Ch10.1 ANOVA The analysis of variance (ANOVA), refers to a collection of experimental situations and statistical procedures for the analysis of quantitative responses from experimental units. The characteristic that differentiates the treatments or populations from one another is called the factor under study, and the different treatments or populations are referred to as the levels of the factor. Single-factor ANOVA focuses on a comparison of more than two population or treatment means. Let I = the number of treatments (populations) being compared Ch10.1

Then the hypotheses of interest are the mean of population 1 or the true average response when treatment 1 is applied the mean of population I or the true average response when treatment I is applied Then the hypotheses of interest are versus at least two of the are different Ch10.1

The observed value of Xi,j when the experiment is performed The random variable that denotes the jth measurement taken from the ith population, or the measurement taken on the jth experimental unit that receives the ith treatment The observed value of Xi,j when the experiment is performed Assumptions The I population or treatment distributions are all normal with the same variance Each Xi,j is normally distributed with Ch10.1

Mean square for treatments: Mean square for error: The Test Statistic The test statistic for single-factor ANOVA is F = MSTr/MSE. Ch10.1

F Distributions and Test When H0 is true, When H0 is false, F Distributions and Test Let F = MSTr/MSE be the statistic in a single-factor ANOVA problem involving I populations or treatments with a random sample of J observations from each one. When H0 is true (basic assumptions true) , F has an F distribution with v1= I – 1 and v2= I(J – 1). The rejection region specifies a test with Ch10.1

Total sum of squares (SST) Formulas for ANOVA Total sum of squares (SST) Treatment sum of squares (SSTr) Error sum of squares (SSE) Ch10.1

Fundamental Indentity SST = SSTr + SSE Mean Squares Ch10.1

ANOVA Table Source of Variation df Sum of squares Mean Square f Treatments I – 1 SSTr MSTr MSTr/MSE Error I(J – 1) SSE MSE Total IJ – 1 SST Ch10.1

Example 10.1 (pp.379, #8) A study of the properties of metal plate-connected trusses used for roof support yielded the following observations on axial stiffness index (kips/in) for plate length 4, 6, 8, 10, 12 in. 4: 309.2 409.5 311.0 326.5 316.8 349.8 309.7 6: 402.1 347.2 361.0 404.5 331.0 348.9 381.7 8: 392.4 366.2 351.0 357.1 409.9 367.3 382.0 10: 346.7 452.9 461.4 433.1 410.6 384.2 362.6 12: 407.4 441.8 419.9 410.7 473.4 441.2 465.8 Does variation in plate length have any effect on true average axial stiffness? State and test the relevant hypothesis using ANOVA with α=0.01. Ch10.1