a study was conducted in which the teachers were asked to rate students for particular trait on a ten point scale.with the help of data given below find out whether significant difference exists in the rating of the students by the teachers.
teachers students
A B C D E F G
X 7 3 7 1 5 5 5
Y 5 5 9 4 3 5 4
Z 6 3 7 1 3 5 3
To find out whether a significant difference exists in the rating of the students by the teachers, we can perform a statistical analysis using the data provided.
1. First, we need to choose an appropriate statistical test. Since we have multiple groups (teachers) and ratings are measured on a numerical scale, we can use a one-way analysis of variance (ANOVA) test.
2. Next, we need to calculate the mean ratings for each group (teacher). The mean rating for each teacher can be calculated by summing up the ratings for each student and dividing by the number of students.
Teachers A: (7 + 3 + 7 + 1 + 5 + 5 + 5) / 7 = 4.57
Teachers B: (5 + 5 + 9 + 4 + 3 + 5 + 4) / 7 = 5.14
Teachers C: (6 + 3 + 7 + 1 + 3 + 5 + 3) / 7 = 3.86
3. We also need to calculate the overall mean rating across all teachers. This is done by summing up all the ratings and dividing by the total number of ratings.
Overall mean: (7 + 3 + 7 + 1 + 5 + 5 + 5 + 5 + 5 + 9 + 4 + 3 + 5 + 4 + 6 + 3 + 7 + 1 + 3 + 5 + 3) / (7 + 7 + 7) = 4.57
4. Now, we can calculate the sum of squares within groups (SSW) and sum of squares between groups (SSB) which are required for the ANOVA test.
SSW can be calculated by summing up the squared differences between each individual rating and its corresponding group mean, for all groups.
SSB can be calculated by summing up the squared differences between each group mean and the overall mean, multiplied by the number of ratings in each group.
5. After calculating SSW and SSB, we can then calculate the degrees of freedom (df) for both SSW and SSB. The df for SSW is (total number of ratings - total number of groups), and the df for SSB is (number of groups - 1).
6. We can then calculate the mean squares within groups (MSW) by dividing SSW by the respective df.
7. Similarly, we can calculate the mean squares between groups (MSB) by dividing SSB by the respective df.
8. Finally, we can calculate the F-statistic by dividing MSB by MSW. We can then compare this value to a critical value from an F-distribution table with the appropriate degrees of freedom.
9. If the calculated F-value is greater than the critical value, we can conclude that a significant difference exists in the rating of the students by the teachers. Otherwise, there is no significant difference.
Performing these calculations, you should be able to determine whether a significant difference exists in the ratings of the students by the teachers based on the data provided.