Download 13.1.1 Steps of ANOVAs - University of Northern Colorado

One-Way ANOVA
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Testing Claims about Differences between
Three or More Means
Step 0: Verify Assumptions
The hypothesis test of the difference of three or more population means has four assumptions.
1. The k samples are each obtained using simple random sampling.
2. The k samples data independent of each other within and among the samples.
3. The k populations are normally distributed.
4. The k populations have equal variances.
Step 1: State the Hypothesis
A claim is made regarding the three or more population means. This claim is used to determine
the following null and alternative hypotheses.
H0: µ1 = µ2 = µ3 = … = µk
H1: At least one of the population means is different from the others.
Step 2: Select a Level of Significance
The selection of the level of significance α is done based on the seriousness of making a Type I
error. (The typical value of α is 0.05.)
Step 3: Calculate the Test Statistic
The test statistic follows the F-distribution with k – 1 degrees of freedom in the numerator and
n – k degrees of freedom in the denominator. The F-statistic is calculated as follow:
a. Find the sample mean of the combined data: b. Find the sample mean for each individual sample: , , , …, c. Find the sample standard deviations of each individual sample: , , , …, d. Compute the mean square due to treatment:
∑ ( − )
=
−1
e. Compute the mean square due to error:
∑( − 1)
=
− f. Compute the test statistic:
=
Robert A. Powers
University of Northern Colorado
One-Way ANOVA
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Step 4: Determine the Decision Criterion
The Classical Approach: Find the Critical Value
The level of significance is used to determine the critical value, represented by the F-value in
the figure below. All one-way ANOVA tests use a right-tailed test, so the critical value is
,, with k – 1 degrees of freedom in the numerator and n – k degrees of freedom in
the denominator.
Fα , k −1, n − k
The Modern Approach: Find the p-Value
Based on the test statistic , determine the probability that the ratio of the mean square due
to treatment and the mean square due to error is more extreme than is found. This is
represented by the shaded region under the F-distribution with k – 1 degrees of freedom in
the numerator and n – k degrees of freedom in the denominator in the figure below.
F0
Step 5: Make a Decision
Reject the null hypothesis if the test statistics lies in the critical region or the probability
associated with the test statistic is less than the level of significance.
Do not reject the null hypothesis if the test statistic does not lie in the critical region or the
probability associated with the test statistic is greater than or equal to the level of significance.
Step 6: State the Conclusion
State the conclusion of the hypothesis test based on the decision made and with respect to the
original claim.
Reject H0
Do Not
Reject H0
Original Claim is H0
There is sufficient evidence (at the α
level) to reject the claim that … .
There is not sufficient evidence (at the
α level) to reject the claim that … .
Robert A. Powers
Original Claim is H1
There is sufficient evidence (at the α
level) to support the claim that … .
There is not sufficient evidence (at the
α level) to support the claim that … .
University of Northern Colorado