Answer :
Sure, let's go through the steps to test if the mean percentage of plagiarized words is different across the five semesters using ANOVA (Analysis of Variance).
### Step 1: State the Hypotheses
- Null Hypothesis (H₀): The mean percentage of plagiarized words is the same for all five semesters.
- Alternative Hypothesis (H₁): At least one semester has a different mean percentage of plagiarized words compared to the others.
### Step 2: Gather the Data
Given data:
- Sample sizes ([tex]\(n\)[/tex]): [36, 41, 31, 31, 33]
- Sample means: [6.33, 3.33, 1.77, 1.81, 1.50]
- Sample standard deviations: [3.75, 3.09, 3.25, 3.12, 2.38]
### Step 3: Calculate the Overall Mean
The overall mean percentage of plagiarized words is calculated by averaging the sample means:
[tex]\[ \text{Overall Mean} = \frac{6.33 + 3.33 + 1.77 + 1.81 + 1.50}{5} = 2.948 \][/tex]
### Step 4: Calculate SSB (Sum of Squares Between Groups)
The formula for SSB is:
[tex]\[ SSB = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2 \][/tex]
Where [tex]\( n_i \)[/tex] is the sample size of the [tex]\(i\)[/tex]-th group, [tex]\( \bar{X}_i \)[/tex] is the mean of the [tex]\(i\)[/tex]-th group, and [tex]\( \bar{X} \)[/tex] is the overall mean.
[tex]\[ SSB = 36(6.33 - 2.948)^2 + 41(3.33 - 2.948)^2 + 31(1.77 - 2.948)^2 + 31(1.81 - 2.948)^2 + 33(1.50 - 2.948)^2 = 570.1 \][/tex]
### Step 5: Calculate SSW (Sum of Squares Within Groups)
The formula for SSW is:
[tex]\[ SSW = \sum_{i=1}^{k} (n_i - 1) s_i^2 \][/tex]
Where [tex]\(s_i\)[/tex] is the standard deviation of the [tex]\(i\)[/tex]-th group.
[tex]\[ SSW = (36-1)3.75^2 + (41-1)3.09^2 + (31-1)3.25^2 + (31-1)3.12^2 + (33-1)2.38^2 = 1664.28 \][/tex]
### Step 6: Calculate Degrees of Freedom
- Between Groups ([tex]\( df_{B} \)[/tex]): [tex]\(k - 1 = 5 - 1 = 4\)[/tex]
- Within Groups ([tex]\( df_{W} \)[/tex]): [tex]\( \sum (n_i - 1) \)[/tex]
[tex]\[ df_{W} = (36-1) + (41-1) + (31-1) + (31-1) + (33-1) = 171 \][/tex]
### Step 7: Calculate Mean Squares
- Mean Square Between Groups ([tex]\( MSB \)[/tex]): [tex]\( \frac{SSB}{df_{B}} \)[/tex]
[tex]\[ MSB = \frac{570.1}{4} = 142.53 \][/tex]
- Mean Square Within Groups ([tex]\( MSW \)[/tex]): [tex]\( \frac{SSW}{df_{W}} \)[/tex]
[tex]\[ MSW = \frac{1664.28}{171} = 9.73 \][/tex]
### Step 8: Calculate the F-statistic
[tex]\[ F = \frac{MSB}{MSW} = \frac{142.53}{9.73} = 14.3 \][/tex]
### Step 9: Determine the p-value
Using the F-distribution and the degrees of freedom, dfb = 4 and dfw = 171, the p-value can be found using statistical software or F-tables.
[tex]\[ P\text{-value} \approx 0.0000 \][/tex]
### Step 10: Conclusion
At the [tex]\( \alpha = 0.05 \)[/tex] level of significance, we compare the p-value to [tex]\( \alpha \)[/tex]:
- Since [tex]\( p \approx 0.0000 < 0.05 \)[/tex], we reject the null hypothesis.
Conclusion: There is significant evidence to suggest that the mean percentage of plagiarized words is not the same for all five semesters. This indicates that the use of an online plagiarism detection system may have an effect on reducing plagiarism over time.
