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Suppose we want to compare three teaching methods based on the mean test scores for each method. The competing theories would be:H0: There is no difference between the three teaching methods with respect to mean test scoreH1: There is a difference between the three teaching methods with respect to the mean test score
Consider these boxplots for the three methods test scores
Now what if they looked like this?
Think About It
There is an obvious difference between Scenario 1 and Scenario 2. What is that difference?
Just looking at the boxplots, which of the two scenarios do you think would provide more evidence that at least one of the population means is different from the others? Explain why.
Notice that there are two types of variation we are looking at, the between variability and the within variability. The test statistic we’ll use to judge if at least one of the means is significantly different is based on these measures.
Analysis of variance is a statistical technique for comparing the means for several populations, where the levels of a single explanatory variable define the populations.
1. The populations have normal distributions.
2. The populations have the same variance 2 (or standard deviation ).
3. The samples are random and independent of each other.
4. The different samples are from populations that are categorized in only one way.
1. The F-distribution is not symmetric; it is skewed to the right.
2. The values of F can be 0 or positive, they cannot be negative.
3. There is a different F-distribution for each pair of degrees of freedom for the numerator and denominator. Let I be the number of means (populations drawn from) and n = total sample size across all populations sampled.
Degrees of freedom numerator is given by: I – 1 (number of groups – 1)
Degrees of freedom denominator is given by: n – I (number of observations – number of groups)
The distribution of the F statistic is the the F-distribution. The distribution is indexed by a pair of degrees of freedom, one for the numerator and one for the denominator.
Think About It
If all the sample means were the same, what would be the value of the numerator of the F-statistic?
If all of the sample means were very spread out and very different from each other, what would the magnitude of the variation be compared to your last answer?
What values of the F-statistic support the alternate hypothesis that at least one population has a different mean?
Let’s Do It
Page 751: LDI 12.1
Assumptions of ANOVA
Page 753: Grey Box
Page 754: Example 12.4
Let’s Do It
LDI 12.2 (Note how to calculate Fcdf)
LDI 12.3 (What is MSB? What is MSW?)
SSB = Sum of Squares Between
SSW = Sum of Squares Within
SST = Sum of Squares Total
Factor = Between
Error = Within
How to do an F-test (ANOVA) on the TI
Example 12.8 and LDI 12.4
Now, using the summary statistics for each treatment do the calculation by A1ANOVA on the TI.