Key | Third Hourly | Math 1107 | Fall Semester 2010

 

Protocol

 

You will use only the following resources: Your individual calculator; Your individual tool-sheet (one (1) 8.5 by 11 inch sheet); Your writing utensils; Blank Paper (provided by me); This copy of the hourly. Do not share these resources with anyone else. Show complete detail and work for full credit.  Follow case study solutions and sample hourly keys in presenting your solutions.

 

Work all four cases. Using only one side of the blank sheets provided, present your work. Do not write on both sides of the sheets provided, and present your work only on these sheets. Do not share information with any other students during this hourly. Neither give nor receive information with any other students during this hourly.

 

When you are finished: Prepare a Cover Sheet: Print your name on an otherwise blank sheet of paper. Then stack your stuff as follows: Cover Sheet (Top), Your Work Sheets, The Test Papers and Your Tool-sheet. Then hand all of this in to me.

 

Before you begin work, sign and acknowledge

 

I agree to follow this protocol. ______ (ÜInitial Here)

 

 

________________________________________________________________________

Name (Print Clearly)                                       Signature                                         Date

 

Case One | Confidence Interval, Mean | FICO Credit Scores

 

Fair Isaac Corporation developed a consumer credit score, a number that summarizes the risk present in lending money to a consumer. The consumer credit score ranges from 300 to 850. Credit bureau scores are often called “FICO scores” because most credit bureau scores used in the U.S. are produced from software developed by Fair Isaac and Company. FICO scores are provided to lenders by the major credit reporting agencies. Suppose that credit scores are observed for a random sample of US residents taken in July of 2007:

 

350, 375, 450, 535, 576, 585, 590, 625, 640, 661, 669, 670, 673, 675, 679, 680, 685, 688, 691, 695, 700, 701 705, 706, 707, 713, 723, 727, 740, 743, 752, 755, 757, 759, 761, 774, 774, 775, 783, 785, 791, 794, 798, 801 805, 810, 815, 827, 830, 845

 

Compute and interpret a 97% confidence interval for the population mean FICO score for US Residents as of July 2007. Show your work. Completely discuss and interpret your results, as indicated in class and case study summaries.

 

                                  

 

From 2.20 0.013903 0.97219, Z = 2.20.

 

lower97  = m – ( Z*sd  / sqrt(n) ) ≈ 702.96 – ( 2.2*106.193/sqrt(50) ) ≈ 669.9[670]

upper97 = m + ( Z*sd  / sqrt(n) ) ≈ 702.96 + ( 2.2*106.193/sqrt(50) ) ≈ 736.0[736]

Our population is the population of US Residents, and our population mean is the population mean FICO credit score as of July 2007.

Each member of the Family of Samples is a single random sample of 50 US Residents, from whom a FICO credit score is obtained during July 2007.

From each member sample, compute: sample mean age at injury m, sample standard deviation sd, then from the means table row  2.20 0.013903 0.97219, use Z = 2.20. Then compute the interval

 

[lower97 = m – ( Z*sd  / sqrt(n) ), upper97 = m + ( Z*sd  / sqrt(n) )].

 

Doing this for each member of the family of samples yields a family of intervals.

 

Approximately 97% of the intervals in the family cover the unknown population mean. If our interval resides in this 97% supermajority of member intervals, then the true population mean FICO credit score for US Residents as of July 2007 is between 670 and 736.

 

 

Case Two | Hypothesis Test, Median | Fictitious Spotted Toad

The Fictitious Spotted Toad is a native species of Toad Island, and is noteworthy for the both the quantity and quality of its spots. Consider a random sample of toads, in which the number of spots per toad is noted:

9, 10, 11, 12, 12, 13, 15, 16, 18, 18, 19, 20, 20, 22, 23, 24, 24, 25, 29, 31, 33, 33, 33, 35, 37, 40, 43, 45, 47, 56, 56, 60, 63, 65, 66

 

Test the following: null (H0): The median number of spots per Fictitious Spotted Toad is 20 (h = 20) against the alternative (H1): h < 20.  Show your work. Completely discuss and interpret your results, as indicated in class and case study summaries.

