A researcher has constructed the following ANOVA table. Some numbers are left blank for you to figure out.
| Source | SS | DF | MS | F | p-value |
| :----- | :---- | :- | :----- | :---------- | :---------- |
| Factor | | 3 | 45 | --------- | ---------- |
| Error | 337 | 12 | | | |
| Total | 472 | | | | |
Answer the following questions.
(d) What's the p-value?
(e) What should be the conclusion?
A. < 0.05; We reject [tex]$H _0$[/tex]. We conclude that the four group means are significantly different.
B. > 0.05; We do NOT reject [tex]$H _0$[/tex]. We conclude that the four group means are NOT significantly different.
C. > 0.05; We reject [tex]$H _0$[/tex]. We conclude that the four group means are significantly different.
D. < 0.05; We do NOT reject [tex]$H _0$[/tex]. We conclude that the four group means are NOT significantly different.
Answer (1)
Calculate Factor SS: F a c t or SS = 3 × 45 = 135 .
Calculate Total DF: T o t a l D F = 12 + 3 = 15 .
Calculate Error MS: E rror MS = 12 337 = 28.0833 .
Calculate F statistic and p-value: F = 28.0833 45 = 1.6024 , p ≈ 0.2406 . Since 0.05"> p > 0.05 , we do NOT reject H 0 . We conclude that the four group means are NOT significantly different.
Explanation
Understand the problem We are given an ANOVA table with some missing values. Our goal is to find the p-value and determine the correct conclusion based on it.
Gather the data First, we need to calculate the missing values in the ANOVA table. We have:
Factor DF = 3 Factor MS = 45 Error SS = 337 Error DF = 12 Total SS = 472
Calculate missing values
Calculate Factor SS: F a c t or SS = F a c t or D F × F a c t or MS = 3 × 45 = 135
Calculate Total DF: T o t a l D F = E rror D F + F a c t or D F = 12 + 3 = 15
Calculate Error MS: E rror MS = E rror D F E rror SS = 12 337 = 28.0833
Calculate the F statistic: F = E rror MS F a c t or MS = 28.0833 45 = 1.6024
Calculate the p-value. Using the F-statistic, Factor DF, and Error DF, we find the p-value to be approximately 0.2406.
Draw conclusion Now we compare the p-value to 0.05. Since the p-value (0.2406) is greater than 0.05, we do not reject the null hypothesis. Therefore, we conclude that the four group means are not significantly different.
Examples
ANOVA is used in many fields, such as medicine, engineering, and business, to compare the means of different groups. For example, in a clinical trial, ANOVA can be used to compare the effectiveness of different treatments. In manufacturing, it can be used to compare the quality of products produced by different machines. In marketing, it can be used to compare the sales of different products in different regions.
