Chapter 10, Problem 14E

(a)

To determine

To graph: A scatter plot of each explanatory variable with AvgDebt and to find whether these relationships look linear or not.

Explanation of Solution

Given: The data for the average debt, percentage of students who borrow, admittance rate, four-year graduation rate, in-state tuition after aid, out-of-state tuition after aid, average aid per student and the number of students per faculty member 40 colleges in US.

College	Admit	Yr4Grad	StudPerFac	InAfterAid	OutAfterAid	AvgAid	PercBorrow	AvgDebt
The University of Arizona, Tucson	75	34	20	9565	25775	9261	44	20074
Stony Brook University, State University of New York, Stony Brook	41	43	19	11193	20643	7275	59	19807
University of Illinois at Springfield, Springfield	59	50	14	11489	20639	8213	61	17335
State University of New York College at Oneonta, Oneonta	41	48	17	12649	21699	5311	75	17564
Iowa State University of Science and Technology, Ames	81	37	17	9738	21610	6413	69	30062
University of South Florida, Tampa	45	24	28	10598	19788	5966	52	21679
University of New Hampshire, Durham	73	61	18	21947	35267	3955	76	35671
University of North Florida, Jacksonville	51	20	21	9171	21922	5789	39	15300
University of South Carolina, Columbia	70	46	19	14117	30301	5027	49	21811
State University of New York College at Brockport, Brockport	42	42	18	12888	21938	5140	80	25733
The University of North Carolina at Chapel Hill, Chapel Hill	32	76	14	6548	26374	11080	31	16165
Towson University, Towson	57	40	17	11627	23139	7601	56	19069
University of California, Santa Barbara, Santa Barbara	46	64	17	12711	34521	15389	47	17596
St. Mary's College of Maryland, St. Mary's City	65	74	12	17696	29773	8664	70	17505
University of Georgia, Athens	59	54	19	10288	28498	8970	45	15938
Christopher Newport University, Newport News	60	42	17	15725	24947	5087	51	21572
The Ohio State University, Columbus	68	49	19	12680	27575	7789	59	22830
University of Delaware, Newark	54	62	12	15728	31998	6734	44	17200
California State Polytechnic University, Pomona, Pomona	45	20	26	10701	21861	8280	35	16704
University of Illinois at Urbana–Champaign, Champaign	67	67	16	14809	28951	10885	51	21543
Auburn University, Auburn University	79	36	18	13057	27649	6733	46	24777
The University of Alabama, Tuscaloosa	54	38	19	11584	24884	6680	53	26701
University of Wisconsin–Madison, Madison	58	51	17	12733	28483	5857	48	22837
The University of Texas at Dallas, Richardson	50	42	19	14068	31094	6664	41	17384
State University of New York at Binghamton, Binghamton	40	66	21	13310	21420	6716	50	21110
Rutgers, The State University of New Jersey, New Brunswick/Piscataway, Piscataway	59	53	15	15629	28291	9937	68	16766
University of Science and Arts of Oklahoma, Chickasha	56	18	14	4132	11092	7098	29	15339
Miami University, Oxford	79	68	17	19310	34482	6083	51	27315
University of Maryland, College Park, College Park	44	63	18	12450	29821	6959	43	22696
New College of Florida, Sarasota	53	57	10	5316	28345	10142	36	11458
The University of North Carolina at Asheville, Asheville	77	30	14	7443	21075	6202	48	15443
University of Mary Washington, Fredericksburg	77	68	15	12206	23934	6500	57	15600
University of California, Los Angeles, Los Angeles	23	68	17	15397	38275	12777	45	16733
University of Virginia, Charlottesville	33	85	16	5138	29460	16488	32	19384
University of California, Berkeley, Berkeley	22	69	17	13164	36042	15863	41	16056
University of Oklahoma, Norman	85	33	18	11417	22370	6011	52	21517
The University of North Carolina at Wilmington, Wilmington	57	45	17	7367	19188	6792	25	19277
University of Connecticut, Storrs	54	66	18	14403	31299	8167	63	23237
University of Michigan, Ann Arbor	51	72	15	13170	38842	10783	46	27828
University of Colorado at Boulder, Boulder	83	41	18	14549	35727	7873	43	19758

Graph: To draw the scattered plot of each explanatory variable with AvgDebt of the provided data, below mentioned steps are followed in Minitab.

Step 1: Enter the data into Minitab worksheet.

