is the ultimate goal of a scientific theory to provide an objective description of some aspect, or natural law, of the physical universe, such as the Law of Gravitation, Laws of Thermodynamics, Mendel s Laws of Genetic Inheritance, etc.

TA: Sheng Zhgang (Th 1:20) / 342 (W 1:20) / 343 (W 2:25) / 344 (W 12:05) Haoyang Fan (W 1:20) / 346 (Th 12:05) FINAL EXAM

TA: Sheng Zhgang (Th 1:20) / 342 (W 1:20) / 343 (W 2:25) / 344 (W 12:05) Haoyang Fan (W 1:20) / 346 (Th 12:05) FINAL EXAM STAT 301, Fall 2011 Name Lec 4: Ismor Fischer Discussion Section: Please circle one! TA: Sheng Zhgang... 341 (Th 1:20) / 342 (W 1:20) / 343 (W 2:25) / 344 (W 12:05) Haoyang Fan... 345 (W 1:20) / 346 (Th