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.

Select Responses are in one column for all factor levels.

In Response, enter Hardness.

In Factor, enter Paint.

On the Comparisons tab, select Tukey (family error rate).

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 R^{2} 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

Null hypothesis

H₀: All means are equal

Alternative hypothesis

H₁: At least one mean is different

Equal variances were assumed for the analysis.

Factor Information

Factor

Levels

Values

Paint

4

Blend 1, Blend 2, Blend 3, Blend 4

Analysis of Variance

Source

DF

Adj SS

Adj MS

F-Value

P-Value

Paint

3

281.698

93.8993

6.02

0.0043

Error

20

312.068

15.6034

Total

23

593.766

Model Summary

S

R-sq

R-sq(adj)

R-sq(pred)

3.95012

47.44%

39.56%

24.32%

Means

Paint

N

Mean

StDev

95% CI

Blend 1

6

14.733

3.363

(11.369, 18.097)

Blend 2

6

8.567

5.500

(5.203, 11.931)

Blend 3

6

12.983

3.730

(9.619, 16.347)

Blend 4

6

18.067

2.636

(14.703, 21.431)

Pooled StDev = 3.95012

Grouping Information Using the Tukey Method and 95% Confidence

Paint

N

Mean

Grouping

Blend 4

6

18.067

A

Blend 1

6

14.733

A

B

Blend 3

6

12.983

A

B

Blend 2

6

8.567

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

95% CI

T-Value

Adjusted P-Value

Blend 2-Blend 1

-6.167

2.281

(-12.553, 0.219)

-2.70

0.0606

Blend 3-Blend 1

-1.750

2.281

(-8.136, 4.636)

-0.77

0.8682

Blend 4-Blend 1

3.333

2.281

(-3.053, 9.719)

1.46

0.4779

Blend 3-Blend 2

4.417

2.281

(-1.969, 10.803)

1.94

0.2450

Blend 4-Blend 2

9.500

2.281

(3.114, 15.886)

4.17

0.0025

Blend 4-Blend 3

5.083

2.281

(-1.303, 11.469)

2.23

0.1495

Individual confidence level = 98.89%

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