The Analysis of Variance

• The analysis of variance (ANOVA) is a collection of statistical procedures used to analyze quantitative responses from multiple samples.
• Single-factor ANOVA, also known as single-classification or one-way ANOVA, is used to analyze data sampled from two or more numerical populations.
  It involves a single factor with multiple levels.
• The factor is the characteristic that labels the populations, and the populations are the levels of the factor.
• Examples of single-factor ANOVA:
  • Effects of five different brands of gasoline on automobile engine operating efficiency (mpg).
  • Effects of four different sugar solutions (glucose, sucrose, fructose, and a mixture of the three) on bacterial growth.
  • Effect of hardwood concentration in pulp (%) on tensile strength of bags.
  • Effect of the amount of dye used on the color density of fabric specimens.

11.1 Single-Factor ANOVA

• Single-factor ANOVA compares two or more populations or treatments.
  • $I$ = the number of treatments being compared
  • $μ₁$ = the mean of population 1 (or the true average response when treatment 1 is applied)
  • $μ_I$ = the mean of population I (or the true average response when treatment I is applied)
• Hypotheses of interest:
  • Null Hypothesis: $H<em>o: μ</em>1 = μ<em>2 =···=μ</em>I$
  • Alternative Hypothesis: $H_a$: at least two of the $μ$’s are different
• If $I = 4$, $H_o$ is true only if all four $μ$’s are identical.
• $H_a$ would be true if:
  • $μ<em>1 = μ</em>2 ≠ μ<em>3 = μ</em>4$
  • $μ<em>1 ≠ μ</em>3 = μ<em>4 ≠ μ</em>2$
  • all four $μ$’s differ from each other.
• A test of these hypotheses requires a random sample from each population or treatment.
• Example: An experiment comparing the compression strength of $I = 4$ different types of boxes.