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To determine if there are significant differences in extroversion levels among students in different majors, we can perform a one-way ANOVA (Analysis of Variance). Here’s a step-by-step solution:
### Step 1: State the Hypotheses
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean extroversion scores are equal across all majors.
[tex]\[ H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): At least one mean extroversion score is different.
[tex]\[ H_1: \text{At least one } \mu \text{ is different} \][/tex]
### Step 2: Extract Data and Given Values
We have the means and sample sizes for each major:
- English: [tex]\( \bar{X}_1 = 3.67, n_1 = 60 \)[/tex]
- History: [tex]\( \bar{X}_2 = 3.02, n_2 = 44 \)[/tex]
- Psychology: [tex]\( \bar{X}_3 = 5.10, n_3 = 21 \)[/tex]
- Math: [tex]\( \bar{X}_4 = 3.47, n_4 = 60 \)[/tex]
Additionally, the sum of squares between groups ([tex]\( SS_{B} \)[/tex]) and within groups ([tex]\( SS_{W} \)[/tex]):
- [tex]\( SS_{B} = 62.92 \)[/tex]
- [tex]\( SS_{W} = 1943.06 \)[/tex]
Significance level:
- [tex]\( \alpha = 0.05 \)[/tex]
### Step 3: Calculate Degrees of Freedom
- Degrees of Freedom Between Groups ([tex]\( DF_{B} \)[/tex]):
[tex]\[ DF_{B} = k - 1 \][/tex]
where [tex]\( k \)[/tex] is the number of groups.
[tex]\[ DF_{B} = 4 - 1 = 3 \][/tex]
- Degrees of Freedom Within Groups ([tex]\( DF_{W} \)[/tex]):
[tex]\[ DF_{W} = N - k \][/tex]
where [tex]\( N \)[/tex] is the total number of observations.
[tex]\[ N = n_1 + n_2 + n_3 + n_4 \][/tex]
[tex]\[ N = 60 + 44 + 21 + 60 = 185 \][/tex]
[tex]\[ DF_{W} = 185 - 4 = 181 \][/tex]
### Step 4: Calculate Mean Squares
- Mean Square Between Groups ([tex]\( MS_{B} \)[/tex]):
[tex]\[ MS_{B} = \frac{SS_{B}}{DF_{B}} \][/tex]
[tex]\[ MS_{B} = \frac{62.92}{3} \approx 20.973 \][/tex]
- Mean Square Within Groups ([tex]\( MS_{W} \)[/tex]):
[tex]\[ MS_{W} = \frac{SS_{W}}{DF_{W}} \][/tex]
[tex]\[ MS_{W} = \frac{1943.06}{181} \approx 10.735 \][/tex]
### Step 5: Calculate the F-Statistic
[tex]\[ F = \frac{MS_{B}}{MS_{W}} \][/tex]
[tex]\[ F = \frac{20.973}{10.735} \approx 1.954 \][/tex]
### Step 6: Determine the Critical F-Value
Using the F-distribution table or statistical software, we find the critical F-value for [tex]\( DF_{B} = 3 \)[/tex] and [tex]\( DF_{W} = 181 \)[/tex] at [tex]\( \alpha = 0.05 \)[/tex]:
[tex]\[ F_{\text{critical}} \approx 2.655 \][/tex]
### Step 7: Compare and Conclude
- If [tex]\( F \leq F_{\text{critical}} \)[/tex], we fail to reject the null hypothesis.
- If [tex]\( F > F_{\text{critical}} \)[/tex], we reject the null hypothesis.
Since [tex]\( 1.954 < 2.655 \)[/tex], we do not reject the null hypothesis.
### Conclusion
There is not enough evidence to suggest that there are significant differences in extroversion scores among students in the different majors at the 5% significance level. The mean extroversion scores of students across English, History, Psychology, and Math majors are considered statistically similar.
### Step 1: State the Hypotheses
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean extroversion scores are equal across all majors.
[tex]\[ H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): At least one mean extroversion score is different.
[tex]\[ H_1: \text{At least one } \mu \text{ is different} \][/tex]
### Step 2: Extract Data and Given Values
We have the means and sample sizes for each major:
- English: [tex]\( \bar{X}_1 = 3.67, n_1 = 60 \)[/tex]
- History: [tex]\( \bar{X}_2 = 3.02, n_2 = 44 \)[/tex]
- Psychology: [tex]\( \bar{X}_3 = 5.10, n_3 = 21 \)[/tex]
- Math: [tex]\( \bar{X}_4 = 3.47, n_4 = 60 \)[/tex]
Additionally, the sum of squares between groups ([tex]\( SS_{B} \)[/tex]) and within groups ([tex]\( SS_{W} \)[/tex]):
- [tex]\( SS_{B} = 62.92 \)[/tex]
- [tex]\( SS_{W} = 1943.06 \)[/tex]
Significance level:
- [tex]\( \alpha = 0.05 \)[/tex]
### Step 3: Calculate Degrees of Freedom
- Degrees of Freedom Between Groups ([tex]\( DF_{B} \)[/tex]):
[tex]\[ DF_{B} = k - 1 \][/tex]
where [tex]\( k \)[/tex] is the number of groups.
[tex]\[ DF_{B} = 4 - 1 = 3 \][/tex]
- Degrees of Freedom Within Groups ([tex]\( DF_{W} \)[/tex]):
[tex]\[ DF_{W} = N - k \][/tex]
where [tex]\( N \)[/tex] is the total number of observations.
[tex]\[ N = n_1 + n_2 + n_3 + n_4 \][/tex]
[tex]\[ N = 60 + 44 + 21 + 60 = 185 \][/tex]
[tex]\[ DF_{W} = 185 - 4 = 181 \][/tex]
### Step 4: Calculate Mean Squares
- Mean Square Between Groups ([tex]\( MS_{B} \)[/tex]):
[tex]\[ MS_{B} = \frac{SS_{B}}{DF_{B}} \][/tex]
[tex]\[ MS_{B} = \frac{62.92}{3} \approx 20.973 \][/tex]
- Mean Square Within Groups ([tex]\( MS_{W} \)[/tex]):
[tex]\[ MS_{W} = \frac{SS_{W}}{DF_{W}} \][/tex]
[tex]\[ MS_{W} = \frac{1943.06}{181} \approx 10.735 \][/tex]
### Step 5: Calculate the F-Statistic
[tex]\[ F = \frac{MS_{B}}{MS_{W}} \][/tex]
[tex]\[ F = \frac{20.973}{10.735} \approx 1.954 \][/tex]
### Step 6: Determine the Critical F-Value
Using the F-distribution table or statistical software, we find the critical F-value for [tex]\( DF_{B} = 3 \)[/tex] and [tex]\( DF_{W} = 181 \)[/tex] at [tex]\( \alpha = 0.05 \)[/tex]:
[tex]\[ F_{\text{critical}} \approx 2.655 \][/tex]
### Step 7: Compare and Conclude
- If [tex]\( F \leq F_{\text{critical}} \)[/tex], we fail to reject the null hypothesis.
- If [tex]\( F > F_{\text{critical}} \)[/tex], we reject the null hypothesis.
Since [tex]\( 1.954 < 2.655 \)[/tex], we do not reject the null hypothesis.
### Conclusion
There is not enough evidence to suggest that there are significant differences in extroversion scores among students in the different majors at the 5% significance level. The mean extroversion scores of students across English, History, Psychology, and Math majors are considered statistically similar.
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