Data Matters with Fathom! Dynamic Statistics™ software
Activity 8.1
Your task in this project is to find how F is distributed so you can fill in the following table from Data Matters. Three cells of the table are already filled in. It would be a good idea to do at least two of those first to make sure Fathom is set up correctly.
Table 8.1.3 (from Data Matters)
Critical F-Values for the Analysis of Variance
(If an ANOVA has an F-value greater than
the value in the table, the F is significant.)
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df of Variance Estimated from Between-Group
Variation (df numerator: Number of Groups – 1)
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df of Variance
Estimated from
Within-Group Variation
(df denominator:
Number of
Observations—Number
of Groups)
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1 |
2 |
3 |
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1
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| 2 |
18.56 |
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3
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4
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6.6 |
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60
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100
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2.76 |
Two Groups, Four Observations
Drag a case table onto the workspace. Add an attribute, Group. Add four cases. Enter A for the first two cases and B for the third and fourth cases.
Add a second attribute, Observation. Right-click on Observation and enter the formula randomNormal(3,2). You can set the mean and standard deviation of the population you are drawing from however you like by changing the numbers in the parentheses.
Fathom will calculate the F-values. Once you can see the random numbers, select Analyze, Test Hypothesis, Empty Test, Analysis of Variance. Drag Group onto Grouping Attribute and Observation onto Response Attribute.
Select Analyze, Collect Measures. Drag a case table onto the workspace so you can see what is in the Measures Collection. If you like, drag a graph onto the workspace and drag F into the graph so you can see it growing as the sampling is done.
Double-click on the Measures Collection to set your number of samples and Animation.
After you have the F-values, right-click on their column and select Sort Descending. Then you can scroll down 5% and find the cutoff.
Try a larger number of samples.
My experience is that with less than 10,000 samples, the F-value 5% down the list is more than 1 away from 18.56. When I was creating the entries for that table, I used more than 400,000 samples.
Other Designs
To get the cutoffs for the other sets of degrees of freedom, edit the original case table. Add more groups and/or more observations as needed.
Other Distributions and Other Questions
Does it matter what the mean and/or standard deviation of your original distribution is?
Each F-value that was calculated, was calculated as Fathom was doing an ANOVA. For each of those tests, was the null hypothesis true or false? When Fathom found an F that was greater than 20, was the null hypothesis true or false? If an F above 20 were found, would you reject the null hypothesis?
What happens if you use another distribution? For example, you could replace randomNormal(3,2) with random(1,.5). Does that affect the distribution of F?
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