Notes on one-factor experiments
Notes on one-factor experiments ENGR 0020: Probability and statistics for Engineers I
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This 2 page Class Notes was uploaded by Emily Binakonsky on Friday April 10, 2015. The Class Notes belongs to ENGR 0020: Probability and statistics for Engineers I at University of Pittsburgh taught by Maryam Mofrad in Spring2015. Since its upload, it has received 177 views. For similar materials see Probability and Statistics for Engineers 1 in Engineering and Tech at University of Pittsburgh.
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Date Created: 04/10/15
One Factor Experiments Emily Binakonsky I One factor Experiment a Analysis of Variance Technique i Introduction 1 ANOVA Broad classification of experimental designs and statistical analyses ii Terminology 1 Factor A variable that is changed a Characteristic that differentiates the treatments or populations from one another 2 Level A value assigned to change the factor a The different treatments or populations iii Concrete Analysis Suppose in an industrial experiment that an engineer is interested in how the mean absorption of moisture in concrete varies among 5 different concrete aggregates The samples are exposed to moisture for 48 hours It is decided that 6 samples are to be tested for each aggregate requiring a total of 30 samples to be tested 1 Hypothesis to test 3 H0411 2 5 b HaAt least two of the means are not equal There are six observations per population iv Variation amongst the means can be attributed to 1 Variation among the observations within the same level 2 Variation among different levels b The Strategy of Experimental Design 1 Experimental units are the units that provide the heterogeneity that leads to experimental error a Random assignment to treatment groups eliminates bias b Blocking Division of Experimental units into homogeneous pairs c OneWay Analysis of Variance Completely Randomized Design i Assumptions 1 The k populations are independent and normally distributed with means M1412 akand the same variance 02 2 The assumptions are reasonable because of randomizations ii Hypotheses 1 Howl 2 5 2 HaAt least two of the means are not equal One Factor Experiments Emily Binakonsky Notation and Data Table yij jthobservation from the ith treatment Y1 the total of all observations in the sample from the ith treatment 371 mean of all observations in the sample from the ith treatment Y total of all nk observaitons 37quot mean of all nk observations Null HVpothesis Treatment means are equal Test Based on a comparison of two independent estimates of the common population variance SumofSguares Identity 2 2 1 2521 232101 y 7121le y2 2521 232101 31 SST SSA SSE Total Sum Treatment Sum Error Sum of Squares of Squares of Squares How to Compare the means a The variability due to differences between treatments and error within treatments is found by comparing the magnitude of SSA and SSE i To facilitate this the mean squares of the treatments and error must be computed by normalizing the sum of squares by their respective degrees of freedom SSA Treatment Mean of Squares sf 2 a SSE iii Error Mean of S uares s2 q 2 kn 1 b Fratio for Testing Equality of Means i The F test is used to compare the ratio of the treatment variance with the error variance ii If Fsmtisuc gt Famlmz where 171 k 1 and v2 kn 1 then the factor level means are different for a level a test
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