Introduction to Statistics
Steps in Significance/Hypothesis Testing
Using the F Distribution for One-Way Analysis of Variance
(ANOVA)
Step 1: State null (Ho) and alternative (Ha) hypotheses.
Hypotheses, because they refer to the population, always contain population parameter
symbols (i.e., µ).
The null hypothesis (Ho) is a statement of no difference and contains the "equal to" (=)
sign. In the case of a test about the difference between more than two population
means, if you do not reject Ho, you conclude that the population means are not
significantly different from each other.
For example: Ho: µ1 = µ2 = µ3 = µ4
The alternative hypothesis (Ha) is a statement of difference. In the case of a test about
the difference between more than two population means, if you do not reject Ho, you
conclude that the populations are significantly different from each other. More technically
in one-way ANOVA, you conclude that at least one of the population means is
significantly different from one or more of the other means. There are no directional tests
in one-way ANOVA.
For example: Ha: not all of the µs are equal
Step 2: Compute a test statistic.
The formulas for computing the F test statistic are in the ANOVA table below. Go from
left to right to calculate F and from that, the p-value. "I" refers to the number of groups in
the comparison and "N" is the sum of the sample sizes of all groups (N = ∑ni).
Analysis of Variance Formulas
Source
DF
Sum of
Squares
Mean Square
F
Group
I-1
given
SSG/DFGnumerator
MSG/MSE
Error
N-I
given
SSE/DFEdenominator
P-value
Total
M. Berntson – Grinnell College
Spring 2002 – Steps in Hypothesis Testing Using the F Distribution
1
Step 3: Make a decision about the null hypothesis (Ho).
In this step, you will either reject or not reject Ho. When you reject Ho, you conclude in
favor of the alternative hypothesis (Ha) of significant difference. When you do not reject
Ho, you conclude in favor of the null hypothesis of no significant difference. There are
two decision rules you may follow. You should know how to use each one.
Decision Rule 1 instructions: Find a p-value and compare it to your significance level
(α). P-values are the probabilities associated with the test statistic (observed F) found in
step 2. Probabilities and two different kinds of degrees of freedom organize the values of
F found in the body of Moore's Table D. Use Moore's Table D and the instructions below
to find the p-value. Note that when finding a p-value in Table D, your observed F might
fall between two values of F. This means that your p-value falls between two
probabilities.
Finding P-Values using Table D
Test Statistic
(observed F)
Look for F in body of
table
P-value
given dfnumerator (I - 1) in the table's
columns and dfdenominator (N - I) in the
table's rows, find the probabilities
(this might be a range)
Decision rule 1: If p-value(s) < α, reject Ho (eg: .01 to .025 < .05). If p-value(s) > α, do
not reject Ho (eg: .15 to .20 > .05).
Decision Rule 2 instructions: Find a critical value (F*) and compare it to your test
statistic. Critical values (F*) are the F scores in Table D associated with your level of
significance (α).
Finding Critical Values (F*) Using Table D
Critical Value (F*)
given α, dfnumerator (I - 1) in the table's
columns, and dfdenominator (N - I) in the
table's rows, find the F value
Decision rule 2: If F > F*, reject Ho. If F < F*, do not reject Ho.
M. Berntson – Grinnell College
Spring 2002 – Steps in Hypothesis Testing Using the F Distribution
2