By Myer Kutz
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Extra resources for Mechanical Engineer's Handbook [Vol 2 - Instrum., Sys, Ctls, MEMS]
5. Define H0 and H1. Choose the confidence level ␥ of the test. Form an appropriate probabilistic statement. Using the appropriate statistical distribution, perform the indicated calculation. Make a decision concerning the hypothesis and / or determine confidence limits. Two types of error are possible in statistical testing. A Type I error is that of rejecting a true null hypothesis (rejecting truth). A Type II error is that of accepting a false null hypothesis (embracing fiction). The confidence levels and sample size are chosen to minimize the probability of making a Type I or Type II error.
From the propagation of variance, d2 becomes 2d ϭ 2x1 ϩ 2x2 ϭ 21 22 ϩ n1 n2 (55) 28 Instrument Statics where 12 and 22 are each estimates of the population variance 2. A better estimate of 2 is the combined variance 2c , and it replaces both 21 and 22 in Eq. (55). The combined variance is determined by weighting the individual estimates of variance based on their degrees of freedom according to the relation c2 ϭ ˆ 21v1 ϩ ˆ 22v2 v1 ϩ v2 (56) Then d2 ϭ ͩ ˆ 2c ˆ 2 1 1 ϩ c ϭ c2 ϩ n1 n2 n 1 n2 ͪ (57) Under the hypothesis H0 that 1 ϭ 2 (no effect due to treatment), the resulting probabilistic statement is ͫ Ί P Ϫtc Ί ͬ 1 1 1 1 ϩ Ͻ x1 Ϫ x2 Ͻ tc ϩ ϭ␥ n1 n2 n1 n 2 (58) If the variances of the items being compared are not equal (homogeneous), a modified t- (or d-) statistic is used,9,15 where d depends on confidence level ␥, degrees of freedom v, and a parameter that depends on the ratio of standard deviations according to tan ϭ ˆ 1 / ͙n1 (59) ˆ 2 / ͙n2 The procedure for using the d-statistic is the same as described for the t-statistic.
2. 3. 4. 5. 6. 7. 8. Introduction 33 Dimming of the headlights while starting a car Slowdown of an electric mixer lowered into heavy batter Freezing a showerer by starting the dishwasher Speedup of a vacuum cleaner when the hose plugs Two-minute wait for a fever thermometer to rise Special connectors required for TV antennas Speedup of a fan in the window with the wind against it Shifting of an automatic transmission on a hill These effects happen because one part of a system loads another. Most mechanical engineers would guess that weighing an automobile by placing a bathroom-type scale under its wheels one at a time and summing the four measurements will yield a higher result than would be obtained if the scale was flush with the floor.
Mechanical Engineer's Handbook [Vol 2 - Instrum., Sys, Ctls, MEMS] by Myer Kutz