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ANOVA

ANOVA. Analysis of Variance. Introduction. We’re making paper bags and want to see what we can do to improve the tensile strength of the bags. Conduct an experiment to determine if the level of hardwood in the bags affects tensile strength. Investigate 4 levels of hardwood.

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ANOVA

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  1. ANOVA Analysis of Variance

  2. Introduction • We’re making paper bags and want to see what we can do to improve the tensile strength of the bags. • Conduct an experiment to determine if the level of hardwood in the bags affects tensile strength. • Investigate 4 levels of hardwood. • Select 6 samples at each level • 5% 10% 15% 20%

  3. Introduction Source: Dr. G. Baker, Statistics Dept., University of South Carolina

  4. Introduction • Study of pain threshold / color of a person’s hair • Small sample size • Means differ • Dark brunette: 37.4 • Light brunette: 42.5 • Dark blonde: 51.2 • Light blonde: 59.2 • Are these different or is this caused by random variation?

  5. Blonde v. Brunette Source: Family Weekly, Gainesville, Sun, Gainesville, Florida, February 5, 1978

  6. Definitions • Experiment • A study designed to investigate the effect one variable has on the value of another variable • Dependent variable • The variable of interest that we are measuring. • Sometimes called the response variable. • Dependent variable is quantitative. • Independent variable • The observed (or controlled) variable we use to detect its effect on the independent variable. • Independent variable can be quantitative or qualitative.

  7. Definitions • Independent variable • Sometimes referred to as a factor or explanatory variable. • One or more factors may be involved in a study. • An experiment may involve different factor levels. • Each specific level of a factor (or the intersection of multiple factors) is referred to as a treatment. • When there’s only one factor level in an experiment, the terms factor levels and treatments are used interchangeably.

  8. Definitions

  9. Experiments • Observational study • No control over the factors • Completely randomized design • Select random independent samples for each treatment OR • Randomly assign treatments to selected people • Complete randomization is not possible in observational studies

  10. Hypothesis Testing • H0: µ1 = µ2 = ... = µt * • µt * = the total number of treatments in the experiment • Ha: At least one of the treatment group means differs from the rest. OR At least two of the population means are not equal. • We will be comparing the variation between groups and the variation within groups • Between>within indicates a difference in the means

  11. ANOVA Terminology • Labels for individual responses • = the individual response for the jth observation receiving the ith treatment • = the average of all the observations receiving a treatment • = the average for all the observations in the experiment

  12. Definitions Variation within groups… Variation between groups… d a b c

  13. Variation: Within v. Between a b c

  14. Variation: Within v. Between a c b

  15. Paper Bag Example

  16. Paper Bag Hypothesis • H0: µ.05= µ.10= µ.15 = µ.20 • Ha: At least one mean differs from the other 3 OR At least 2 of the means are not equal

  17. Data

  18. ANOVA Table

  19. ANOVA Table

  20. ANOVA Table from Excel

  21. ANOVA Table from Minitab • One-way ANOVA: C16 versus C15 • Source DF SS MS F P • C15 3 382.79 127.60 19.61 0.000 • Error 20 130.17 6.51 • Total 23 512.96 • S = 2.551 R-Sq = 74.62% R-Sq(adj) = 70.82% • Individual 95% CIs For Mean Based on Pooled StDev • Level N Mean StDev +---------+---------+---------+--------- • 5.00% 6 10.000 2.828 (----*----) • 10.00% 6 15.667 2.805 (----*-----) • 15.00% 6 17.000 1.789 (----*-----) • 20.00% 6 21.167 2.639 (-----*----) • +---------+---------+---------+--------- • 8.0 12.0 16.0 20.0 • Pooled StDev = 2.551

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