Section 15.2 One-Way Analysis of Variance AP Statistics www.toddfadoir.com/apstats

One-Way Analysis of Variance Which of four advertising offers mailed to sample households produces the highest sales in dollars? Which of ten brands of automobile tires wears longest? How long do cancer patients live under each of three therapies for their cancer?

One-Way Analysis of Variance Which of four advertising offers mailed to sample households produces the highest sales in dollars? Which of ten brands of automobile tires wears longest? How long do cancer patients live under each of three therapies for their cancer?

One-Way Analysis of Variance One-Way Analysis of Variance (ANOVA) is good at comparing more than two treatments against each other. When comparing two means, we would typically 2-sample z-test or 2-sample t-test. When comparing more than 2 means, use ANOVA.

Example Pickup trucks and four-wheel-drive sport utility vehicles are replacing cars in American driveways. Do trucks and SUVs have lower gas mileage than cars? We have data on the highway gas mileage (in miles per gallon) for 28 midsize cars, 8 standard size pickup trucks, and 26 SUVs.

Sure, the means are different, but are they significantly so?

The problem with doing multiple comparisions We could do 3 2-sample t-tests, but this is cumbersome. Instead, we will do one test determine whether there any differences amongst the three groups. If there is a difference, then follow up with pair-wise analysis

One-way ANOVA: Midsize, Pickup, SUV Source DF SS MS F P Factor 2 606.37 303.19 40.12 0.000 Error 59 445.90 7.56 Total 61 1052.27 S = 2.749 R-Sq = 57.63% R-Sq(adj) = 56.19% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---+---------+---------+---------+------ Midsize 28 27.107 2.629 (---*----) Pickup 8 23.125 2.588 (-------*------) SUV 26 20.423 2.914 (----*---) ---+---------+---------+---------+------ 20.0 22.5 25.0 27.5 Pooled StDev = 2.749

One-way ANOVA: Midsize, Pickup, SUV Source DF SS MS F P Factor 2 606.37 303.19 40.12 0.000 Error 59 445.90 7.56 Total 61 1052.27 S = 2.749 R-Sq = 57.63% R-Sq(adj) = 56.19% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---+---------+---------+---------+------ Midsize 28 27.107 2.629 (---*----) Pickup 8 23.125 2.588 (-------*------) SUV 26 20.423 2.914 (----*---) ---+---------+---------+---------+------ 20.0 22.5 25.0 27.5 Pooled StDev = 2.749

The idea of analysis of variance Here is the main idea for comparing means: what matters is not how far apart the sample means are but how far apart they are relative to the variability of individual observations.

Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---+---------+---------+---------+------ Midsize 28 27.107 2.629 (---*----) Pickup 8 23.125 2.588 (-------*------) SUV 26 20.423 2.914 (----*---) ---+---------+---------+---------+------ 20.0 22.5 25.0 27.5 Pooled StDev = 2.749

Same means, different variances

The analysis of variance idea Analysis of variance compares the variation due to specific sources with the variation among individuals who should be similar. In particular, ANOVA tests whether several populations have the same mean by comparing how far apart the sample means are with how much variation there is within the samples.

Degree of freedom for ANOVA

Assignment 15.11-15.15 odd 15.30