1. AP Statistics
FINAL EXAM
ANALYSIS OF VARIANCE

2. Objective: To be able to carry out a One-Way ANOVA test.
The Analysis of Variance F-test is used when comparing more than 2 population means.
If we are only comparing 2 means then we use the two-sample t-test.
Suppose:
𝐻 0 : 𝜇 1 = 𝜇 2 = 𝜇 3
𝐻 𝑎 :𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝜇 𝑖 𝑖𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡
We could use the t-procedures 3 different times, but that yields multiple p-values which could lead to a type I error.

3. In addition, using the t-procedures multiple times does not take into account the variability across all means at the same time.
We can’t make multiple comparisons among means by looking at two confidence intervals at a time.
Analysis of Variance (ANOVA)
Test used to compare multiple means.
I. Assumptions:
We have I independent SRS’s, one from each of I populations.
Each population is normally distributed
All populations have the same 𝜎.

4. Unlike the t-test, the F-test is extremely sensitive to non-normal distributions.
Even as sample size increases, the F-test is still effected by a lack of normality.
Skewed distributions tend to have larger standard deviations than normal.
F-Distributions:
Right skewed.
F is always greater than 0.
Area under the curve is 1.
Peak near where F = 1.
Always work in the upper tail and double the upper tail area for two-sided tests.

5. Identified by the degrees of freedom of the numerator and degrees of freedom of the denominator.
Can’t freely change the 𝑑𝑓 𝑁 and 𝑑𝑓 𝐷 . Meaning 𝐹 2,4 ≠ 𝐹 4,2
F-Test
Test used to determine whether or not 2 distributions have the same spread.
Assumptions: 𝑠 1 2 and 𝑠 2 2 are sample variances from 2 independent SRS’s of sizes 𝑛 1 and 𝑛 2 drawn from normal populations.
Hypotheses:
𝐻 𝑜 : 𝜎 1 = 𝜎 2
𝐻 𝑎 : 𝜎 1 ≠ 𝜎 2

6. Rejection Region: I will reject Ho if my p-value < 𝛼.Or
If F > 𝐹 (𝛼, 𝑛 1 −1, 𝑛 2 −1)
Test Statistic and p-value:
F= 𝑙𝑎𝑟𝑔𝑒𝑟 𝑠 2 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑠 ∙𝑃(𝐹> 𝐹 𝑡𝑠 )
Conclusion: Give the results with 2 sentences.

7. Many times will be testing to see if more than 2 standard deviations are the same. To do that we use Hartley’s F-Max test.
Check F = 𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑠 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑠 <2
Basic Idea:
This test is used to compare how far apart the sample means are with how much variation there is within the samples.
For example if the sample means are relatively close to one another but the is very little variation among the samples, we may have significant differences.
On the other hand if the sample means are relatively far apart and there is a lot of variability among the samples the results might not be significant.

8. II. Hypotheses: 𝐻 𝑜 : 𝜇 1 = 𝜇 2 = 𝜇 3 =…= 𝜇 𝑖 𝐻 𝑎 :𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝜇 𝑖 𝑖𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 Define the parameter! III. Rejection Region: Reject Ho if the p-value < 𝛼. Reject Ho if 𝐹> 𝐹 (𝛼, 𝑑𝑓 𝑁 , 𝑑𝑓 𝐷) IV. Test Statistic and p-value: F = 𝑀𝑆𝐺 𝑀𝑆𝐸 MSG = Mean squares for the groups MSE = Mean squares for the errors

9. 𝑷( 𝑭 𝒅𝒇 𝑵 , 𝒅𝒇 𝑫 >𝑭) ANOVA TABLE: Where I = total number of population N = total number of observations “Group” measures variability between group means. SSG = 𝒏 𝒊 ( 𝒙 𝒊 − 𝒙 ) 𝟐 where 𝒙 𝒊 is the mean of the 𝒊 𝒕𝒉 group and 𝒙 is the mean of all observations.
Source df
Sum of squares
Mean squares F
P-value
Group
I – 1
SSG
MSG=SSG / I - 1
MSG / MSE
Error
N – I
SSE
MSE = SSE / N- I
Total
N – 1
SST

10. “Error” measures the variation within the groups
“Error” measures the variation within the groups. SSE = 𝒏 𝒊 ( 𝒙 𝒊𝒋 − 𝒙 𝒊 ) 𝟐 or ( 𝒏 𝒊 −𝟏)∙ 𝒔 𝒊 𝟐 Where 𝒙 𝒊𝒋 is the 𝒋 𝒕𝒉 observation in the 𝒊 𝒕𝒉 group. “Total” measures the total variability across all observations SST = SSG + SSE SST = ( 𝒙 𝒊𝒋 − 𝒙 ) 𝟐 V. Conclusion: Two sentences to give a general conclusion. Using software, a more detailed comparison can be done using Tukey’s pairwise comparisons.

11. Example 1: Black Flag insect company is working on creating the most effective fly paper that is can. To explore the idea that flies are attracted to certain colors, they have run an experiment where 4 different colors of flypaper will be used over a 6 hour period. Six different samples from each color will be placed in randomly selected areas of a room. After the 6 hours have passed, the number of flies stuck to the flypaper will be recorded. Below are the results: Is there a difference in the average number of flies that are trapped on the different colors of fly paper?
Color
Insects
Blue 16 11 20 21 14 7
Green 37 32 29
White 12 17 13
Yellow 45 59 48 46 38 47

12. ANOVA TEST:

13. ANOVA TEST - MINITAB

14. Example 2: Workers for a local company were asked to rate various elements of safety in the workplace. A composite score called the Safety Climate Index was calculated. Its values are between 0 and 100. the workers were classified according to their job category as unskilled, skilled and supervisor. Is there evidence of a difference in the mean responses to the SCI among the three groups of employees?
Job n
Mean SD
Unskilled
448
70.42
18.27
Skilled 91
71.21
18.83
Supervisors 51
80.51
14.58

15. ANOVA TEST

16. ANOVA TEST - MINITAB