1. Lesson 2

2. ANOVA Analysis of Variance

3. One Way ANOVA One variable is measured for many different treatments (population)_ Null Hypothesis: all population means are equal Alternative Hypothesis: not all population means are equal (i.e. at least one is different) If variance is small (the sample means are close) and the null hypothesis is true If variance is large (the sample means are far apart), the alternative hypothesis is true

4. Example 1: Does the weight class of a car make a difference in the number of head injuries sustained by crash test dummies? A random sample of 5 compact, midsize, and full-size cars was obtained and the head injury count for these vehicles was recorded below Calculate the mean and variance for each sample. Calculate the overall mean.

5. CARHEAD INJURY COUNT Compact Chevy Cavalier643 Dodge Neon655 Mazda 626442 Pontiac Sunfire514 Subaru Legacy525

6. Midsize Chevy Camaro469 Dodge Intrepid727 Ford Mustang525 Honda Accord454 Volvo S70259

7. Full-Size Audi A8384 Cadillac Deville656 Ford Crown Vic602 Olds Aurora687 Pontiac Bonneville360

8. Mean 1 – head injuries for compact cars Mean 2 – head injuries for midsize cars Mean 3 – head injuries for full size cars Null hypothesis: means are all equal Alternative hypothesis: at least one mean is different

9. MSTR Mean square due to treatment Estimate of the variance BETWEEN the treatments (populations) –A good estimate of the variance ONLY when the null hypothesis is TRUE –If the null hypothesis is FALSE, MSTR overestimates the variance

10. MSTR Formula Find the difference between each sample mean and the overall mean; square this number Multiply result of 1 st step by n Sum these numbers Divide by the degrees of freedom Steps 1 through 3 are SSTR (sum of the squares due to treatment) and the numerator Step 4 is k-1 and is the denominator

11. Compact: Midsize Fullsize

12. MSE Mean Square Due to Error Within treatment estimate of the variance –Average of the individual population variances –Unaffected by whether the null hypothesis is true or not –Provides an UNBIASED estimate of the population variance

13. MSE = Average of the sample variances

14. F-test F is the test statistic for the equality of k population means F = MSTR / MSE has k-1 df in numerator and n- k df in denominator = 6405 / 20096.023 = 0.319 n.b. If null hypothesis is T, the value of MSTR / MSE should appear to have been selected from this F distribution If null hypothesis is F, MSTR / MSE will be inflated because MSTR overestimates variance

15. Calculate p-value: P(F > F observed) and write conclusion Summarize in an ANOVA table

16. SOURCEDFSSMSFP Factor Error Total SST = SSTR + SSE