 

Null Hypothesis: Median Spot Count = 20 Spots

Alternative  Hypothesis: Median Spot Count < 20 Spots (Guess is too Large)

Error Function: Number of Sample Toads with Strictly Fewer Than 20 Spots

 

9, 10, 11, 12, 12 | 13, 15, 16, 18, 18 | 19, 20, 20, 22, 23 | 24, 24, 25, 29, 31 | 33, 33, 33, 35, 37 | 40, 43, 45, 47, 56 | 56, 60, 63, 65, 66

 

Alternative Hypothesis: “Guess is too Large”

Error: Count Below Guess

Error = #{sample toads with strictly fewer than 20 spots} = 11

n = 35

 

*** Instructor Glitch: I should have either restricted sample size to 30 or less, or provided tables for n = 35. We’ll use the

n = 30 median test table block, as indicated in class, for testing purposes. ***

 

From 30 11 0.95063, p ≈ 0.95. (From 35      11     0.99166, p ≈ 0.99)

 

                                    

 

Null Hypothesis: Median Spot Count = 20 Spots

Alternative  Hypothesis: Median Spot Count < 20 Spots (Guess is too Large)

Error Function: Number of Sample Toads with Strictly Fewer Than 20 Spots

n = sample size = 35

 

Our population consists of Fictitious Spotted Toads. Our null hypothesis is that the population median spot count for Fictitious Spotted Toads is 20 spots per toad.

 

Each member of the family of samples (FoS) is a single random sample of 35 Fictitious Spotted Toads. The FoS consists of all possible samples of this type.

 

From each member of the (FoS), compute an error as the number of sample toads with strictly fewer than 20 spots. Computing this error for each member of the FoS forms a family of errors (FoE).

 

If the true population median spot count for Fictitious Spotted Toads is 20 spots per toad, then approximately 95.063% member samples from the FoS yield errors as bad as or worse than our error. The sample does not appear to present significant evidence against the null hypothesis.

 

Case Three | Confidence Interval, Proportion | FICO Credit Scores

 

Using the context and data of Case One, consider the proportion of US residents whose FICO scores are 750 or higher as of July 2007. Compute and interpret a 92% confidence interval for the population proportion of US residents whose FICO scores are 750 or higher as of July 2007. Show your work. Completely discuss and interpret your results, as indicated in class and case study summaries.

 

350, 375, 450, 535, 576 | 585, 590, 625, 640, 661 | 669, 670, 673, 675, 679 | 680, 685, 688, 691, 695 | 700, 701 705, 706, 707

713, 723, 727, 740, 743 | 752, 755, 757, 759, 761 | 774, 774, 775, 783, 785 | 791, 794, 798, 801 805 | 810, 815, 827, 830, 845

 

       e           p              sdp          Z     lower92    upper92

50    20    0.35897    0.054315    1.8    0.26121    0.45674

 

sample size n = 50

event=“US Resident with FICO credit score at 750 or higher as of July 2007”

event count = e = 20

p = e /n = 20/50 = 0.40

1 – p = 1 – (.40) = 0.60

sdp = sqrt(p*(1 – p)/n) = sqrt( .4*.6 / 50 ) ≈ 0.069282

 

from 1.80 0.035930 0.92814, Z ≈ 1.80

 

 lower92 = p – ( 1.8*sdp ) ≈ 0.40 – ( 1.8* 0.069282 ) ≈ 0.275292

upper92 = p + ( 1.8*sdp ) ≈ 0.40 + ( 1.8* 0.069282 ) ≈ 0.524708

Our population is the population of US Residents, and our population proportion is the population proportion of US Residents with FICO credit scores at 750 or higher as of July 2007.

Each member of the Family of Samples is a single random sample of 50 US Residents, from whom FICO credit scores are obtained during July 2007.

