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### 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.
- Select 6 samples at each level
- 5% 10% 15% 20%

Introduction

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

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?

Blonde v. Brunette

Source: Family Weekly, Gainesville, Sun, Gainesville, Florida, February 5, 1978

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.

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.

- Sometimes referred to as a factor or explanatory variable.

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

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

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

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

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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