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7 anova1 - 1 One-Way Analysis of Variance(ANOVA One-Way...

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1 One-Way Analysis of Variance (ANOVA) One-Way Analysis of Variance (ANOVA) is a method for comparing the means of a populations. This kind of problem arises in two different settings 1. When a independent random samples are drawn from a populations. 2. When the effects of a different treatments on a homogeneous group of experimental units is studied, the group of experimental units is subdivided into a subgroups and one treatment is applied to each subgroup. The a subgroups are then viewed as independent random samples from a populations. 3. Assumptions required for One-Way ANOVA (a) Random samples are independently selected from a (treatments) populations. (b) The a populations are approximately normally distributed. (c) All a population variances are equal. 4. The assumptions are conveniently summarized in the following statistical model: X ij = μ i + e ij where e ij are independent N (0 , σ 2 ), i = 1 , 2 , . . . , a , j = 1 , 2 , . . . , n i 5. Example: Tests were conducted to compare three top brands of golf balls for mean dis- tance traveled when struck by a driver. A robotic golfer was employed with a driver to hit a random sample of 5 golf balls of each brand in a random sequence. Distance traveled, in yards, for each hit is shown in the table below. Brand A Brand B Brand C 251.2 263.2 269.7 245.1 262.9 263.2 248.0 265.0 277.5 251.1 254.5 267.4 260.5 264.3 270.5 Suppose we want to compare the mean distance traveled by the three brands of golfballs based on the three samples. One-Way ANOVA provides a method to accomplish this. 6. The hypotheses of interest in One-Way ANOVA are: H 0 : μ 1 = μ 2 = ... = μ a H A : μ i 6 = μ j for some i, j (a) In the above example, a = 3. So the mean distance traveled by the three brands of golfballs are equal according to H 0 . (b) According to H A , at least one mean is not equal to the others. 7. The total variability in the response, X ij is partitioned into between treatment and within treatment (error) components. When these component values are squared and summed over all the observations, terms called sums of squares are produced. There is an additive relation which states that the total sum of squares equals the sum of the treatment and error sum of squares.
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2 SST = SS Tr + SSE The notations SS Tr , SSTr, SS treatment , and SS ( Between ) are synonymous for “treatment sum of squares”. The abbreviations SSE, SS error , SS Error , SS E and SS ( Within ) are synonymous for “error sum of squares”.
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