From each member sample, compute: the sample number e of US Residents with FICO credit scores at 750 or higher as of July 2007, the sample proportion p = e / 50 of US Residents with FICO credit scores at 750 or higher as of July 2007 and  the standard error for proportion sdp = sqrt( p*(1 – p)/50 ). Then from the means table row  1.80 0.035930 0.92814, use Z ≈ 1.80. Then compute the interval

 

[lower92 = p – ( Z*sdp ), upper92 = p + ( Z*sdp )].

 

Doing this for each member of the family of samples yields a family of intervals.

 

Approximately 92% of the intervals in the family cover the unknown population proportion. If our interval resides in this 92% supermajority of member intervals, then between 27.5% and 52.5% of US Residents with FICO credit scores at 750 or higher as of July 2007.

 

 

Case Four | Categorical Goodness of Fit | Traumatic Brain Injury (TBI)

 

The Glasgow Coma Scale (GCS) is the most widely used system for scoring the level of consciousness of a patient who has had a traumatic brain injury. GCS is based on the patient's best eye-opening, verbal, and motor responses. Each response is scored and then the sum of the three scores is computed. Glasgow Coma Scale Categories: Mild (13-15); Moderate (9-12) and Severe/Coma (3-8). Traumatic brain injury (TBI) is an insult to the brain from an external mechanical force, possibly leading to permanent or temporary impairments of cognitive, physical, and psychosocial functions with an associated diminished or altered state of consciousness. Consider a random sample of patients surviving with TBI, with GCS at initial treatment and diagnosis listed below:

 

3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15

 

Test the hypothesis that the TBI survivors are equally distributed as 20% Severe, 30% Moderate and 50% Mild. Show your work. Completely discuss and interpret your results, as indicated in class and case study summaries.

 

3, 3, 3, 4, 4 | 5, 5, 6, 6, 7 | 8 [11] Severe

9, 9, 9, 9, 9 | 10, 10, 10, 11, 11 | 12, 12 [12] Moderate

13, 13, 13, 13, 13 | 13, 13, 13, 13, 13 | 13, 13, 13, 14, 14 | 14, 14, 14, 14, 14 | 14, 14, 14, 14, 14 | 15, 15 [27] Mild

 

n = 11 + 12 + 27 = 23 + 27 = 50

 

Expected Counts from the Null Hypothesis for n=50, equally likely categories (Severe, Mild, Moderate)

 

eSevereTBI = (.20)*50 = 10

eModerateTBI = (.30)*50 = 15

eMildTBI = (.50)*50 = 25

 

Severe

Observed nSevereTBI = 11

Expected eSevereTBI = 10

ErrorSevereTBI = (ObservedSevereTBI –ExpectedSevereTBI)2/ExpectedSevereTBI = (11 – 10 )2/ (10) ≈ 0.10

 

Moderate

Observed nModerateTBI = 12

Expected eModerateTBI = 15

ErrorModerateTBI = (ObservedModerateTBI –ExpectedModerateTBI)2/ExpectedModerateTBI = (12 – 15 )2/ 15 ≈ 0.60

 

Mild

Observed nMildTBI = 27

Expected eMildTBI = 25

ErrorMildTBI = (ObservedMildTBI –ExpectedMildTBI)2/ExpectedMildTBI = (27 – 25 )2/ 25 ≈ 0.16

 

Total Error = ErrorSevereTBI + ErrorModerateTBI + ErrorMildTBI = ErrorSevereTBI ≈ 0.10 + 0.60 + 0.16 = 0.86

 

From 3  0.71335  0.700 and 3  1.02165  0.600, 0.60 < p < 0.70  

 

Interpretation

 

Our population consists of people who have survived traumatic brain injury (TBI). Our categories are based on initial severity of injury based on Glasgow Coma Score (GCS): Severe{3 £ GCS £ 8}, Moderate{9 £ GCS £ 12} and Mild {13 £ GCS £ 15}.Our null hypothesis is that the categories are distributed as 20% Severe, 30% Moderate and 50% Mild.

