The article “Impact of Free Calcium Oxide Content of Fly Ash on Dust and Sulfur Dioxide Emissions in a Lignite-Fired Power Plant” (D. Sotiropoulos, A. Georgakopoulos, and N. Kolovos, Journal of Air and Waste Management, 2005:1042–1049) presents measurements of dust emissions, in mg/m3, for four power plants. Thirty measurements were taken for each plant. The sample means and standard deviations are presented in the following table:
- a. Construct an ANOVA table. You may give a
range for the P-value. - b. Can you conclude that there are differences among the
mean emission levels?
a.
Construct an ANOVA table and give a range for the P-value.
Answer to Problem 8E
The ANOVA table is,
Source | DF | SS | MS | F | P |
Plant | 3 | 12,712.7 | 4,237.6 | 4.8179 | 0.01<P-value<0.001 |
Error | 116 | 102,027.8 | 879.55 | ||
Total | 119 | 114,740.5 |
The range of P-value is
Explanation of Solution
Given info:
The data represents the means and standard deviations of 30 measurements of dust emissions taken on four types of plants.
Calculation:
The ANOVA table can be obtained as follows:
There are four samples, therefore
The total number of observations is,
The degrees of freedom corresponding to the plant is obtained as follows:
The degrees of freedom corresponding to the total is obtained as follows:
The degrees of freedom corresponding to the error is obtained as follows:
Total mean can be obtained as follows:
Substitute
The treatment sum of squares (SSTr) is obtained as follows:
Substitute
The error sum of squares (SSE) is obtained as follows:
Substitute
The total sum of squares (SST) is obtained as follows:
Substitute
The treatment mean square is obtained as follows:
Substitute
The error mean square is obtained as follows:
Substitute
The F-value is obtained as follows:
Substitute
Thus, the F-value is 4.8179.
From Appendix A table A.8, the upper 1% point of the
Therefore, the range of P-value is
Thus, the ANOVA table is,
Source | DF | SS | MS | F | P |
Plant | 3 | 12,712.7 | 4,237.6 | 4.8179 | 0.01<P-value<0.001 |
Error | 116 | 102,027.8 | 879.55 | ||
Total | 119 | 114,740.5 |
b.
Check whether the mean emission levels differ for four types of plants.
Answer to Problem 8E
There is sufficient evidence to conclude that the mean emission levels differ for four types of plants.
Explanation of Solution
Calculation:
State the hypotheses:
Null hypothesis:
Alternative hypothesis:
From part (a), the F-ratio is 4.8179.
P-value:
From part (a), the P-value is between 0.01 and 0.001.
Since, the level of significance is not specified; the prior level of significance
Decision:
If
If
Conclusion:
Here, the P-value is less than the level of significance.
That is,
By rejection rule, reject the null hypothesis.
There is sufficient evidence to conclude that the mean emission levels differ for four types of plants at
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