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Part 1 Solution

- Taking into consideration the different sections of the students and their grades in final exam, the study tests whether there is the group mean scores for the three sections (1,2 and 3) are equal.
- H0: The group means scores are equal

H1: The groups mean scores are not equal

However, this can be rewritten as

H0: µ_{1 }= µ_{2 }= µ_{3}

H1: µ_{1} ≠ µ_{2 }≠ µ_{3}

- The appropriate statistical test that can be ran in this case is a one-way factor ANOVA (Analysis of Variance) to establish if there is any significant difference in the group means for the three sections.
- Results for the ANOVA test and the Post-hoc analysis

Table 1

*ANOVA test results*

ANOVA | |||||

Final Exam Points | |||||

Sum of Squares | df | Mean Square | F | Sig. | |

Between Groups | 183.752 | 2 | 91.876 | 1.469 | .235 |

Within Groups | 6378.438 | 102 | 62.534 | ||

Total | 6562.190 | 104 |

Table 2

*Post Hoc Tests*

Multiple Comparisons | ||||||

Dependent Variable: Final Exam Points Tukey HSD | ||||||

(I) Section of Class | (J) Section of Class | Mean Difference (I-J) | Std. Error | Sig. | 95% Confidence Interval | |

Lower Bound | Upper Bound | |||||

1 | 2 | 2.042 | 1.870 | .521 | -2.41 | 6.49 |

3 | 3.303 | 1.947 | .211 | -1.33 | 7.93 | |

2 | 1 | -2.042 | 1.870 | .521 | -6.49 | 2.41 |

3 | 1.261 | 1.870 | .779 | -3.19 | 5.71 | |

3 | 1 | -3.303 | 1.947 | .211 | -7.93 | 1.33 |

2 | -1.261 | 1.870 | .779 | -5.71 | 3.19 |

Interpretation and Conclusion

From the ANOVA results, it is clear that there was no statistical significance to claim that the group means were different from each other. In other words, there were no statistical evidence to claim group mean differences for the three sections since the P-value = 0.235. The F-value was given as (F(2, 102) = 1.469, P= 0.235). Such results indicate that, there was no need to carry out he Post Hoc test since the group means are the same. Accordingly, the Post-Hoc test is carried out where there is significance different between the groups – hence this test aims to identify where exactly the difference is. In other words, it tells the analyst which groups differed. Therefore, the Post-Hoc test should be run where the results are statistically significance showing statistical evidence in difference between group means. Also, the plot indicates that the group sizes are unequal, hence making use of the Harmonic mean to represent the sample size (Harmonic mean = 34.784). Therefore, this reveals that the groups are heterogeneous and each mean score for the final exam is given separately as shown in the SPSS output file attached. To conclude on this test, for the three section of classes (1, 2 and 3) there is no statistical evidence to claim that their mean final exam points are different.

Solution 2

- When comparing the ethnicities of students and their respective sections, the question is trying to find if the final course score is different among the three sections and across the different ethnicities
- Sets of hypothesis

Set 1

H0: The mean final course scores is equal among the three sections

H1: The mean final course scores is different among the three sections

Set 2

H0: The mean final course scores is equal among the different ethnicities

H1: The Mean final Course scores is different among the different ethnicities

- In this case, two-factor ANOVA is appropriate in this scenario as there are two sets of hypothesis.

Table 3

* Two-way ANOVA*

Tests of Between-Subjects Effects | |||||

Dependent Variable: Final Course Percent | |||||

Source | Type III Sum of Squares | df | Mean Square | F | Sig. |

Corrected Model | 2127.302^{a} | 13 | 163.639 | 1.129 | .346 |

Intercept | 301720.719 | 1 | 301720.719 | 2081.881 | .000 |

section | 141.678 | 2 | 70.839 | .489 | .615 |

ethnicit | 180.606 | 4 | 45.152 | .312 | .870 |

section * ethnicit | 920.876 | 7 | 131.554 | .908 | .504 |

Error | 13188.355 | 91 | 144.927 | ||

Total | 693088.000 | 105 | |||

Corrected Total | 15315.657 | 104 | |||

a. R Squared = .139 (Adjusted R Squared = .016) |

Table 4

*Post – Hoc Tests*

Tests of Between-Subjects Effects | |||||

Dependent Variable: Final Course Percent | |||||

Source | Type III Sum of Squares | df | Mean Square | F | Sig. |

Corrected Model | 2127.302^{a} | 13 | 163.639 | 1.129 | .346 |

Intercept | 301720.719 | 1 | 301720.719 | 2081.881 | .000 |

section | 141.678 | 2 | 70.839 | .489 | .615 |

ethnicit | 180.606 | 4 | 45.152 | .312 | .870 |

section * ethnicit | 920.876 | 7 | 131.554 | .908 | .504 |

Error | 13188.355 | 91 | 144.927 | ||

Total | 693088.000 | 105 | |||

Corrected Total | 15315.657 | 104 | |||

a. R Squared = .139 (Adjusted R Squared = .016) |

Table 5

Post Hoc tests for ethnicity

Multiple Comparisons | ||||||

Dependent Variable: Final Course Percent Tukey HSD | ||||||

(I) Ethnicity | (J) Ethnicity | Mean Difference (I-J) | Std. Error | Sig. | 95% Confidence Interval | |

