Question

This is the final grade and number of absences for a set of students. Regress grade on absences. Use a 95% confidence level. Give the equation of estimation. Interpret the equation. According to the regression, now much does a tardy (= 1/2 an absence) change your grade? Evaluate the model. What evaluation criterion did you use? Could this be a case of reverse causality? If so, give an example of how the causation could run in the opposite direction.

Student	Grade	Absences
1	57	6
2	87.2	0
3	87.6	2.5
4	66.2	6
5	94.2	1
6	96.1	0
7	74.8	2.5
8	86.6	0
9	74.6	4.5
10	90.7	1
11	85.5	1
12	83.4	2.5
13	92.8	1
14	76.7	4
15	78.9	1.5
16	84.6	0
17	84.7	1.5
18	86.3	2.5
19	95.7	0
20	95.3	2
21	87.9	0
22	84.7	0
23	81.6	2
24	70.5	5.5
25	76.7	1
26	90.1	0
27	95.1	1
28	98.2	0
29	66.5	4
30	87.1	0
31	69.8	4.5
32	77.2	2
33	81	0.5
34	76.6	0
35	84.2	0
36	79.1	1.5
37	84.5	3.5
38	71.4	2.5
39	68.3	5
40	92.2	0
41	69.2	5

Answer 1

Regressing Grade on Absences in Excel (go to Data tab -> Data Analysis -> Regression, and choose Grade column as Y-values and Absences column as X-values, Confidence level as 95%) gives us the following output:

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	89.7151986	1.36692369	65.63292396	1.53875E-41	86.95033447	92.48006274
Absences	-3.968040552	0.511113307	-7.763524244	1.96325E-09	-5.001864797	-2.934216306

Equation of estimation obtained using the coefficients above:

Grade = -3.968 * Absences + 89.715

The equation suggests that when there are no absences (Absences=0), the projected Grade is 89.715.

However, the Grade comes down by 3.968 points for every Absence day. Hence, Grade and Absence are negatively correlated.

A tardy (1/2 an absence) brings down the grade by 3.968 * 0.5 = 1.984 points

Looking at the p-value of 1.96325E-09, which is << 0.01, we can say that absences are a very significant predictor of Grades at 99% confidence level. Hence, the model is a very nice estimate of the Grades in terms of Absences.

The model was evaluated as above using the p-value of slope (absences) coefficient, which is much less than the critical p-value, hence we can reject the null hypothesis that the Grades and Absences are uncorrelated ( = 0), at 99% confidence level, in favour of the alternative hypothesis ( != 0).

This could as well be a case of Reverse Causality, where lower Grades lead to more Absences (for example, because of lack of interest in such students in attending classes, owing to lower grades in earlier exams).