Step 2: Go to Graph, select Matrix Plot, select Each Y versus each X and select Simple.

Step 3: Select AvgDebt as Y variable and Admit, Yr4Grad, StudPerFac, InAfterAid, OutAfterAid, AvgAid, PercBorrow.

Step 4: Press “OK”.

The obtained scattered plot is as follows:

Interpretation: The obtained scattered plot shows that the relationship between AvgDebt and AvgAid is negative linear while with all other variable it is positively linear.

(b)

To determine

To find: The Curves which fit the each of the predictors separately and to create a table which lists the explanatory variable, model standard deviation $s$ and the P-value for the test of the linear association.

Answer to Problem 14E

Solution: The table that shows the models for each explanatory variable, model standard deviation $s$ and the P-value for the test of the linear association:

Explanatory variable	Model	Model standard deviation s	P-value
Admit,	$\text{AvgDebt}=1\text{4851 }+\text{ 96}\text{.4}\left(\text{Admit}\right)$	$\text{4556}\text{.91}$	$\text{0}\text{.039}$
Yr4Grad	$\text{AvgDebt}=\text{19582}+\text{14}\text{.4}\left(\text{Yr4Grad}\right)$	$\text{4816}\text{.64}$	$\text{0}\text{.754}$
StudPerFac	$\text{AvgDebt}=\text{14307}+\text{346}\left(\text{StudPerFac}\right)$	$\text{4683}\text{.84}$	$\text{0}\text{.138}$
InAfterAid	$\text{AvgDebt}=\text{12594}+\text{0}\text{.633}\left(\text{InAfterAid}\right)$	$\text{4209}\text{.25}$	$\text{0}\text{.001}$
OutAfterAid	$\text{AvgDebt}=\text{15713}+\text{0}\text{.170}\left(\text{OutAfterAid}\right)$	$\text{4709}\text{.49 }$	$\text{0}\text{.181}$
AvgAid	$\text{AvgDebt}=\text{24952}-\text{0}\text{.575}\left(\text{AvgAid}\right)$	$\text{4507}\text{.29}$	$\text{0}\text{.024}$
PercBorrow	$\text{AvgDebt}=\text{11818}+\text{169}\left(\text{PercBorrow}\right)$	$\text{4281}\text{.04 }$	$\text{0}\text{.003}$

Explanation of Solution

Given: The data for the average debt, percentage of students who borrow, admittance rate, four-year graduation rate, in-state tuition after aid, out-of-state tuition after aid, average aid per student and the number of students per faculty member 40 colleges in US.

