Three universities have decided to administer the same comprehensive examination on their students. From each institution, a random sample of undergraduate students have been selected and given the test. The following table shows the scores of the students from each university
University UpNorth University @Middle University DownSouth
56 62 94
85 97 72
65 91 93
86 82 78
93 54
77
At the .01 significance level, test to see if there is any significant difference in the average scores of the students from the three universities.
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To test if there is any significant difference in the average scores of the students from the three universities, we can use a one-way analysis of variance (ANOVA).
The null hypothesis (H0) is that there is no significant difference in the average scores of the students from the three universities. The alternative hypothesis (H1) is that there is a significant difference in the average scores of the students from the three universities.
The steps to perform the ANOVA test are as follows:
1. Calculate the sum of squares between groups (SSB), the sum of squares within groups (SSW), and the total sum of squares (SST).
2. Calculate the degrees of freedom between groups (dfB), the degrees of freedom within groups (dfW), and the total degrees of freedom (dfT).
3. Calculate the mean square between groups (MSB) and the mean square within groups (MSW).
4. Calculate the F statistic (F = MSB/MSW).
5. Compare the calculated F statistic with the critical F value from the F distribution table at the .01 significance level.
If the calculated F statistic is greater than the critical F value, we reject the null hypothesis and conclude that there is a significant difference in the average scores of the students from the three universities. If the calculated F statistic is less than or equal to the critical F value, we do not reject the null hypothesis and conclude that there is no significant difference in the average scores of the students from the three universities.
Note: The actual calculations require statistical software or a calculator capable of performing ANOVA. The data provided is not sufficient to perform these calculations manually.
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