Studied by 3 peopleAnalysis of Variance
For hypothesis tests comparing averages between more than two groups
ANOVA Test
determine the existence of a statistically significant difference among several group means.
Variances
helps determine if the means are equal or not
Ho
μ1 = μ2 = μ3 = ... = μk
Ha
At least two of the group means μ1, μ2, μ3, ..., μk are not equal. That is, μi ≠ μj for some i ≠ j.
The null hypothesis
is simply that all the group population means are the same.
The alternative hypothesis
is that at least one pair of means is different.
Ho is true
All means are the same; the differences are due to random variation.
Ho is NOT true
All means are not the same; the differences are too large to be due to random variation.
F-distribution
theoretical distribution that compares two populations
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.).
Variance within samples
An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
SSbetween
the sum of squares that represents the variation among the different samples
SSwithin
the sum of squares that represents the variation within samples that is due to chance.
MS means
"mean square."
MSbetween
is the variance between groups
MSwithin
is the variance within groups.
k
the number of different groups
nj
the size of the jth group
sj
the sum of the values in the jth group
n
total number of all the values combined (total sample size
x
one value→ ∑x = ∑sj
Sum of squares of all values from every group combined
∑x2
Between-group variability
SStotal = ∑x2 – (∑𝑥2) / n
Total sum of squares
∑x^2 – (∑𝑥)^2n / n
Explained variation
sum of squares representing variation among the different samples→ SSbetween = ∑[(𝑠𝑗)^2 / 𝑛𝑗]−(∑𝑠𝑗)^2 / 𝑛
Unexplained variation
sum of squares representing variation within samples due to chance→ 𝑆𝑆within = 𝑆𝑆total – 𝑆𝑆between
df's for different groups (df's for the numerator)
df = k – 1
dfwithin = n – k
Equation for errors within samples (df's for the denominator)
MSbetween = 𝑆𝑆between / 𝑑𝑓between
Mean square (variance estimate) explained by the different groups
MSwithin = 𝑆𝑆within / 𝑑𝑓within
Mean square (variance estimate) that is due to chance (unexplained)
Null hypothesis is true
MSbetween and MSwithin should both estimate the same value.
The alternate hypothesis
at least two of the sample groups come from populations with different normal distributions.
The null hypothesis
all groups are samples from populations having the same normal distribution
F-Ratio Formula when the groups are the same size
𝐹 = 𝑛⋅𝑠𝑥^2 / 𝑠^2 pooled
n
the sample size
dfnumerator
k – 1
dfdenominator
n – k
s2 pooled
the mean of the sample variances (pooled variance)
sx¯^2
the variance of the sample means
F is close to one
the evidence favors the null hypothesis (the two population variances are equal)
F is much larger than one
then the evidence is against the null hypothesis
Analysis of Variance
For hypothesis tests comparing averages between more than two groups
ANOVA Test
determine the existence of a statistically significant difference among several group means.
Variances
helps determine if the means are equal or not
Ho
μ1 = μ2 = μ3 = ... = μk
Ha
At least two of the group means μ1, μ2, μ3, ..., μk are not equal. That is, μi ≠ μj for some i ≠ j.
The null hypothesis
is simply that all the group population means are the same.
The alternative hypothesis
is that at least one pair of means is different.
Ho is true
All means are the same; the differences are due to random variation.
Ho is NOT true
All means are not the same; the differences are too large to be due to random variation.
F-distribution
theoretical distribution that compares two populations
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.).
Variance within samples
An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
SSbetween
the sum of squares that represents the variation among the different samples
SSwithin
the sum of squares that represents the variation within samples that is due to chance.
MS means
"mean square."
MSbetween
is the variance between groups
MSwithin
is the variance within groups.
k
the number of different groups
nj
the size of the jth group
sj
the sum of the values in the jth group
n
total number of all the values combined (total sample size
x
one value→ ∑x = ∑sj
Sum of squares of all values from every group combined
∑x2
Between-group variability
SStotal = ∑x2 – (∑𝑥2) / n
Total sum of squares
∑x^2 – (∑𝑥)^2n / n
Explained variation
sum of squares representing variation among the different samples→ SSbetween = ∑[(𝑠𝑗)^2 / 𝑛𝑗]−(∑𝑠𝑗)^2 / 𝑛
Unexplained variation
sum of squares representing variation within samples due to chance→ 𝑆𝑆within = 𝑆𝑆total – 𝑆𝑆between
df's for different groups (df's for the numerator)
df = k – 1
dfwithin = n – k
Equation for errors within samples (df's for the denominator)
MSbetween = 𝑆𝑆between / 𝑑𝑓between
Mean square (variance estimate) explained by the different groups
MSwithin = 𝑆𝑆within / 𝑑𝑓within
Mean square (variance estimate) that is due to chance (unexplained)
Null hypothesis is true
MSbetween and MSwithin should both estimate the same value.
The alternate hypothesis
at least two of the sample groups come from populations with different normal distributions.
The null hypothesis
all groups are samples from populations having the same normal distribution
F-Ratio Formula when the groups are the same size
𝐹 = 𝑛⋅𝑠𝑥^2 / 𝑠^2 pooled
n
the sample size
dfnumerator
k – 1
dfdenominator
n – k
s2 pooled
the mean of the sample variances (pooled variance)
sx¯^2
the variance of the sample means
F is close to one
the evidence favors the null hypothesis (the two population variances are equal)
F is much larger than one
then the evidence is against the null hypothesis
Note
Flashcard