A study was conducted in which the teachers were asked to rate students for a 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 determine whether a significant difference exists in the rating of the students by the teachers, we can perform a statistical analysis called a one-way analysis of variance (ANOVA).
Here's how you can calculate the ANOVA for the given data:
Step 1: Calculate the mean for each group (teacher):
- For Teacher X: Sum of ratings = 7 + 3 + 7 + 1 + 5 + 5 + 5 = 33. Mean = 33 / 7 = 4.71
- For Teacher Y: Sum of ratings = 5 + 5 + 9 + 4 + 3 + 5 + 4 = 35. Mean = 35 / 7 = 5
- For Teacher Z: Sum of ratings = 6 + 3 + 7 + 1 + 3 + 5 + 3 = 28. Mean = 28 / 7 = 4
Step 2: Calculate the overall mean (grand mean):
- Sum of all ratings = 33 + 35 + 28 = 96.
- Total number of ratings = 7 + 7 + 7 = 21.
- Grand mean = 96 / 21 = 4.57
Step 3: Calculate the sum of squares (SS) for each group and the total:
- SS for Teacher X = (7 - 4.71)^2 + (3 - 4.71)^2 + (7 - 4.71)^2 + (1 - 4.71)^2 + (5 - 4.71)^2 + (5 - 4.71)^2 + (5 - 4.71)^2 = 16.76
- SS for Teacher Y = (5 - 5)^2 + (5 - 5)^2 + (9 - 5)^2 + (4 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (4 - 5)^2 = 14
- SS for Teacher Z = (6 - 4)^2 + (3 - 4)^2 + (7 - 4)^2 + (1 - 4)^2 + (3 - 4)^2 + (5 - 4)^2 + (3 - 4)^2 = 24
- Total sum of squares (SST) = SS for Teacher X + SS for Teacher Y + SS for Teacher Z = 16.76 + 14 + 24 = 54.76
Step 4: Calculate the sum of squares between groups (SSB):
- SSB = (7 * (4.71 - 4.57)^2) + (7 * (5 - 4.57)^2) + (7 * (4 - 4.57)^2) = 0.61
Step 5: Calculate the sum of squares within groups (SSW):
- SSW = SST - SSB = 54.76 - 0.61 = 54.15
Step 6: Calculate the degrees of freedom (df) for both SSB and SSW:
- dfSSB = Number of groups - 1 = 3 - 1 = 2
- dfSSW = Total number of ratings - Number of groups = 21 - 3 = 18
Step 7: Calculate the mean squares (MSB and MSW):
- MSB = SSB / dfSSB = 0.61 / 2 = 0.305
- MSW = SSW / dfSSW = 54.15 / 18 = 3.008
Step 8: Calculate the F-statistic (F):
- F = MSB / MSW = 0.305 / 3.008 = 0.101
Step 9: Determine the critical F-value:
- At a significance level of α = 0.05 and with dfSSB = 2 and dfSSW = 18, the critical F-value is 3.55.
Step 10: Compare the F-statistic to the critical F-value:
- If F > critical F-value, there is a significant difference in the ratings.
- If F < critical F-value, there is no significant difference in the ratings.
In this case, F = 0.101 and the critical F-value is 3.55. Since F < critical F-value, we can conclude that there is no significant difference in the rating of the students by the teachers.
Note: The calculations provided here are based on the assumption that the data meets the necessary assumptions for ANOVA (e.g., independence, normality, and homogeneity of variances).