12+Analysis+of+variance

# In this case the experiment has two factors brand of

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Unformatted text preview: bot II type. In this case the experiment has two factors: brand of golf ball and type of robot. Below the diagram provides an overview of the experimental process and a summary of the introduced terminology. Population of experimental units Sampling experiment ¨¨rr rr ¨¨ Sample rr ¨ ¨¨ ¨ of experimental rr ¨ rr ¨¨ r ¨ rr units ¨¨ r rr¨¨ ¨r rr ¨¨ ¨ r ¨ rr ¨¨ ¨ Apply Apply factor-level combination 1 ¨ factor-level combination p ¨ ¨ % Treatment 1 sample • • c • r rr j r Treatment p sample c Responses for Treatment 1 Responses for Treatment p 2 A. Zhensykbaev ANOVA Model of one-factor analysis. Let us given k treatments T1 , ..., Tk and corresponding samples are: k X1 = {x11 , . . . , x1n1 }, . . . , Xk = {xk1 , . . . , xknk }, n= ni , i=1 xij is j -th measurement in i-th sample, ni is the size of i-th sample, xi - its ¯ sample mean, µi - mean of treatment Ti , x - overall mean response of all ¯ sample measurements k 1 x= ¯ xi . ¯ k i=1 We present these measurements as the table Treatment 1 Treatment 2 . . . Treatment k x11 x21 ... xk 1 x12 x22 ... xk 2 . . . . . . . . . . . . x1n1 x 2n 2 ... xknk Means x1 ¯ x2 ¯ ... xk ¯ In general the model of one-factor analysis is yij = µi + εij (i = 1 : k, j = 1 : ni ), where errors εij are independent. Assumptions: 1. The probability distribution of the populations of responses associated with each treatment are normal with mean =0. 2. Population variances are supposed to be equal. 3. The samples experiment units selected for each treatment are random and independent. The null hypothesis is H0 = {µ1 = · · · = µk }. 3 A. Zhensykbaev ANOVA Alternative hypothesis is Ha = {at least two means diﬀer}. Test statistics is M ST , M SE where M ST - mean square for treatments F= SST M ST = , k−1 k ni (¯i − x)2 , x ¯ SST = i=1 M SE - mean square for error SSE , M SE = n−k k ni (xij − xi )2 , ¯ SSE = i=1 j =1 SST - sum of squares for treatments, SSE - sum of squares for error. RR = {F > Fα }, where Fα is deﬁned from the Tables V...
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## This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.

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