Task 1 (Boxplot and t tests)
1(a) The relevant boxplot is shown below.
The relevant five number summary is indicated below.
It is apparent from the shape of both the boxplots that the GPA distribution for neither of the genders is symmetric and hence there is skew involved. This is also confirmed from the fact that the presence of outliers on both ends for both genders since there are certain students who tend to perform very well and very miserably. Owing to the skewed nature of the data, the median in both cases is the appropriate measure of central tendency with IQR (Inter-quartile range) being the suitable measure for variation.
(b) The requisite hypotheses are as highlighted below.
Null Hypothesis: Average GPA of males does not significantly deviate from average GPA of females
Alternative Hypothesis: Average GPA of males does significantly deviate from average GPA of females
Based on the alternative hypothesis, it is apparent that thee given test would be a two tail test. Also, the relevant test statistics would be T since the population standard deviation is not known for either of the two genders. Further, the two samples are independent, hence 2 sample independent t test has been deployed whose excel output is highlighted as follows.
Two tail p value is 0.1972 which is greater than the level of significance (0.05). Hence, the available evidence is not sufficient to reject the null hypothesis and accept the alternative hypothesis. Therefore, no significant difference exists in the average GPA of the two genders.
2(a) The requisite hypotheses are as highlighted below.
Null Hypothesis: µPG=µUG which implies that no significant difference exists between the average GPA of students having post graduate parents and under graduate parents.
Alternate Hypothesis: µPG>µUG which implies that average GPA of students having post graduate parents is higher than the corresponding average GPA for students with under graduate parents.
Based on the alternative hypothesis, it is apparent that thee given test would be a one tail test. Also, the relevant test statistics would be T since the population standard deviation is not known for either of the two groups of students. Further, the two samples are independent, hence 2 sample independent t test has been deployed whose excel output is highlighted as follows.
One tail p value is 0.1972 which is greater than the level of significance (0.05). Hence, the available evidence is not sufficient to reject the null hypothesis and accept the alternative hypothesis. Therefore, no significant difference exists in the average GPA of students with post graduate parents and under graduate parents.
(b) The requisite hypotheses are as highlighted below.
Null Hypothesis: µUG=µS which implies that no significant difference exists between the average GPA of students having under graduate parents and secondary & below qualification parents
Alternate Hypothesis: µUG>µS which implies that average GPA of students having under graduate parents is higher than the corresponding average GPA for students with secondary & below qualification parents.
Based on the alternative hypothesis, it is apparent that thee given test would be a one tail test. Also, the relevant test statistics would be T since the population standard deviation is not known for either of the two groups of students. Further, the two samples are independent, hence 2 sample independent t test has been deployed whose excel output is highlighted as follows.
One tail p value is 0.000 which is lower than the level of significance (0.05). Hence, the available evidence is sufficient to reject the null hypothesis and accept the alternative hypothesis. Therefore, the average GPA of students with under graduate parents tends to exceed those students who have parents with secondary or less qualification.
Task 2: Regression Analysis
3) The requisite correlation matrix is indicated below.
From the above, it is apparent that GPA has the highest correlation with the maths score at high school Next in line is the ATAR score which also has a correlation coefficient in excess of 0.4. The lowest correlation amongst the given factors is witnessed for English scores at high school.
4) (i) HS_SCI is a predictor of GPA considering that there is moderate correlation between the variables. It is also confirmed from the regression output where the slope of this variable is found to be significant.
(ii) HS_ENG is a predictor of GPA considering that there is moderate correlation between the variables. It is also confirmed from the regression output where the slope of this variable is found to be significant.
(iii) HS_MATH is a predictor of GPA considering that there is moderate correlation between the variables. It is also confirmed from the regression output where the slope of this variable is found to be significant.
(iv) ATAR is a predictor of GPA considering that there is moderate correlation between the variables. It is also confirmed from the regression output where the slope of this variable is found to be significant.
5) Step 1 output is indicated below.
Step 2 output is indicated below.
Step 3 output is indicated below.
Step 4 output is indicated below.
Step 5 output is indicated below.
Step 6 output is indicated below.
6) The output for step 5 is indicated below.
Interpretation of Slopes
HS_SCI – The slope of this implies that a unit change in the science score at high school would change the GPA of the student by 0.08. The direction of change in both variables would be same.
HS_ENG – The slope of this implies that a unit change in the English score at high school would change the GPA of the student by 0.04. The direction of change in both variables would be same.
HS_ MATH – The slope of this implies that a unit change in the Math score at high school would change the GPA of the student by 0.25. The direction of change in both variables would be same.
PARENT EDUC – The slope of this implies that a unit change in the parent highest education qualification would change the GPA of the student by 0.28. The direction of change in both variables would be same.
GENDER – For a female student, the GPA would be higher by 0.11 in comparison with a male student assuming all other factors are same.
Significance of Slopes
Null Hypothesis: The slope is not significant and can be assumed to be zero.
Alternative Hypothesis: The slope is significant and hence cannot be assumed to be zero.
The relevant test is a two tail test with the test statistics of choice being T. The respective values of the various t statistics coupled with the p values have already been estimated in the regression model as indicated below.
The decision rule is that the slope of a given variable would be considered significant only if the corresponding p value would be lower than 0.05 (significance level). Considering the p value of the various slope coefficients, it is apparent that the p value corresponding to two variables namely PARENT EDUC and HS_MATH is lower than 0.05. Hence, the slopes of PARENT EDUC and HS_MATH are significant while the other slopes are not significant.
7) The negative coefficient of ATAR in step 6 does surprise me since it would be expected that students have higher value of ATAR would have a higher GPA. It is apparent that the p value corresponding to the slope of ATAR is 0.91 which is indicative of the fact that the slope of ATAR variable is not significant and can be assumed to be zero. Thus, it is apparent that the inclusion of ATAR does not improve the model fit and infact worsens the same which is apparent from the adjusted R square value.
Task 3: Summary Report
Based on the given sample data and suitable inferential test, it can be inferred that SES has a limited impact on the academic performance of students. It is essential that a minimum level of under graduation must be achieved by the parents as higher qualification does not improve performance but lower qualification does diminish academic performance of students. Also, it can be concluded that the academic performance of science students is not dependent on the underlying gender.
With regards to regression analysis, it is apparent that as one moves from Step to Step 3, progressively there is a decline in the slope coefficient attached to variable HS_SCI. With regards to slope significance, it is apparent that from Step to Step 3, the slope of HS_SCI moves from being significant to insignificance. If ATAR is considered as a standalone variable, then it is significant as explained in Task 2. However, when the other variables are already present, then ATAR does not improve the fit and is an insignificant variable.
Amongst the existing models, Step 4 would be the preferred choice since the predictive capability of this is the highest as indicated from the adjusted R2 value. However, if a new model may be constructed, then it would be best to have only two independent variables namely HS_MATH and PARENT_EDUC. Considering the Step 4 as the final model, the fit remains poor only since the independent variables collectively offer explanation to 23.22% of the total variations observed in GPA.
There is requirement to introduce more independent variables having a significant relationship with GPA while removing the insignificant variables. These new independent variables may include the IQ level of students, attendance in class, time spent on social media, amount of hours studied. These individual related variables would be able to enhance the predictability of GPA and hence need to be introduced.
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