Chapter 11

Question 1

What is ANOVA

Answer 1

Analysis of Variance

Def: Allows statistical comparison among samples taken from many populations

The comparison is typically the result of an experiment.

Question 2

What is the basis for an ANOVA experiment

Answer 2

What is a “factor”

Question 3

Define the levels of a factor

Answer 3

The levels of a factor are the groups that comprise the experiment and analysis. They provide the basis for comparison.

Question 4

What are the three types of experiments that can be conducted with ANOVA?

And which one will we use?

Answer 4

1. Completely randomized design: an experiment with only one factor
2. Factoral design: An experiment in which more than one factor is considered (Two-way ANOVA)
3. Randomized block design: An experiment in which the members of each group have been placed in blocks either by being matched or subjected to repeated measurements as was done the two populations of a paired t test

We will only use Completely randomized design

Question 5

What is the Two Part Process of ANOVA

Answer 5

1. Determine if there is a significant difference among the group means (if you reject H₀, continue with #2)
2. Identify the groups whose means are significantly different from the other group means

Question 6

What is the purpose of ANOVA

Answer 6

To reach conclusions about possible differences month the means of each group

Question 7

Define the Completely Randomized Design

Answer 7

The ANOVA method that analyzes a single factor.

This design is executed using the statistical method one-way ANOVA

Question 8

Define SST

Answer 8

The sum of squares total represents total variation. It is partitioned into 2 groups

1. Within-group variation (SSW) measures random variation
2. Among-group variation (SSA) measures differences from group to group

Question 9

Define n and c

Answer 9

• ‘n’* represents the number of values in all groups
• ‘c’* represents the number of groups

Question 10

Review “Partitioning the Total Variation”

Answer 10

SST = SSA + SSW

Question 11

What are the assumptions of ANOVA

Answer 11

The c groups represent populations who(se)

1. Values are randomly and independently selected
2. Follow a normal distribution
3. Have equal variances

Question 12

What is the H₀ for ANOVA

Answer 12

H₀ = μ₁ = μ₂ = . . . = μ_c

Question 13

What is H₁ for ANOVA

Answer 13

H₁: not all μj are equal

where j = 1, 2, . . . , c

Question 14

Total variation is represented by what

Answer 14

The sum of squares total (SST)

Question 15

What is the grand mean

Answer 15

The mean of all the values in all the groups combined

X_double-bar

Question 16

Define SSW

Answer 16

“Within-group variation”

Measures random variation

Question 17

Define SSA

Answer 17

“Among-group” variation

Measures differences from group to group

Question 18

Review the equation for Total Variation in One-Way ANOVA

Answer 18

Question 19

Review equation for Among-group variation in One-way ANOVA

Answer 19

Question 20

Review Within-group variation in One-way ANOVA

Answer 20

Question 21

Review the equation for the Mean Squares in One-way ANOVA

Answer 21

Note: the mean square is just another term for variances that is used in ANOVA

Question 22

Define the F Test for Differences Among More than Two Means

Answer 22

Tests to see if there is a significant difference b/t the means.

Note: It does not tell you where the differences are

Question 23

Review the One-way ANOVA F_STAT Test Statistic

Answer 23

Question 24

Review the Area of Rejection and Non-rejection

Answer 24

Note: start drawing these for HW and tests

Question 25

Review the ANOVA Summary Table

Answer 25

Summarized the results of a one-way ANOVA

Question 26

What are the One-way ANOVA F Test Assumptions

Answer 26

1. Randomness and independence of the sample selected 2. Normality of the *c* groups from which samples are selected Note: use a Normal Probability Plot to determine normality. (see image) 3. Homogeneity of variance (the variances of the *c* groups are equal) Note: this will require the Levene test

Question 27

What is the Levene Test

Answer 27

A Test for Homogeneity of Variance 1. Compute the absolute value of the difference between each value and the median of the group 2. Perform a one-way ANOVA on these *absolute differences* Note: most statisticians suggest using a level of significance of α = .05 when performing the ANOVA

Question 28

Define the Turkey-Kramer Procedure

Answer 28

Procedure to construct multiple comparisons to test H₀ that the differences in the means of all pairs of “items” are equal to 0 Determines with of the *c* means are significantly different

Question 29

What are the four steps of the Turkey-Kramer Procedure

Answer 29

1. Compute the absolute mean differences (|x̄_j - x̄_j'|; *where j, j' refers to group j & j', and j ≠ j'*) among all pairs of sample means [*c(c - 1)* / 2 pairs] 2. Compute the critical range (Using equation 11.6, attached) 3. Compare each of the *c\**(*c* - 1) / 2 pairs of means against its corresponding critical range. (Declare a specific pair significantly different if the absolute difference in the sample means,|x̄_j - x̄_j'| , is greater than the critical range) 4. Interpret the results

Question 30

How many degrees of freedom are there in determining * the among-group variation * the within-group variation * the total variation

Answer 30

Among-group variation: *c* - 1 Within-group variation: *n* - *c* Total variation: *n* - 1 Where c = number of groups; n = total number of values in all groups

Question 31

Define the Mean Squares in One-way ANOVA

Answer 31

Mean square among (MSA) = SSA / c-1 Mean square within (MSW) = SSW / n - c Mean square total (MST) = SST / n - 1

Question 32

How do you find the value of F_STAT

Answer 32

MSA / MSW

Question 33

How do you find F_α

Answer 33

Use the **Critical Values of F** Table Numerator = *c* - 1 Denominator = *n* - *c*

Question 34

How do you find Q_α

Answer 34

Use the **Critical Values of the Sudentized Range, Q** Table Numerator = *c* Denominator = *n* - *c*