Question 713:

Asked on May 3, 2009

Tags: ANOVA , Eta-Squared , Effect Size , ANOVA Summary Data

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1. Q713_1_ANOVA_Summary713.xls

We are given the results of three means and variances from 3 samples. We can use this summary information to construct a 1-Way ANOVA. Recall that the variance is just the square of the standard deviation. To arrive at the standard deviation, we just take the square root. We now have the standard deviations of 1.414, 1.2247 and 1.5811 respectively.

Part A.
Using the steps of hypothesis testing we have:

1. The null hypothesis is there is no difference between any of the mean times of socializing.  The alternative hypothesis is that at least one mean is different than the others.
2. We will set the alpha level to .05.
3. The 1-Way ANOVA (using summary information) will be conducted. We will reject the NULL hypothesis if the p-value from the test statistic F is less than the alpha of .05.
4. Computing the F-statistic is best done using software to run the ANOVA. We get an F-ratio of 12.5 on 2 and 72 degrees of freedom with an associated p-value of less than .001
1. To calculate these values by hand takes several steps. See the attached excel file for detailed computations. The steps are:
2. Multiply the mean for each group by their sample size. For college X you get 125.
   
3. Multiply the sample size times the mean squared plus the sample size minus one times the variance. For college X you should get 25*25 + 24*2 = 673
4. Square the result from step 2 and divide by the sample size. For college X you should get 15625/25 = 625.
5. Now add up all the values for each college from the previous three steps. You should get 375, 2069 and 1925.
6. We can build the ANOVA table now with the values we have. The Sum of Squares Between groups is found as the sum of step 4 minus the sum from step 2 squared divided by the total sample size of 75 = 1925-(375^2)/75 = 50.
7. The Sum of Squares Error is just the sum from step 3 minus the sum from step 4 = 2069-1925 = 144.
   
8. The total Sum of Squares is the Between plus within Sum of Squares = 144+50 = 194
9. The MeanSquare Between is found as SS Between divided by the degrees of freedom (groups -1) = 50/2 = 25.
   
10. The Mean Square within is SS Withing divided by the degrees of freedom within (total sample size -1 minus between degrees of freedom = 74-2 = 72) = 144/72 = 2.
    
11. The F-ratio is the MS Between divided by MS within = 25/2 = 12.5 which is the F-ratio we found if we had access to the software.
12. The p-value can be found using an F-table with 2 and 72 df for an alpha of .05 = <.00001.
    
5. Since p is less than .01 we reject the Null Hypothesis and conclude there is a difference in the average time spent socializing between schools.

Part B.
To compute the effect size, we'll use the Eta-Squared statistic. Eta-Squared is calculated as the ratio of Sum of Squares Treatment to Sum of Squares Total. We get this information from the summary table of the ANOVA. These values are 50/194 = .2577. We can treat Eta-Squared, like an r-squared statistic and say that 25.77% of the variation in mean time spent socializing can be accounted for by the different schools.

Part C.
When the average time spent socializing at three schools was analyzed, it was found that there is a difference in average times based on the school a student attends. That is, some schools have a higher average socialization time than others. This effect remained even when taking chance into account. What's more, knowing which school the student attends accounts for around 25% of the variation in socialization time, which means it's a pretty important factor.