Question 1195794: 9.A random sample of 24 management students in each batch was taken. The total (100%) scores for those students were recorded as follows.
1st year students 2nd year students 3rd year students
82 83 38
83 73 59
97 68 55
93 61 66
55 77 45
67 54 52
63 69 52
63 51 61
Required: At the 0.01 α, do the three batch students lead to different levels of score?
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. Set up Hypotheses**
* **Null Hypothesis (H0):** The mean scores of the three batches of students are equal. (μ1 = μ2 = μ3)
* **Alternative Hypothesis (H1):** At least one of the batch means is different from the others.
**2. Calculate Necessary Statistics**
* **Calculate the mean score for each batch:**
* 1st Year: (82 + 83 + 97 + 93 + 55 + 67 + 63 + 63) / 8 = 73.125
* 2nd Year: (83 + 73 + 68 + 61 + 77 + 54 + 69 + 51) / 8 = 65.75
* 3rd Year: (38 + 59 + 55 + 66 + 45 + 52 + 52 + 61) / 8 = 53.5
* **Calculate the overall mean:**
* (Sum of all scores) / (Total number of students)
* (Sum of all scores for 1st Year + Sum of all scores for 2nd Year + Sum of all scores for 3rd Year) / (24 students)
* **Calculate Sum of Squares Between Groups (SSB):**
* SSB = Σ(n_i * (x̄_i - x̄_grand)²), where:
* n_i is the sample size of each group (8 in this case)
* x̄_i is the mean of each group
* x̄_grand is the overall mean
* **Calculate Sum of Squares Within Groups (SSW):**
* SSW = Σ(Σ(x_ij - x̄_i)²), where:
* x_ij is the score of each individual student
* x̄_i is the mean of the group that student belongs to
* **Calculate Degrees of Freedom:**
* Between groups: df_between = k - 1 = 3 groups - 1 = 2
* Within groups: df_within = N - k = 72 - 3 = 69 (where N is the total number of students)
* **Calculate Mean Squares:**
* MSB = SSB / df_between
* MSW = SSW / df_within
* **Calculate F-statistic:**
* F = MSB / MSW
**3. Determine Critical Value**
* **Find the critical F-value** from the F-distribution table using:
* α = 0.01
* df_numerator = df_between = 2
* df_denominator = df_within = 69
**4. Decision**
* **Compare the calculated F-statistic to the critical F-value:**
* If F-statistic > F-critical, reject the null hypothesis (H0).
* If F-statistic ≤ F-critical, fail to reject the null hypothesis.
**Interpretation**
* If you reject the null hypothesis, it means there is sufficient evidence at the 0.01 significance level to conclude that there is a significant difference in the mean scores between at least two of the three student batches.
**Note:**
* This analysis requires the use of statistical software (like R, Python, or statistical packages like SPSS or Excel) to perform the calculations efficiently and accurately.
* The ANOVA test only tells you that there is a significant difference between groups. To determine which specific groups differ, you would need to perform post-hoc tests (such as Tukey's HSD).
**Let me know if you'd like help with the actual calculations using statistical software.**
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