Question

Question 1

Simon Dlamini is enrolled for a course in quantitative methods at his home university. Simon lives on the Atlantic seaboard and loves surfing. Every swell is a good reason not to study – and as a result he is not performing too well in his course.

The course is assessed via three assessments: a take home assignment (20%), a mid-term test (30%) and a final test (50%). He has already received his home assignment and mid-term test results and knows what he should get for his final test in order to pass the course. Simon managed to get his hand on the results of a previous class, and based on these results he would like to estimate what his final test mark could be. The results of the previous class are shown in the table below.

	Max = 20	Max = 30	Max = 50		Max = 20	Max = 30	Max = 50
Student no	Assignment	Midterm	Final test	Student no	Assignment	Midterm	Final test
1	12	11	25	18	14	20	44
2	15	28	45	19	11	20	43
3	13	19	37	20	13	15	32
4	19	26	38	21	15	10	26
5	20	22	40	22	19	30	48
6	11	13	22	23	17	16	38
7	15	10	32	24	13	24	29
8	10	11	30	25	18	20	34
9	13	16	27	26	10	16	27
10	14	15	34	27	12	10	25
11	19	16	40	28	11	18	30
12	17	16	28	29	12	13	25
13	10	12	24	30	14	15	20
14	12	22	26	31	15	22	40
15	10	13	29	32	12	12	24
16	16	23	45	33	18	21	33
17	13	28	46	34	14	23	37

1. Decide which linear regression model (with the assignment grade (A) or Midterm (M) or both as independent variable) is the better model to describe the final test score and motivate your selection.

The remaining answers must be based on the model that you have selected in a.

2. What is the total variation in final test grade that is explained by the independent variable(s) that you have selected?

3. Is the overall regression model statistically significant? Test at the 5% level of significance using the model that you have selected in a. For this test formulate the appropriate null and alternative hypothesis, determine the region of acceptance, use an appropriate test statistic and draw the statistical and management conclusions.

4. Write down the linear regression equation in algebraic format using A for the assignment grade, M for the midterm grade and F for the final test grade.

5. Compute Simon’s expected final test grade (out of 50) if he scored 12/20 for his assignment and 13/30 for his midterm test.

6. Compute the 95% confidence interval of the mean test grade for Simon (with his 12/20 assignment grade and 13/30 midterm test grade). Comment on his likelihood of passing the course.

Answer 1

I've used R software to do the analysis of this dataset.

R code and output:

a. Here we consider the "Final Test" as the dependent variable and "Mid Term" and "Assignment" as the independent variables to conduct the regression analysis.

b. Here Multiple R² values come out to be 0.5871, so 58.71% of the total variability is explained by the independent variables.

c. r²_F.MA is the multiple correlation coefficient of F on M and A.

We reject the null hypothesis if observed F statistic > F_2,31(0.05). [ at a 5% level of significance]

Here p-value = 1.111 x 10^-6 < 0.05, so we reject the null hypothesis and conclude on the basis of the given data that the overall regression model is significant.

d. The linear regression equation is given by

e. The Simson's expected final test grade is = 8.9686 + 0.5832 (12) + 0.8909 (13) = 27.55

f.

Hence, the CI is (25.06415 , 30.0332).