Chapter 13: F Distribution and One-way Anova

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84 Terms

1
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Analysis of Variance
For hypothesis tests comparing averages between more than two groups
2
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ANOVA Test
determine the existence of a statistically significant difference among several group means.
3
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Variances
helps determine if the means are equal or not
4
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Ho
μ1 \= μ2 \= μ3 \= ... \= μk
5
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Ha
At least two of the group means μ1, μ2, μ3, ..., μk are not equal. That is, μi ≠ μj for some i ≠ j.
6
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The null hypothesis
is simply that all the group population means are the same.
7
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The alternative hypothesis
is that at least one pair of means is different.
8
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Ho is true
All means are the same; the differences are due to random variation.
9
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Ho is NOT true
All means are not the same; the differences are too large to be due to random variation.
10
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F-distribution
theoretical distribution that compares two populations
11
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Variance between samples
An estimate of σ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.).
12
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Variance within samples
An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
13
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SSbetween
the sum of squares that represents the variation among the different samples
14
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SSwithin
the sum of squares that represents the variation within samples that is due to chance.
15
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MS means
"mean square."
16
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MSbetween
is the variance between groups
17
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MSwithin
is the variance within groups.
18
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k
the number of different groups
19
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nj
the size of the jth group
20
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sj
the sum of the values in the jth group
21
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n
total number of all the values combined (total sample size
22
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x
one value→ ∑x \= ∑sj
23
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Sum of squares of all values from every group combined
∑x2
24
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Between-group variability
SStotal \= ∑x2 – (∑𝑥2) / n
25
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Total sum of squares
∑x^2 – (∑𝑥)^2n / n
26
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Explained variation
sum of squares representing variation among the different samples→ SSbetween \= ∑[(𝑠𝑗)^2 / 𝑛𝑗]−(∑𝑠𝑗)^2 / 𝑛
27
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Unexplained variation
sum of squares representing variation within samples due to chance→ 𝑆𝑆within \= 𝑆𝑆total – 𝑆𝑆between
28
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df's for different groups (df's for the numerator)
df \= k – 1
29
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dfwithin \= n – k
Equation for errors within samples (df's for the denominator)
30
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MSbetween \= 𝑆𝑆between / 𝑑𝑓between
Mean square (variance estimate) explained by the different groups
31
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MSwithin \= 𝑆𝑆within / 𝑑𝑓within
Mean square (variance estimate) that is due to chance (unexplained)
32
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Null hypothesis is true
MSbetween and MSwithin should both estimate the same value.
33
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The alternate hypothesis
at least two of the sample groups come from populations with different normal distributions.
34
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The null hypothesis
all groups are samples from populations having the same normal distribution
35
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F-Ratio Formula when the groups are the same size
𝐹 \= 𝑛⋅𝑠𝑥^2 / 𝑠^2 pooled
36
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n
the sample size
37
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dfnumerator
k – 1
38
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dfdenominator
n – k
39
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s2 pooled
the mean of the sample variances (pooled variance)
40
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sx¯^2
the variance of the sample means
41
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F is close to one
the evidence favors the null hypothesis (the two population variances are equal)
42
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F is much larger than one
then the evidence is against the null hypothesis
43
New cards
Analysis of Variance
For hypothesis tests comparing averages between more than two groups
44
New cards
ANOVA Test
determine the existence of a statistically significant difference among several group means.
45
New cards
Variances
helps determine if the means are equal or not
46
New cards
Ho
μ1 \= μ2 \= μ3 \= ... \= μk
47
New cards
Ha
At least two of the group means μ1, μ2, μ3, ..., μk are not equal. That is, μi ≠ μj for some i ≠ j.
48
New cards
The null hypothesis
is simply that all the group population means are the same.
49
New cards
The alternative hypothesis
is that at least one pair of means is different.
50
New cards
Ho is true
All means are the same; the differences are due to random variation.
51
New cards
Ho is NOT true
All means are not the same; the differences are too large to be due to random variation.
52
New cards
F-distribution
theoretical distribution that compares two populations
53
New cards
Variance between samples
An estimate of σ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.).
54
New cards
Variance within samples
An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
55
New cards
SSbetween
the sum of squares that represents the variation among the different samples
56
New cards
SSwithin
the sum of squares that represents the variation within samples that is due to chance.
57
New cards
MS means
"mean square."
58
New cards
MSbetween
is the variance between groups
59
New cards
MSwithin
is the variance within groups.
60
New cards
k
the number of different groups
61
New cards
nj
the size of the jth group
62
New cards
sj
the sum of the values in the jth group
63
New cards
n
total number of all the values combined (total sample size
64
New cards
x
one value→ ∑x \= ∑sj
65
New cards
Sum of squares of all values from every group combined
∑x2
66
New cards
Between-group variability
SStotal \= ∑x2 – (∑𝑥2) / n
67
New cards
Total sum of squares
∑x^2 – (∑𝑥)^2n / n
68
New cards
Explained variation
sum of squares representing variation among the different samples→ SSbetween \= ∑[(𝑠𝑗)^2 / 𝑛𝑗]−(∑𝑠𝑗)^2 / 𝑛
69
New cards
Unexplained variation
sum of squares representing variation within samples due to chance→ 𝑆𝑆within \= 𝑆𝑆total – 𝑆𝑆between
70
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df's for different groups (df's for the numerator)
df \= k – 1
71
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dfwithin \= n – k
Equation for errors within samples (df's for the denominator)
72
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MSbetween \= 𝑆𝑆between / 𝑑𝑓between
Mean square (variance estimate) explained by the different groups
73
New cards
MSwithin \= 𝑆𝑆within / 𝑑𝑓within
Mean square (variance estimate) that is due to chance (unexplained)
74
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Null hypothesis is true
MSbetween and MSwithin should both estimate the same value.
75
New cards
The alternate hypothesis
at least two of the sample groups come from populations with different normal distributions.
76
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The null hypothesis
all groups are samples from populations having the same normal distribution
77
New cards
F-Ratio Formula when the groups are the same size
𝐹 \= 𝑛⋅𝑠𝑥^2 / 𝑠^2 pooled
78
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n
the sample size
79
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dfnumerator
k – 1
80
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dfdenominator
n – k
81
New cards
s2 pooled
the mean of the sample variances (pooled variance)
82
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sx¯^2
the variance of the sample means
83
New cards
F is close to one
the evidence favors the null hypothesis (the two population variances are equal)
84
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F is much larger than one
then the evidence is against the null hypothesis