Question

A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course. The default level of significance is .05 in excel. Use the default level of significance.

Student	Hours Studying	Final Grade
1	42	92
2	58	95
3	32	81
4	39	78
5	37	75
6	51	88
7	49	85
8	45	85

1.  Is β₁ statistically significant? State the null and alternative hypothesis, the t-test, p-value, and your decision.

2. Using the F test, test the overall fit of the model. You must state the null and alternative hypothesis, F-test, significance F, and decision.

3. What is the predicted final grade if a student studied 38 hours?

4. How much does final grade change if a student studied an additional 1.5 hours? HINT: Find the marginal effect on y for a 1-unit change in x.

5. In terms of this problem, how would you interpret the intercept – assuming the intercept is statistically significant? Interpretations are always relative to the data - in this case student final grade and hours of study.

6.If the regression output p-value was .01 and significance F was .01 would that change your conclusion in Question #7 and Question #8 above? Why or why not?

7. From your regression output, reference the 95% confidence intervals for β₁ . Would a student have more incentive to study 1 additional hour if β₁ = .08 or if β₁ = 1? Why? HINT: Find the marginal effect using the 2 different β₁ and compare the change in y.

8.Hubert, a basket weaving major, would like to use your model to predict his final grade. Hubert tells you that he studies 25 hours. Should you use your model to predict Hubert’s final grade? Why or why not?

Answer 1

1.  Is β₁ statistically significant? State the null and alternative hypothesis, the t-test, p-value, and your decision.

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	58.00609	9.755659	5.945891	0.001011	34.13485	81.87732	34.13485	81.87732
Hours studied	0.608927	0.217674	2.797424	0.03127	0.076297	1.141557	0.076297	1.141557

Null hypothesis:

Alternate Hypothesis:

Level of significance:

Test statistic:

p-value =0.0312. Since the p-value <0.05, we reject the Null significance and conclude that the parameter  β_{1 is significant in hypothesis model and cannot be dropped.}

2. Using the F test, test the overall fit of the model. You must state the null and alternative hypothesis, F-test, significance F, and decision.

Null Hypothesis:

The model with no independent variables fits the data as well as the model(the liner regression is absent)

The model fits the data better than the intercept-only model.(Liner regression is significant.

Level of significance:

The test statistic:

F-test based on the ANOVA table:

ANOVA
	df	SS	MS	F	Significance F
Regression	1	182.7543	182.7543	7.825578	0.03127
Residual	6	140.1207	23.35345
Total	7	322.875

The p-value or significance F is 0.0313<0.05, we reject the null hypothesis and conclude that the liner regression fits the data well.

3. What is the predicted final grade if a student studied 38 hours?

The fitted equation to the data is final greade=58.0061+0.6089*hours studies.

The predicted score for 38 hours: final grade=58.0061+0.6089*38

final grade=81.1443

4. How much does final grade change if a student studied an additional 1.5 hours? HINT: Find the marginal effect on y for a 1-unit change in x.

The regression coefficient in the equation is 0.6089. This is the increase in the final grade for every hour studied. Therefore for 1.5 hours, the change in grade is 1.5*0.6089=0.9134

5. In terms of this problem, how would you interpret the intercept – assuming the intercept is statistically significant? Interpretations are always relative to the data - in this case student final grade and hours of study.

Here the intercept is 58.0061. This is the score one can expect when no hours of study are put.

6.If the regression output p-value was .01 and significance F was .01 would that change your conclusion in Question #7 and Question #8 above? Why or why not?

If the significance F-was changed to 0.01, then the conclusions will reverse since the p-value of t and F is 0.0313>0.01 hence the decision will be reversed.

7. From your regression output, reference the 95% confidence intervals for β₁ . Would a student have more incentive to study 1 additional hour if β₁ = .08 or if β₁ = 1? Why? HINT: Find the marginal effect using the 2 different β₁ and compare the change in y.

The students will have more incentive when is more. Since this value gives the marginal increase in scores and hence the higher values of will give more incentives to the student.

For the if , the score as per the model is 58.0861 and if the score is 58.58.615 for an additional hour of study.

Therefore when the value is more than 1, it gives more incentives.

8.Hubert, a basket weaving major, would like to use your model to predict his final grade. Hubert tells you that he studies 25 hours. Should you use your model to predict Hubert’s final grade? Why or why not?

For this study,the scores predicted is Final score=58.0061+0.6089*25=73.2286. This model may not give a good prediction as we know that spending 25 hours is a great effort and the scores predicted seems to be less unless Hubert is not an intelligent student.