Six Easy Steps for an ANOVA 1) State the hypothesis 2) Find the F-critical value 3) Calculate the F-value 4) Decision 5) Create the summary table 6) Put answer into words

Example Want to examine the effects of feedback on self-esteem. Three different conditions -- each have five subjects 1) Positive feedback 2) Negative feedback 3) Control Afterward all complete a measure of self-esteem that can range from 0 to 10.

Example: Question: Is the type of feedback a person receives significantly (.05) related their self-esteem?

Results

Step 1: State the Hypothesis H1: The three population means are not all equal H0: pos = neg = cont

Step 2: Find F-Critical Step 2.1 Need to first find dfbetween and dfwithin dfbetween = k - 1 (k = number of groups) dfwithin = N - k (N = total number of observations) dftotal = N - 1 Check yourself dftotal = dfbetween + dfwithin

Step 2: Find F-Critical Step 2.1 Need to first find dfbetween and dfwithin dfbetween = 2 (k = number of groups) dfwithin = 12 (N = total number of observations) dftotal = 14 Check yourself 14 = 2 + 12

Step 2: Find F-Critical Step 2.2 Look up F-critical using table F

Step 2: Find F-Critical Step 2.2 Look up F-critical using table F

Step 3: Calculate the F-value Has 4 Sub-Steps 3.1) Calculate the needed ingredients 3.2) Calculate the SS 3.3) Calculate the MS 3.4) Calculate the F-value

Step 3.1: Ingredients X X2 Tj2 N n

Step 3.1: Ingredients

X X = 85 Xp = 40 Xn = 25 Xc = 20

X2 X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20 X2n = 135 X2c = 90

T2 = (X)2 for each group X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20 T2n = 625 T2c = 400 T2p = 1600

Tj2 X = 85 X2 = 555 Tj2 = 2625 Xp = 40 Xn = 25 Xc = 20 T2n = 625 T2c = 400 T2p = 1600

N X = 85 X2 = 555 Tj2 = 2625 N = 15 Xp = 40 Xn = 25 Xc = 20 T2n = 625 T2c = 400 T2p = 1600

n X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Xp = 40 Xn = 25 Xc = 20 X2n = 135 X2c = 90 X2p = 330 T2n = 625 T2c = 400 T2p = 1600

Step 3.2: Calculate SS SStotal X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 85 73.33 555 15 SStotal X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SStotal 85 73.33 555 15

Step 3.2: Calculate SS SSWithin X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 2625 30 555 5 SSWithin X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SSWithin 2625 30 555 5

Step 3.2: Calculate SS SSBetween X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 43.33 2625 85 5 15 SSBetween X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SSBetween 43.33 2625 85 5 15

Step 3.2: Calculate SS Check! SStotal = SSBetween + SSWithin

Step 3.2: Calculate SS Check! 73.33 = 43.33 + 30

Step 3.3: Calculate MS

Step 3.3: Calculate MS 43.33 21.67 2

Calculating this Variance Ratio

Step 3.3: Calculate MS 30 2.5 12

Step 3.4: Calculate the F value

Step 3.4: Calculate the F value 21.67 8.67 2.5

Step 4: Decision If F value > than F critical Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Step 4: Decision If F value > than F critical Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Step 5: Create the Summary Table

Step 6: Put answer into words Question: Is the type of feedback a person receives significantly (.05) related their self-esteem? H1: The three population means are not all equal The type of feedback a person receives is related to their self-esteem

Practice You are interested in comparing the performance of three models of cars. Random samples of five owners of each car were used. These owners were asked how many times their car had undergone major repairs in the last 2 years.

Results

Practice Is there a significant (.05) relationship between the model of car and repair records?

Step 1: State the Hypothesis H1: The three population means are not all equal H0: V = F = G

Step 2: Find F-Critical Step 2.1 Need to first find dfbetween and dfwithin Dfbetween = 2 (k = number of groups) dfwithin = 12 (N = total number of observations) dftotal = 14 Check yourself 14 = 2 + 12

Step 2: Find F-Critical Step 2.2 Look up F-critical using table F on pages 370 - 373. F (2,12) = 3.88

Step 3.1: Ingredients X = 60 X2 = 304 Tj2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SStotal X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 60 64 304 15 SStotal X = 60 X2 = 304 Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SStotal 60 64 304 15

Step 3.2: Calculate SS SSWithin X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 1400 24 304 5 SSWithin X = 60 X2 = 304 Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SSWithin 1400 24 304 5

Step 3.2: Calculate SS SSBetween X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 40 1400 60 5 15 SSBetween X = 60 X2 = 304 Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SSBetween 40 1400 60 5 15

Step 3.2: Calculate SS Check! SStotal = SSBetween + SSWithin

Step 3.2: Calculate SS Check! 64 = 40 + 24

Step 3.3: Calculate MS

Step 3.3: Calculate MS 40 20 2

Calculating this Variance Ratio

Step 3.3: Calculate MS 24 2 12

Step 3.4: Calculate the F value

Step 3.4: Calculate the F value 20 10 2

Step 4: Decision If F value > than F critical Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Step 4: Decision If F value > than F critical Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Step 5: Create the Summary Table

Step 6: Put answer into words Question: Is there a significant (.05) relationship between the model of car and repair records? H1: The three population means are not all equal There is a significant relationship between the type of car a person drives and how often the car is repaired

Conceptual Understanding Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.

Conceptual Understanding Fcrit = 3.18 Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05. Fcrit (2, 57) = 3.15

Conceptual Understanding Distinguish between: Between-group variability and within-group variability What do they measure? How do they work together?

Conceptual Understanding Distinguish between: Between-group variability and within-group variability Between concerns the differences between the mean scores in various groups Within concerns the variability of scores within each group

Between and Within Group Variability Between-group variability Within-group variability

Between and Within Group Variability sampling error + effect of variable sampling error

Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?

Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00? F ratio will approach 1.00 when the null hypothesis is true F ratio will be greater than 1.00 when the null hypothesis is not true

Conceptual Understanding Without computing the SS within, what must its value be? Why?

Conceptual Understanding The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)