Our Family of Samples (FoS) consists of every possible random sample of 50 surviving TBI cases. Under the null hypothesis, within each member of the FoS, we expect approximately:

eSevereTBI = (.20)*50 = 10

eModerateTBI = (.30)*50 = 15

eMildTBI = (.50)*50 = 25

 

From each member sample of the FoS, we compute sample counts and errors for each level of severity:

 

Severe

Observed nSevereTBI

Expected eSevereTBI = 10

ErrorSevereTBI = (ObservedSevereTBI –ExpectedSevereTBI)2/ExpectedSevereTBI

 

Moderate

Observed nModerateTBI

Expected eModerateTBI = 15

ErrorModerateTBI = (ObservedModerateTBI –ExpectedModerateTBI)2/ExpectedModerateTBI

 

Mild

Observed nMildTBI

Expected eMildTBI = 25

ErrorMildTBI = (ObservedMildTBI –ExpectedMildTBI)2/ExpectedMildTBI

 

Then add the individual errors for the total error as

 

Total Error = ErrorSevereTBI + ErrorModerateTBI + ErrorMildTBI = ErrorSevereTBI

 

Computing this error for each member sample of the FoS, we obtain a Family of Errors (FoE).

 

If the TBI severity levels (for survivors with TBI) based on initial severity of injury based on Glasgow Coma Score (GCS): Severe{3 £ GCS £ 8}, Moderate{9 £ GCS £ 12} and Mild {13 £ GCS £ 15} are distributed as 20% Severe, 30% Moderate and 50% Mild, then between 60% and 70% of the member samples of the Family of Samples yields errors as large as or larger than that of our single sample. Our sample does not present significant evidence against the null hypothesis.

 

Table 1. Means and Proportions

Z(k) PROBRT PROBCENT

0.05 0.48006 0.03988

0.10 0.46017 0.07966

0.15 0.44038 0.11924

0.20 0.42074 0.15852

0.25 0.40129 0.19741

0.30 0.38209 0.23582

0.35 0.36317 0.27366

0.40 0.34458 0.31084

0.45 0.32636 0.34729

0.50 0.30854 0.38292

0.55 0.29116 0.41768

0.60 0.27425 0.45149

0.65 0.25785 0.48431

0.70 0.24196 0.51607

0.75 0.22663 0.54675

0.80 0.21186 0.57629

0.85 0.19766 0.60467

0.90 0.18406 0.63188

0.95 0.17106 0.65789

1.00 0.15866 0.68269

Z(k) PROBRT PROBCENT

1.05 0.14686 0.70628

1.10 0.13567 0.72867

1.15 0.12507 0.74986

1.20 0.11507 0.76986

1.25 0.10565 0.78870

1.30 0.09680 0.80640

1.35 0.088508 0.82298

1.40 0.080757 0.83849

1.45 0.073529 0.85294

1.50 0.066807 0.86639

1.55 0.060571 0.87886

1.60 0.054799 0.89040

1.65 0.049471 0.90106

1.70 0.044565 0.91087

1.75 0.040059 0.91988

1.80 0.035930 0.92814

1.85 0.032157 0.93569

1.90 0.028717 0.94257

1.95 0.025588 0.94882

2.00 0.022750 0.95450

Z(k) PROBRT PROBCENT

2.05 0.020182 0.95964

2.10 0.017864 0.96427

2.15 0.015778 0.96844

2.20 0.013903 0.97219

2.25 0.012224 0.97555

2.30 0.010724 0.97855

2.35 0.009387 0.98123

2.40 0.008198 0.98360

2.45 0.007143 0.98571

2.50 0.006210 0.98758

2.55 0.005386 0.98923

2.60 0.004661 0.99068

2.65 0.004025 0.99195

2.70 .0034670 0.99307

2.75 .0029798 0.99404

2.80 .0025551 0.99489

2.85 .0021860 0.99563

2.90 .0018658 0.99627

2.95 .0015889 0.99682

3.00 .0013499 0.99730

Table 2. Medians

 n error base p-value

25 1 1.00000

25 2 1.00000

25 3 0.99999

25 4 0.99992

25 5 0.99954

25 6 0.99796

25 7 0.99268

25 8 0.97836

25 9 0.94612

25 10 0.88524

25 11 0.78782

25 12 0.65498

25 13 0.50000

25 14 0.34502

25 15 0.21218

25 16 0.11476

25 17 0.05388

25 18 0.02164

 

