Practice• Does drinking milkshakes affect (alpha

= .05) your weight?

• To see if milkshakes affect a persons weight you collected data from 5 sets of twins. You randomly had one twin drink water and the other twin drank milkshakes. After 3 months you weighed them.

Results

Water

Twin A 186

Twin B 200

Twin C 190

Twin D 162

Twin E 175

Milkshakes

195

202

196

165

183

HypothesisTwo-tailed

• Alternative hypothesis– H1: water = milkshake

• Null hypothesis– H0: water = milkshake

Step 2: Calculate the Critical t

• N = Number of pairs

• df = N - 1

• 5 - 1 = 4 = .05

• t critical = 2.776

Step 3: Draw Critical Region

tcrit = 2.776tcrit = -2.776

Step 4: Calculate t observed

tobs = (X - Y) / SD

3.04 =

(D)

-9

-2

-6

-3

-8

D = -28

D2 =194

N = 6

-28194

5

5 - 1

Step 4: Calculate t observed

tobs = (X - Y) / SD

1.36=3.04 / 5

N = number of pairs

Step 4: Calculate t observed

-4.11 = (182.6 – 188.2) / 1.36

X = 182.6

Y = 188.2

SD = 1.36

Step 5: See if tobs falls in the critical region

tcrit = 2.776tcrit = -2.776

tobs = -4.11

Step 6: Decision

• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:

– Fail to reject H0

Step 7: Put answer into words

• Reject H0, and accept H1

• Milkshakes significantly ( = .05) affect a persons weight.

What if. . .

• You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person misses class)

• You would simply do a two-sample t-test– two-tailed

• Easy!

But, what if. . .

• You were asked to determine if psychology, sociology, and biology majors have significantly different class attendance

• You can’t do a two-sample t-test – You have three samples

• No such thing as a three sample t-test!

One-Way ANOVA

• ANOVA = Analysis of Variance

• This is a technique used to analyze the results of an experiment when you have more than two groups

Example• You measure the number of days 7

psychology majors, 7 sociology majors, and 7 biology majors are absent from class

• You wonder if the average number of days each of these three groups was absent is significantly different from one another

Hypothesis

• Alternative hypothesis (H1)

• H1: The three population means are not all equal

Hypothesis

• Alternative hypothesis (H1)

socio = bio

Hypothesis

• Alternative hypothesis (H1)

socio = psych

Hypothesis

• Alternative hypothesis (H1)

psych = bio

Hypothesis

• Alternative hypothesis (H1)

psych = bio = soc

Hypothesis

• Alternative hypothesis (H1)

– Notice: It does not say where this difference is at!!

Hypothesis

• Null hypothesis (H0)

psych = socio = bio

– In other words, all three means are equal to one another (i.e., no difference between the means)

Results

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

Logic

• Is the same as t-tests

• 1) calculate a variance ratio (called an F; like t-observed)

• 2) Find a critical value

• 3) See if the the F value falls in the critical area

Between and Within Group Variability

• Two types of variability

• Between– the differences between the mean scores of the

three groups– The more different these means are, the more

variability!

Results

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

Between Variability

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

S2 = .66

Between Variability

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

+ 5

Between Variability

Sociology Psychology Biology9 1 18 2 17 0 28 2 08 4 29 3 07 2 1

X = 8.00 X = 2.00 X = 1.00

Between Variability

Sociology Psychology Biology9 1 18 2 17 0 28 2 08 4 29 3 07 2 1

X = 8.00 X = 2.00 X = 1.00

S2 = 9.55

Between Group Variability

• What causes this variability to increase?

• 1) Effect of the variable (college major)

• 2) Sampling error

Between and Within Group Variability

• Two types of variability

• Within– the variability of the scores within each group

Results

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

Within Variability

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

S2 =.57

Within Variability

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

S2 =.57 S2 =1.43 S2 =.57

Within Group Variability

• What causes this variability to increase?

• 1) Sampling error

Between and Within Group Variability

Between-group variability

Within-group variability

Between and Within Group Variability

sampling error + effect of variable

sampling error

Between and Within Group Variability

sampling error + effect of variable

sampling error

Thus, if null hypothesis was true this would result in a value of 1.00

Between and Within Group Variability

sampling error + effect of variable

sampling error

Thus, if null hypothesis was not true this value would be greater than 1.00