Key

The Comprehensive Final Examination

Math 1107

Spring Semester 2009

CJ Alverson

 

Protocol

 

You will use only the following resources: Your individual calculator; Your individual tool-sheets (two (2) 8.5 by 11 inch sheets); Your writing utensils; Blank Paper (provided by me); This copy of the hourly and

the tables provided by me. 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. When you’re done: Print your name on a blank sheet of paper. Place your toolsheet, test and work under this sheet, and turn it all in to me. Do not share information with any other students during this hourly.

 

Sign and Acknowledge:  I agree to follow this protocol. Initial:___

 

______________________________________________________________________________________

Name (PRINTED)                                          Signature                                          Date

 

Case One | Color Slot Machine | Conditional Probabilities

 

Here is our slot machine – on each trial, it produces a color sequence, using the table below:

 

Sequence*

Probability

BBRRYRRR

.10

GGRGBRRB

.10

BBYYGGBR

.15

GRRGBRGG

.10

BGYRYGYY

.25

YYGRRBBY

.10

YBYYBGRR

.20

Total

1.00

*B-Blue, G-Green, R-Red, Y-Yellow

 

Compute the following conditional probabilities:

 

Pr{ Blue Shows Exactly Twice | Yellow Shows} 

Pr{ Yellow Shows | “BR” Shows } 

Pr{ Blue Shows | Green Shows}

 

Show all work for full credit.

 

Key

 

Pr{ Blue Shows Exactly Twice | Yellow Shows} 

 

Prior

 

Pr{Yellow Shows} 

 

 

Sequence*

Probability

BBRRYRRR

.10

BBYYGGBR

.15

BGYRYGYY

.25

YYGRRBBY

.10

YBYYBGRR

.20

Total

0.80

 

Pr{Yellow Shows} = Pr{One of BBRRYRRR, BBYYGGBR, BGYRYGYY, YYGRRBBY or BYYBGRR Shows} =

Pr{BBRRYRRR}+Pr{BBYYGGBR}+Pr{BGYRYGYY}+Pr{YYGRRBBY}+Pr{BYYBGRR} =

.10 + .15 + .25 + .10 + .20 = 0.80

 

Joint

 

Pr{ Blue Shows Exactly Twice and Yellow Shows} 

 

 

Sequence*

Probability

BBRRYRRR

.10

YYGRRBBY

.10

YBYYBGRR

.20

Total

0.40

 

Pr{ Blue Shows Exactly Twice and Yellow Shows} = Pr{One of BBRRYRRR, YYGRRBBY or BYYBGRR Shows} =

Pr{BBRRYRRR}+Pr{YYGRRBBY}+Pr{BYYBGRR} =

.10 +.10 + .20 = 0.40

 

Conditional = Joint / Prior

 

Pr{ Blue Shows Exactly Twice | Yellow Shows} = Pr{ Blue Shows Exactly Twice and Yellow Shows}/ Pr{Yellow Shows} = .4/.8 = .5

 

Pr{ Yellow Shows | “BR” Shows } 

 

Prior

 

Pr{ “BR” Shows } 

 

Sequence*

Probability

BBRRYRRR

.10

GGRGBRRB

.10

BBYYGGBR

.15

GRRGBRGG

.10

Total

.45

 

Pr{ “BR” Shows } = Pr{One of BBRRYRRR, GGRGBRRB, BBYYGGBR or GRRGBRGG Shows} =

Pr{BBRRYRRR}+Pr{GGRGBRRB}+Pr{BBYYGGBR}+Pr{GRRGBRGG} = .1+.1+.15+.1 = .45

 

Joint

 

Pr{ Yellow Shows and  “BR” Shows } 

 

Sequence*

Probability

BBRRYRRR

.10

BBYYGGBR

.15

Total

.25

 

Pr{ Yellow Shows and “BR” Shows } = Pr{One of BBRRYRRR or BBYYGGBR Shows} =

Pr{BBRRYRRR}+ Pr{BBYYGGBR}= .1+.15 = .25

 

Conditonal = Joint / Prior

 

Pr{ Yellow Shows | “BR” Shows } = Pr{ Yellow Shows and “BR” Shows }/Pr{ “BR” Shows } = .25/.45 » .5556

 

 

Pr{ Blue Shows | Green Shows}

 

Prior

 

Pr{ Green Shows}

 

Sequence*

Probability

GGRGBRRB

.10

BBYYGGBR

.15

GRRGBRGG

.10

BGYRYGYY

.25

YYGRRBBY

.10

YBYYBGRR

.20

Total

.90

 

Pr{Green Shows} = Pr{One of GGRGBRRB, BBYYGGBR, GRRGBRGG, BGYRYGYY, YYGRRBBY or YBYYBGRR Shows} = 

Pr{GGRGBRRB}+Pr{BBYYGGBR}+Pr{GRRGBRGG}+Pr{BGYRYGYY}+Pr{YYGRRBBY}+Pr{YBYYBGRR} =

.10+.15+.10+.25+.10+.20=.90

 

Joint

 

Pr{ Blue Shows and Green Shows}

Sequence*

Probability

GGRGBRRB

.10

BBYYGGBR

.15

GRRGBRGG

.10

BGYRYGYY

.25

YYGRRBBY

.10

YBYYBGRR

.20

Total

.90

 