### Step 1: State the Hypotheses
- Null Hypothesis (H₀): The mean percentage of plagiarized words is the same for all five semesters.
- Alternative Hypothesis (H₁): At least one semester has a different mean percentage of plagiarized words compared to the others.
### Step 2: Gather the Data
Given data:
- Sample sizes ([tex]\(n\)[/tex]): [36, 41, 31, 31, 33]
- Sample means: [6.33, 3.33, 1.77, 1.81, 1.50]
- Sample standard deviations: [3.75, 3.09, 3.25, 3.12, 2.38]
### Step 3: Calculate the Overall Mean
The overall mean percentage of plagiarized words is calculated by averaging the sample means:
[tex]\[ \text{Overall Mean} = \frac{6.33 + 3.33 + 1.77 + 1.81 + 1.50}{5} = 2.948 \][/tex]
### Step 4: Calculate SSB (Sum of Squares Between Groups)
The formula for SSB is:
[tex]\[ SSB = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2 \][/tex]
Where [tex]\( n_i \)[/tex] is the sample size of the [tex]\(i\)[/tex]-th group, [tex]\( \bar{X}_i \)[/tex] is the mean of the [tex]\(i\)[/tex]-th group, and [tex]\( \bar{X} \)[/tex] is the overall mean.
[tex]\[ SSB = 36(6.33 - 2.948)^2 + 41(3.33 - 2.948)^2 + 31(1.77 - 2.948)^2 + 31(1.81 - 2.948)^2 + 33(1.50 - 2.948)^2 = 570.1 \][/tex]
### Step 5: Calculate SSW (Sum of Squares Within Groups)
The formula for SSW is:
[tex]\[ SSW = \sum_{i=1}^{k} (n_i - 1) s_i^2 \][/tex]
Where [tex]\(s_i\)[/tex] is the standard deviation of the [tex]\(i\)[/tex]-th group.
[tex]\[ SSW = (36-1)3.75^2 + (41-1)3.09^2 + (31-1)3.25^2 + (31-1)3.12^2 + (33-1)2.38^2 = 1664.28 \][/tex]
### Step 6: Calculate Degrees of Freedom
- Between Groups ([tex]\( df_{B} \)[/tex]): [tex]\(k - 1 = 5 - 1 = 4\)[/tex]
- Within Groups ([tex]\( df_{W} \)[/tex]): [tex]\( \sum (n_i - 1) \)[/tex]
[tex]\[ df_{W} = (36-1) + (41-1) + (31-1) + (31-1) + (33-1) = 171 \][/tex]
### Step 7: Calculate Mean Squares
- Mean Square Between Groups ([tex]\( MSB \)[/tex]): [tex]\( \frac{SSB}{df_{B}} \)[/tex]
[tex]\[ MSB = \frac{570.1}{4} = 142.53 \][/tex]
- Mean Square Within Groups ([tex]\( MSW \)[/tex]): [tex]\( \frac{SSW}{df_{W}} \)[/tex]
[tex]\[ MSW = \frac{1664.28}{171} = 9.73 \][/tex]
### Step 8: Calculate the F-statistic
[tex]\[ F = \frac{MSB}{MSW} = \frac{142.53}{9.73} = 14.3 \][/tex]
### Step 9: Determine the p-value
Using the F-distribution and the degrees of freedom, dfb = 4 and dfw = 171, the p-value can be found using statistical software or F-tables.
[tex]\[ P\text{-value} \approx 0.0000 \][/tex]
### Step 10: Conclusion
At the [tex]\( \alpha = 0.05 \)[/tex] level of significance, we compare the p-value to [tex]\( \alpha \)[/tex]:
- Since [tex]\( p \approx 0.0000 < 0.05 \)[/tex], we reject the null hypothesis.
Conclusion: There is significant evidence to suggest that the mean percentage of plagiarized words is not the same for all five semesters. This indicates that the use of an online plagiarism detection system may have an effect on reducing plagiarism over time.
