6 - TEXAS A&M UNIVERSITY Statistical Data Analysis,...

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TEXAS A&M UNIVERSITY DEPARTMENT OF MECHANICAL ENGINEERING TEXAS A&M UNIVERSITY COLLEGE STATION, TX 77843-3123 979 845 1251 FAX 979 845 3081 1 of 14 R. Langari, 9/20/10 Statistical Data Analysis, Part III Review of Confidence Intervals Assume we have measurements with a sample mean of and sample standard decision, . If is sufficiently high, i.e. 100 or more: with 95% confidence (1) For smaller we need to use the formula where is the confidence level (2) Note that comes from the so called T-Table given below: Table 1: T-Table ( ) (D.O.F.) 1 3.078 6.134 12.706 63.657 2 1.886 2.920 4.303 9.925 3 1.638 2.353 3.182 5.841 4 1.533 2.132 2.776 4.604 5 1.476 2.015 2.571 4.032 6 1.440 1.943 2.447 3.707 N x S x N x 1.96 S x N ------- μ x x 1.96 S x N + N xt α 2 ⁄ν , S x N μ x α 2 , S x N + c 1 α = t α 2 , t αν , ν α 0.1 = α 0.05 = α 0.025 = α 0.005 = DEPARTMENT OF MECHANICAL ENGINEERING TEXAS A&M UNIVERSITY COLLEGE STATION, TX 77843-3123 979 845 1251 FAX 409 845 3081 2 of 14 Statistical Data Analysis, Part III R. Langari, 9/20/10 7 1.415 1.895 2.365 3.5 8 1.397 1.860 2.306 3.355 9 1.383 1.833 2.262 3.25 10 1.372 1.812 2.228 3.169 11 1.363 1.796 2.201 3.106 12 1.356 1.782 2.179 3.055 13 1.305 1.771 2.160 3.012 14 1.345 1.761 2.145 2.977 15 1.341 1.753 2.131 2.947 16 1.337 1.746 2.120 2.921 17 1.333 1.704 2.110 2.898 18 1.330 1.734 2.101 2.878 19 1.328 1.729 2.093 2.861 20 1.325 1.725 2.086 2.845 21 1.323 1.721 2.080 2.831 22 1.321 1.717 2.074 2.819 23 1.319 1.714 2.069 2.807 24 1.318 1.711 2.064 2.797 25 1.316 1.708 2.060 2.787 26 1.315 1.706 2.056 2.779 27 1.314 1.703 2.052 2.771 28 1.313 1.701 2.048 2.763 29 1.311 1.699 2.045 2.756 30 1.301 1.697 2.042 2.75 infinity 1.282 1.645 1.96 2.576 ) t ,
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TEXAS A&M UNIVERSITY DEPARTMENT OF MECHANICAL ENGINEERING TEXAS A&M UNIVERSITY COLLEGE STATION, TX 77843-3123 979 845 1251 FAX 409 845 3081 3 of 14 Statistical Data Analysis, Part III R. Langari, 9/20/10 Statistical Comparison of two Sets of Samples (T-Test) Consider two sets of measurements (from two batches of samples.) Each set produces and , , as its respective mean and standard deviation. Let us define as the null hypothesis: the two samples are effectively the same, and as the alternative hypothesis: the two samples are essentially different. What is the probability of error in our rejecting in favor of ? Let us define and (3) If then the chance of us erroneously rejecting is less than . In most cases we have where (4) i.e, that chances of us erroneously rejecting is less than but greater than . x i S x i i 12 , = H 0 H 1 H o H 1 t x 1 x 2 x 1 x 2 S x 1 2 N 1 S x 2 2 N 2 + --------------------------------------------- = ν int S x 1 2 N 1 S x 2 2 N 2 + () 2 S x 1 2 N 1 2 N 1 1 ------------------------ S x 2 2 N 2 2 N 2 1 + -------------------------------------------------------- = t x 1 x 2 t α 2 v , > H 0 α t α 1 2 ⁄ν , t x 1 x 2 t α 2 2 , << α 2 α 1 < H 0 α 1 α 2 DEPARTMENT OF MECHANICAL ENGINEERING TEXAS A&M UNIVERSITY COLLEGE STATION, TX 77843-3123 979 845 1251 FAX 409 845 3081 4 of 14
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This note was uploaded on 12/26/2010 for the course MEEN 260 taught by Professor Langari during the Fall '08 term at Texas A&M.

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6 - TEXAS A&amp;M UNIVERSITY Statistical Data Analysis,...

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