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 conduct a statistical test called the analysis of variance (ANOVA). ANOVA compares the means of three or more groups to determine if there are significant differences among them.
In this case, we have three groups (teachers A, B, and C) and seven students (A, B, C, D, E, F, and G). The ratings given by each teacher to each student are as follows:
Teachers:
A: 7 3 7 1 5 5 5
B: 5 5 9 4 3 5 4
C: 6 3 7 1 3 5 3
To conduct the ANOVA, we need to perform the following steps:
Step 1: Calculate the mean rating for each teacher:
The mean rating for each teacher can be calculated by adding up all the ratings given by that teacher and dividing it by the number of students. The mean for teachers A, B, and C are as follows:
Mean(A) = (7 + 3 + 7 + 1 + 5 + 5 + 5)/7
Mean(B) = (5 + 5 + 9 + 4 + 3 + 5 + 4)/7
Mean(C) = (6 + 3 + 7 + 1 + 3 + 5 + 3)/7
Step 2: Calculate the overall mean rating:
The overall mean rating is the average of all the ratings given by all the teachers. To calculate the overall mean, add up all the ratings and divide it by the total number of ratings.
Step 3: Calculate the sum of squares between groups (SSbetween):
SSbetween is a measure of how much the means of the groups (teachers) differ from each other. It is calculated by summing the squared difference between each group mean and the overall mean, multiplied by the number of ratings in each group.
Step 4: Calculate the sum of squares within groups (SSwithin):
SSwithin is a measure of the variability within each group. It is calculated by summing the squared difference between each individual rating and their respective group mean.
Step 5: Calculate the mean squares between groups (MSbetween):
MSbetween is obtained by dividing SSbetween by the degrees of freedom between groups, which is one less than the number of groups.
Step 6: Calculate the mean squares within groups (MSwithin):
MSwithin is obtained by dividing SSwithin by the degrees of freedom within groups, which is the total number of ratings minus the number of groups.
Step 7: Calculate the F-statistic:
The F-statistic is calculated by dividing the mean squares between groups (MSbetween) by the mean squares within groups (MSwithin).
Step 8: Compare the F-statistic with the critical F-value:
The F-statistic is compared with the critical F-value at a certain significance level (e.g., 0.05). If the calculated F-statistic is greater than the critical F-value, then there is a significant difference in the ratings by the teachers.
By following these steps and calculating the necessary values, you can determine whether a significant difference exists in the rating of the students by the teachers.
To determine whether a significant difference exists in the rating of the students by the teachers, we can conduct a statistical test called Analysis of Variance (ANOVA). ANOVA compares the means of multiple groups and determines if there is a significant difference between them.
Here are the steps to perform ANOVA:
Step 1: Formulate the null and alternative hypotheses:
- Null Hypothesis (H0): There is no significant difference in the rating of students by the teachers.
- Alternative Hypothesis (Ha): There is a significant difference in the rating of students by the teachers.
Step 2: Calculate the mean and variance for each group (teacher):
Teacher A:
Mean (X): (7 + 3 + 7 + 1 + 5 + 5 + 5) / 7 = 4.71
Variance (σ²): [(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] / 7 = 3.86
Teacher B:
Mean (X): (5 + 5 + 9 + 4 + 3 + 5 + 4) / 7 = 5.14
Variance (σ²): [(5 - 5.14)² + (5 - 5.14)² + (9 - 5.14)² + (4 - 5.14)² + (3 - 5.14)² + (5 - 5.14)² + (4 - 5.14)²] / 7 = 2
Teacher C:
Mean (X): (6 + 3 + 7 + 1 + 3 + 5 + 3) / 7 = 4
Variance (σ²): [(6 - 4)² + (3 - 4)² + (7 - 4)² + (1 - 4)² + (3 - 4)² + (5 - 4)² + (3 - 4)²] / 7 = 1.71
Step 3: Calculate the overall mean and total variation:
Overall Mean (X̄): (4.71 + 5.14 + 4) / 3 = 4.95
Total Variation (SST): [(7 - 4.95)² + (3 - 4.95)² + (7 - 4.95)² + (1 - 4.95)² + (5 - 4.95)² + (5 - 4.95)² + (5 - 4.95)²] + [(5 - 4.95)² + (5 - 4.95)² + (9 - 4.95)² + (4 - 4.95)² + (3 - 4.95)² + (5 - 4.95)² + (4 - 4.95)²] + [(6 - 4.95)² + (3 - 4.95)² + (7 - 4.95)² + (1 - 4.95)² + (3 - 4.95)² + (5 - 4.95)² + (3 - 4.95)²] = 38.86
Step 4: Calculate the between-group sum of squares (SSB):
SSB = (n1 * (X1 - X̄)²) + (n2 * (X2 - X̄)²) + (n3 * (X3 - X̄)²)
= (7 * (4.71 - 4.95)²) + (7 * (5.14 - 4.95)²) + (7 * (4 - 4.95)²)
= 0.378
Step 5: Calculate the within-group sum of squares (SSW):
SSW = (n1 - 1) * σ1² + (n2 - 1) * σ2² + (n3 - 1) * σ3²
= (7 - 1) * 3.86 + (7 - 1) * 2 + (7 - 1) * 1.71
= 79.8
Step 6: Calculate the F-statistic:
F-statistic = (SSB / dfB) / (SSW / dfW)
= (0.378 / 2) / (79.8 / 18)
= 0.189 / 4.433
= 0.0428
Step 7: Determine the critical value of F at a specific significance level (α) and degrees of freedom (dfB and dfW). Check if the calculated F-statistic is less than the critical value.
To determine the critical value, additional information is needed such as the sample size of each group and the significance level (α). Please provide those details and I can help you determine the critical value and whether there is a significant difference in the ratings of the students by the teachers.