Lower Bound | Upper Bound | |||||

AMERICAN INDIAN | ASIAN | -6.10 | 6.019 | .849 | -22.85 | 10.65 |

AFRO-AMERICAN | -3.88 | 5.918 | .965 | -20.35 | 12.59 | |

CAUCASIAN | -5.38 | 5.675 | .877 | -21.17 | 10.42 | |

HISPANIC | 2.02 | 6.493 | .998 | -16.05 | 20.09 | |

ASIAN | AMERICAN INDIAN | 6.10 | 6.019 | .849 | -10.65 | 22.85 |

AFRO-AMERICAN | 2.22 | 3.645 | .973 | -7.93 | 12.36 | |

CAUCASIAN | .72 | 3.235 | .999 | -8.28 | 9.73 | |

HISPANIC | 8.12 | 4.519 | .382 | -4.46 | 20.70 | |

AFRO-AMERICAN | AMERICAN INDIAN | 3.88 | 5.918 | .965 | -12.59 | 20.35 |

ASIAN | -2.22 | 3.645 | .973 | -12.36 | 7.93 | |

CAUCASIAN | -1.49 | 3.043 | .988 | -9.96 | 6.97 | |

HISPANIC | 5.90 | 4.383 | .663 | -6.30 | 18.10 | |

CAUCASIAN | AMERICAN INDIAN | 5.38 | 5.675 | .877 | -10.42 | 21.17 |

ASIAN | -.72 | 3.235 | .999 | -9.73 | 8.28 | |

AFRO-AMERICAN | 1.49 | 3.043 | .988 | -6.97 | 9.96 | |

HISPANIC | 7.40 | 4.049 | .365 | -3.87 | 18.67 | |

HISPANIC | AMERICAN INDIAN | -2.02 | 6.493 | .998 | -20.09 | 16.05 |

ASIAN | -8.12 | 4.519 | .382 | -20.70 | 4.46 | |

AFRO-AMERICAN | -5.90 | 4.383 | .663 | -18.10 | 6.30 | |

CAUCASIAN | -7.40 | 4.049 | .365 | -18.67 | 3.87 | |

Based on observed means. The error term is Mean Square (Error) = 144.927. |

Interpretation and conclusion

From the above analysis, it is clear that there is no statistical evidence that the final course score is different among the sections and across the ethnicities. Accordingly, for the first set of hypothesis the F = 0.489 with a P-value of 0.615 > 0.05 indicating, hence the difference between the means are not statistically significant – accept the null hypothesis. Similarly, for the second set of hypothesis, F = 0.312 with a P-value = 0.870 indicating that, the null hypothesis is accepted and claimed that there difference between group means is not statistically significant. Therefore, in each of these two tests, there is no need of carrying out a Post-Hoc tests as there is no evidence that either of the groups have their mean scores different from each other. Evidently, it can be concluded that the final course scores is not different across the ethnicities and class sections.

Solution 3

- The two variables that the question is investigating are the Quiz 1 and the Total Points. In this case, the total points scored are influenced by what the students scored in Quiz 1 indicating that, the Total points is the dependent variable while the QUIZ 1 is the independent variable.

Given the relationship between these two variables is given by a R-value of (0.823), this value is strong enough to make predictions of the total points scored from quiz 1 scores. It indicates that 82.3% of the total points scored are influenced by changes in the QUIZ 1 scores. In other words, if a simple regression model is formulated, it is easier to forecast future total points based on the QUIZ 1 scores. To add on this, 17.7% of the changes in the Total Points scored are influenced by other factors with an exception of the QUIZ 1 score.

- A regression equation for the above scenario is given as

*Total Points = 62.698 + 5.072(Quiz 1) + µ* (Where µ is the error term)

- Using the prediction equation above, assuming the error term has insignificant effect, a student scoring 6.5 in Quiz one is most likely to score:

Total Points = 62.698 + 5.072 * 6.5

Total Points = 95.666

Solution 4

- Correlation between Quiz 3 and Quiz 5 is given by r-value = 0.493 indicating a weak positive correlation which is statistically significant at alpha = 0.01 indicating that it can be used in making prediction.
- Regression analysis between Quiz 3 and Quiz 5 – coefficients are given as

Coefficients^{a} | ||||||

Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||

B | Std. Error | Beta | ||||

1 | (Constant) | 4.859 | .545 | 8.921 | .000 | |

Quiz 3 Points | .377 | .066 | .493 | 5.745 | .000 | |

a. Dependent Variable: Quiz 5 Points |

- A prediction equation between

The model assumes that the Quiz 5 scores are dependent on the quiz 3 scores, therefore, Quiz 5 scores is the dependent variable while Quiz 3 is the independent variable.

*Quiz 5 = 4.859 + 0.377 Quiz 3 + µ* (µ is error term)

- From the above equation, if a student scores 8.5 in Quiz 3;

Quiz 5 = 4.859 + 0.377 (8.5) + u (however, u is assumed to be zero in the model)

Quiz 5 = 8.0635 points

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