 

n error base p-value

25 19 0.00732

25 20 0.00204

25 21 0.00046

25 22 0.00008

25 23 0.00001

25 24 0.00000

25 25 0.00000

30 1 1.00000

30 2 1.00000

30 3 1.00000

30 4 1.00000

30 5 0.99997

30 6 0.99984

30 7 0.99928

30 8 0.99739

30 9 0.99194

30 10 0.97861

30 11 0.95063

 

n error base p-value

30 12 0.89976

30 13 0.81920

30 14 0.70767

30 15 0.57223

30 16 0.42777

30 17 0.29233

30 18 0.18080

30 19 0.10024

30 20 0.04937

30 21 0.02139

30 22 0.00806

30 23 0.00261

30 24 0.00072

30 25 0.00016

30 26 0.00003

30 27 0.00000

30 28 0.00000

30 29 0.00000

30 30 0.00000

 

 

n     error      base p-value__

35       1     1.00000

35       2     1.00000

35       3     1.00000

35       4     1.00000

35       5     1.00000

35       6     0.99999

35       7     0.99994

35       8     0.99975

35       9     0.99906

35      10     0.99701

35      11     0.99166

35      12     0.97952

35      13     0.95523

35      14     0.91227

35      15     0.84475

35      16     0.75022

35      17     0.63206

35      18     0.50000

35      19     0.36794

35      20     0.24978

35      21     0.15525

35      22     0.08773

35      23     0.04477

35      24     0.02048

35      25     0.00834

35      26     0.00299

35      27     0.00094

35      28     0.00025

35      29     0.00006

35      30     0.00001

35      31     0.00000

35      32     0.00000

35      33     0.00000

35      34     0.00000

35      35     0.00000

 

 

 

 

Table 3. Categories/Goodness of Fit

categories ERROR p-value

3  0.00000  1.000

3  0.21072  0.900

3  0.44629  0.800

3  0.71335  0.700

3  1.02165  0.600

3  1.38629  0.500

3  1.59702  0.450

3  1.83258  0.400

3  2.09964  0.350

3  2.40795  0.300

3  2.77259  0.250

3  3.21888  0.200

3  3.48594  0.175

3  3.79424  0.150

3  4.15888  0.125

3  4.60517  0.100

3  4.81589  0.090

3  5.0515  0.080

3  5.3185  0.070

3  5.6268  0.060

3  5.9915  0.050

3  6.4378  0.040

3  7.0131  0.030

3  7.8240  0.020

3  9.2103  0.010

categories ERROR p-value

4  0.0000  1.000

4  0.5844  0.900

4  1.0052  0.800

4  1.4237  0.700

4  1.8692  0.600

4  2.3660  0.500

4  2.6430  0.450

4  2.9462  0.400

4  3.2831  0.350

4  3.6649  0.300

4  4.1083  0.250

4  4.6416  0.200

4  4.9566  0.175

4  5.3170  0.150

4  5.7394  0.125

4  6.2514  0.100

4  6.4915  0.090

4  6.7587  0.080

4  7.0603  0.070

4  7.4069  0.060

4  7.8147  0.050

4  8.3112  0.040

4  8.9473  0.030

4  9.8374  0.020

4  11.3449  0.010 

Categories ERROR p-value

5  0.0000  1.000

5  1.0636  0.900

5  1.6488  0.800

5  2.1947  0.700

5  2.7528  0.600

5  3.3567  0.500

5  3.6871  0.450

5  4.0446  0.400

5  4.4377  0.350

5  4.8784  0.300

5  5.3853  0.250

5  5.9886  0.200

5  6.3423  0.175

5  6.7449  0.150

5  7.2140  0.125

5  7.7794  0.100

5  8.0434  0.090

5  8.3365  0.080

5  8.6664  0.070

5  9.0444  0.060

5  9.4877  0.050

5  10.0255  0.040

5  10.7119  0.030

5  11.6678  0.020

5  13.2767  0.010