Chapter 13: F Distribution and One-way Anova

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

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