College	Admit	Yr4Grad	StudPerFac	InAfterAid	OutAfterAid	AvgAid	PercBorrow	AvgDebt
The University of Arizona, Tucson	75	34	20	9565	25775	9261	44	20074
Stony Brook University, State University of New York, Stony Brook	41	43	19	11193	20643	7275	59	19807
University of Illinois at Springfield, Springfield	59	50	14	11489	20639	8213	61	17335
State University of New York College at Oneonta, Oneonta	41	48	17	12649	21699	5311	75	17564
Iowa State University of Science and Technology, Ames	81	37	17	9738	21610	6413	69	30062
University of South Florida, Tampa	45	24	28	10598	19788	5966	52	21679
University of New Hampshire, Durham	73	61	18	21947	35267	3955	76	35671
University of North Florida, Jacksonville	51	20	21	9171	21922	5789	39	15300
University of South Carolina, Columbia	70	46	19	14117	30301	5027	49	21811
State University of New York College at Brockport, Brockport	42	42	18	12888	21938	5140	80	25733
The University of North Carolina at Chapel Hill, Chapel Hill	32	76	14	6548	26374	11080	31	16165
Towson University, Towson	57	40	17	11627	23139	7601	56	19069
University of California, Santa Barbara, Santa Barbara	46	64	17	12711	34521	15389	47	17596
St. Mary's College of Maryland, St. Mary's City	65	74	12	17696	29773	8664	70	17505
University of Georgia, Athens	59	54	19	10288	28498	8970	45	15938
Christopher Newport University, Newport News	60	42	17	15725	24947	5087	51	21572
The Ohio State University, Columbus	68	49	19	12680	27575	7789	59	22830
University of Delaware, Newark	54	62	12	15728	31998	6734	44	17200
California State Polytechnic University, Pomona, Pomona	45	20	26	10701	21861	8280	35	16704
University of Illinois at Urbana–Champaign, Champaign	67	67	16	14809	28951	10885	51	21543
Auburn University, Auburn University	79	36	18	13057	27649	6733	46	24777
The University of Alabama, Tuscaloosa	54	38	19	11584	24884	6680	53	26701
University of Wisconsin–Madison, Madison	58	51	17	12733	28483	5857	48	22837
The University of Texas at Dallas, Richardson	50	42	19	14068	31094	6664	41	17384
State University of New York at Binghamton, Binghamton	40	66	21	13310	21420	6716	50	21110
Rutgers, The State University of New Jersey, New Brunswick/Piscataway, Piscataway	59	53	15	15629	28291	9937	68	16766
University of Science and Arts of Oklahoma, Chickasha	56	18	14	4132	11092	7098	29	15339
Miami University, Oxford	79	68	17	19310	34482	6083	51	27315
University of Maryland, College Park, College Park	44	63	18	12450	29821	6959	43	22696
New College of Florida, Sarasota	53	57	10	5316	28345	10142	36	11458
The University of North Carolina at Asheville, Asheville	77	30	14	7443	21075	6202	48	15443
University of Mary Washington, Fredericksburg	77	68	15	12206	23934	6500	57	15600
University of California, Los Angeles, Los Angeles	23	68	17	15397	38275	12777	45	16733
University of Virginia, Charlottesville	33	85	16	5138	29460	16488	32	19384
University of California, Berkeley, Berkeley	22	69	17	13164	36042	15863	41	16056
University of Oklahoma, Norman	85	33	18	11417	22370	6011	52	21517
The University of North Carolina at Wilmington, Wilmington	57	45	17	7367	19188	6792	25	19277
University of Connecticut, Storrs	54	66	18	14403	31299	8167	63	23237
University of Michigan, Ann Arbor	51	72	15	13170	38842	10783	46	27828
University of Colorado at Boulder, Boulder	83	41	18	14549	35727	7873	43	19758

Calculation: Use following steps to find the models in Minitab:

Step 1: Enter the data into Minitab worksheet.

Step 2: Go to Stat, select Regression, again select Regression.

Step 3: Select AvgDebt as Response and all Predictors one by one.

The models for each explanatory variable, model standard deviation $s$ and the P-value for the test of the linear association is shown in the below table:

Explanatory variable	Model	Model standard deviation s	P-value
Admit,	$\text{AvgDebt}=1\text{4851 }+\text{ 96}\text{.4}\left(\text{Admit}\right)$	$\text{4556}\text{.91}$	$\text{0}\text{.039}$
Yr4Grad	$\text{AvgDebt}=\text{19582}+\text{14}\text{.4}\left(\text{Yr4Grad}\right)$	$\text{4816}\text{.64}$	$\text{0}\text{.754}$
StudPerFac	$\text{AvgDebt}=\text{14307}+\text{346}\left(\text{StudPerFac}\right)$	$\text{4683}\text{.84}$	$\text{0}\text{.138}$
InAfterAid	$\text{AvgDebt}=\text{12594}+\text{0}\text{.633}\left(\text{InAfterAid}\right)$	$\text{4209}\text{.25}$	$\text{0}\text{.001}$
OutAfterAid	$\text{AvgDebt}=\text{15713}+\text{0}\text{.170}\left(\text{OutAfterAid}\right)$	$\text{4709}\text{.49 }$	$\text{0}\text{.181}$
AvgAid	$\text{AvgDebt}=\text{24952}-\text{0}\text{.575}\left(\text{AvgAid}\right)$	$\text{4507}\text{.29}$	$\text{0}\text{.024}$
PercBorrow	$\text{AvgDebt}=\text{11818}+\text{169}\left(\text{PercBorrow}\right)$	$\text{4281}\text{.04 }$	$\text{0}\text{.003}$

Interpretation: The regression line for the variable AvgAid is having a negative slope which verifies the conclusion drawn from the scattered plot in part (a).

(c)

To determine

The variable which appears to be best single predictor of average debt.

Answer to Problem 14E

Solution: InAfterAid is best single predictor of average debt.

Explanation of Solution

The model of each explanatory variable, model standard deviation $s$, and the P-value for the test of the linear association obtained in part (b).

The independent variable with smallest P-value is the significant predictor of the model. As the P-value is minimum for the explanatory variable InAfterAid which is $\text{0}\text{.001}$, InAfterAid is appears to be the best single predictor of the average debt.