In a study to examine the effect of school size on student performance the follo

Question

In a study to examine the effect of school size on student performance the following model was formulated relating the percentage of students receiving a passing score on a mathematics test (math) to the average annual teacher total compensation (salary plus benefits, measured in $s) (totcomp), the number of staff in the school per one thousand pupils (staff) and school size, measured by student enrolment (enrol); u represents a generic stochastic error term. 1. math-A+ A totcompt ?2 staff + ?3 errol + u The following Stata results were obtained using data for 408 schools. . reg math totcomp staff enrol Number of obs F 3, 404) Prob >E R-squared Adj R-squared0.0470 Root MSE 408 7.70 0.0001 0.0541 Source I df MS 3 807.644779 Residual 42394.2462 404 104.936253 Model 2422.93434 Total 44817.1805 407 110.115923 - 10.244 Math I Coef Std. Err (95% Conf . Interval] 0006559 .1261884 0002255 14.29284 totcomp0004586 0001004 039814 staff 0479199 enrol-.0001976 .0002152 .57 0.00 1.20 0.229 -0.92 0.359 0.37 0.710 0002613 .0303487 .0006207 -9.744801 cons 2.274021 6.113794 Interpret the estimated slope coefficients in the fitted model. Do they have the expected signs? Based on the reported t-values, are the slope coefficients statistically significant? Are they economically (quantitatively) significant? (a) 10 marks Interpret the reported p-value (0.359) for enrol. How would you use this p-value if you were interested in testing Ho: ? = 0 against a one-sided alternative, H?:?

Explanation / Answer

Part (a)

In general, the slope coefficients represent the expected change (+ for increase and – for decrease ) in the predicted (dependent) variable per unit change in the respective predictor (independent) variable.

Thus, from the Stata results summary we conclude:

i). Per dollar increase in average annual teacher compensation, the test score is expected to go up by just 0.0004586 ANSWER 1

ii). When number of staff per 1000 pupil increase by 1, the expected increase in test score is just 0.0479199 ANSWER 2

iii) For every increase of 1 enrolment, the test score is likely to dip by 0.0001976 ANSWER 3

Yes; all the slope coefficients have expected signs ANSWER 4 since

i) higher pay normally invites quality faculty which is likely to enhance student knowledge which in turn would hike up the test score,

ii) more the staff more the attention and care students get which will have a positive impact on their performance,

iii) more the enrolment less the attention and care per student and that will have a negative impact on the student’s performance.

Only ?1 has a high t-value of 4.57 which is significant and consequently has a very low (virtually zero) p-value. Other two coefficients have insignificant t-values and accompanied by high p-value. Thus, only teacher monetary compensation is statistically significant. ANSWER 5

Note the underline above. ?1 esimate is just 0.0004586 implying that the annual monetary compensation must be up $2500 to increase test score by 1, making it not all practically significant. ANSWER 6

Part (b)

A p-value of 0.359 for ‘enrol’ implies that the probability of the obtained t-value under the hypothesis of ?3 = 0 is quite high [A probability value around the level of significance of 1%, 5% or even 10% is considered low.] Thus, it is only very likely that ?3 = 0 inplying that contribution of enrolment is not at all significant. ANSWER 7

Part (c)

95% CI for ?3 is reported to be: [- 0.0006207, 0.0002255] which includes 0.0005 implying that the hypothesis ?3 = 0.0005 is acceptable at 5% level of significance. ANSWER 8

Part (d)

F 3, 404 is the F-statistic for testing the hypothesis that the stipulated regression model. Its value is reported in the ANOVA result as 7.7 which is quite significant. Note that the corresponding p-value of 0.0001 is quite low which again is a confirmation that the F-value is significant. This should imply that the stipulated model is NOT statistically adequate. ANSWER 9

The corresponding hypotheses are: Null H0: Regression equation is adequate Vs Alternative HA: Regression equation is NOT adequate. ANSWER 10

DONE

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