Example of One-Way ANOVA

A chemical engineer wants to compare the hardness of four blends of paint. Six samples of each paint blend were applied to a piece of metal. The pieces of metal were cured. Then each sample was measured for hardness. In order to test for the equality of means and to assess the differences between pairs of means, the analyst uses one-way ANOVA with multiple comparisons.

  1. Open the sample data, PaintHardness.MTW.
  2. Open the One-Way ANOVA dialog box.
    • Mac: Statistics > ANOVA > One-Way ANOVA
    • PC: STATISTICS > ANOVA > One-Way ANOVA
  3. Select Responses are in one column for all factor levels.
  4. In Response, enter Hardness.
  5. In Factor, enter Paint.
  6. On the Comparisons tab, select Tukey (family error rate).
  7. Click OK.

Interpret the results

The p-value for the paint hardness ANOVA is less than 0.05. This result indicates that the mean differences between the hardness of the paint blends is statistically significant. The engineer knows that some of the group means are different.

The engineer uses the Tukey comparison results to formally test whether the difference between a pair of groups is statistically significant. The graph and the table that include the Tukey simultaneous confidence intervals show that the confidence interval for the difference between the means of Blend 2 and 4 is 3.114 to 15.886. This range does not include zero, which indicates that the difference between these means is significant. The engineer can use this estimate of the difference to determine whether the difference is practically significant.

The confidence intervals for the remaining pairs of means all include zero, which indicates that the differences are not significant.

The low predicted R2 value indicates that the model generates imprecise predictions for new observations. The imprecision may be due to the small size of the groups. Thus, the engineer should be wary about using the model to make generalizations beyond the sample data.

Method
H₀: All means are equal
H₁: At least one mean is different
Equal variances were assumed for the analysis.
Factor Information
Factor
Levels
Values
Blend 1, Blend 2, Blend 3, Blend 4
Analysis of Variance
Source
DF
Adj SS
Adj MS
F-Value
P-Value
 
 
 
 
 
Model Summary
S
R-sq
R-sq(adj)
R-sq(pred)
Means
Paint
N
Mean
StDev
Pooled StDev = 3.95012
Grouping Information Using the Tukey Method and 95% Confidence
Paint
N
Mean
Grouping
A
A
B
A
B
B
Means that do not share a letter are significantly different.
Tukey Simultaneous Tests for Differences of Means
Difference of Levels
Difference of Means
SE of Difference
T-Value
Adjusted P-Value
Individual confidence level = 98.89%
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