Question

A sample of 31 people took a written driver’s license exam. Two variables were measured on them: The result of the exam (0 = fail, 1 = pass), and how much time (in hours) the person studied for the exam. Using the data, fit an appropriate regression model to determine whether time spent studying is a useful predictor of the chance of passing the exam. Formally assess the overall fit of the model. Formally assess whether time spent studying is a useful predictor (as always, providing numerical justification (test statistic and P-value) for your conclusion). Carefully interpret what the estimated model tells you about how the chance of passing the exam changes as the time spent studying changes. A prospective examinee named Matthew spent 3.0 hours studying for his written exam. Estimate (with a point estimate and with a 90% interval) his probability of passing the exam. Based on this estimate, predict whether he will pass or fail.

SAS code:

DATA three;
INPUT result hours;
/* result=0 is fail; result=1 is pass */
cards;
0 0.8
0 1.6
0 1.4
1 2.3
1 1.4
1 3.2
0 0.3
1 1.7
0 1.8
1 2.7
0 0.6
0 1.1
1 2.1
1 2.8
1 3.4
1 3.6
0 1.7
1 0.9
1 2.2
1 3.1
0 1.4
1 1.9
0 0.4
0 1.6
1 2.5
1 3.2
1 1.7
1 1.9
0 2.2
0 1.3
1 1.5
;
run;

Answer 1

Here binary logistic regression model is to be applied.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.625077
R Square	0.390721
Adjusted R Square	0.369711
Standard Error	0.696174
Observations	31
ANOVA
	df	SS	MS	F	Significance F
Regression	1	9.013302	9.013302	18.59724	0.00017
Residual	29	14.05509	0.484658
Total	30	23.06839
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.246154	0.193084	6.45395	4.62E-07	0.851253	1.641055	0.851253	1.641055
X Variable 1	1.092735	0.253391	4.312451	0.00017	0.574493	1.610977	0.574493	1.610977

The log value stats are shown below

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.600745
R Square	0.360895
Adjusted R Square	0.338857
Standard Error	0.210159
Observations	31
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.723275	0.723275	16.37595	0.000352
Residual	29	1.28084	0.044167
Total	30	2.004115
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.034185	0.058288	0.586479	0.562091	-0.08503	0.153396	-0.08503	0.153396
X Variable 1	0.309546	0.076493	4.046721	0.000352	0.1531	0.465991	0.1531	0.465991

for upper 90%

	Upper 90.0%
Intercept	0.133223
X Variable 1	0.439517

So we have 1.7 hrs for Matthew as the minimum, at 90% he should pass at 3 hrs