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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
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