Pr{Blue and Green Shows} = Pr{One of GGRGBRRB, BBYYGGBR, GRRGBRGG, BGYRYGYY, YYGRRBBY or YBYYBGRR Shows} = 

Pr{GGRGBRRB}+Pr{BBYYGGBR}+Pr{GRRGBRGG}+Pr{BGYRYGYY}+Pr{YYGRRBBY}+Pr{YBYYBGRR} =

.10+.15+.10+.25+.10+.20=.90

 

Conditional = Joint/Prior

 

Pr{ Blue Shows | Green Shows}

 

Pr{ Blue Shows | Green Shows} = Pr{ Blue Shows and Green Shows} / Pr{ Green Shows} = .9/.9 = 1

 

Case Two | Descriptive Statistics | Sea Weaselsä

Sea Weaselsä are instant pets, distributed as a kit - Sea Weasel Eggsä , Sea Weasel Water Conditionerä and Sea Weasel Foodä . A Sea Weaselä Kit is started by placing the Sea Weasel Eggsä , Sea Weasel Water Conditionerä and some water in a container. The eggs will then hatch, producing Sea Weaselsä , and some of them will survive. A random sample of Sea Weaselä Kits is selected, each selected kit is started, and the number of surviving Sea Weaselsä is counted one week after start. Here are the Sea Weaselä counts per kit

20 23 25 26 27 32 35 41 42 42 45 47 48 51 56 60 61 62 65 70 73 76 84 89 91 92 96 98 104 109 117 124 130 132 135 138 140 142 145 151 160 162 165 166 167 171 174 174 176 178 179 181 182 183 187 275

 

Compute and interpret the following statistics: sample size, p00, p25, p50, p75, p100, (p75 – p25),

(p100 – p25), (p50-p25).  Show complete detail and work for full credit. Follow case study solutions and sample hourly keys in presenting your solutions.

 

Numbers

 

                                     Quantile      Estimate

                                     100% Max         275.0

                                     99%              275.0

                                     95%              183.0

                                     90%              179.0

                                     75% Q3           163.5

                                     50% Median       101.0

                                     25% Q1            53.5

                                     10%               32.0

                                     5%                25.0

                                     1%                20.0

                                     0% Min            20.0

 

n    p00     p25    p50     p75     p100     R21    R31     R41

56     20    53.5    101    163.5     275    47.5    110    221.5

 

R21 = P50 – P25 = 101 – 53.5 = 47.5

R31 = P75 – P25 = 163.5 – 53.5 = 110

R41 = P100 – P25 = 275 – 53.5 = 221.5

 

Interpretation

 

There are 56 Sea Weasel kits in our sample.

The kit in the sample with the smallest yield yields 20 Sea Weasels.

Approximately 25% of the kits in the sample yield 53.5 or fewer Sea Weasels.

Approximately 50% of the kits in the sample yield 101 or fewer Sea Weasels.

Approximately 75% of the kits in the sample yield 163.5 or fewer Sea Weasels.

The kit in the sample with the largest yield yields 275 Sea Weasels.

 

Approximately 25% of the kits in the sample yield between 53.5 and 101 Sea Weasels – the largest possible difference in kit yield between any pair of kits in the lower middle quarter sample is 47.5 Sea Weasels.

 

Approximately 50% of the kits in the sample yield between 53.5 and 163.5 Sea Weasels – the largest possible difference in kit yield between any pair of kits in the middle half sample is 110 Sea Weasels.

 

Approximately 75% of the kits in the sample yield between 53.5 and 275 Sea Weasels – the largest possible difference in kit yield between any pair of kits in the upper ¾ sample is 221.5 Sea Weasels.

 

Case Three | Confidence Interval: Population Proportion | Sea Weaselsä

 

Using the data and context from Case Two, compute and interpret a 95% confidence interval for the population proportion of  Sea Weaselä Kits with yields of 150 or more Sea Weaselsä. Show all work for full credit. Fully discuss the results. This discussion must include a clear discussion of the population and the population proportion, the family of samples, the family of intervals and the interpretation of the interval.

 

Numbers

 

20 23 25 26 27 32 35 41 42 42 45 47 48 51 56 60 61 62 65 70 73 76 84 89 91 92 96 98 104 109 117 124 130 132 135 138 140 142 145 151 160 162 165 166 167 171 174 174 176 178 179 181 182 183 187 275

 

Event = “Sea Weasel kit yields 150 or more Sea Weasels”

e = sample event count = 17

n = sample size = 56

p = e/n = 17/56 » 0.30357

sdp = sqrt(p*(1 – p)/n) = sqrt((17/56)*(39/56)/56) » 0.061443

lower95 = p – (2*sdp) » .30357 – (2*0.061443) » 0.18068

upper95 = p + (2*sdp) » .30357 + (2*0.061443) » 0.42646

From this row: 2.00 0.022750 0.95450, Z » 2.00

 

Report the interval as [0.181, 0.426].

     

Interpretation

 

Each member of the Family of Samples (FoS) is a single random sample of 56 Sea Weasel kits – the Family of Samples consists of every possible sample of this type.

 

From each member of the FoS, compute e = number of kits in the sample yielding 150 or more weasels. Then compute p=e/56, sdp = sqrt(p*(1 – p)/n), and then the interval  [ lower bound = p – (2*sdp), p + (2*sdp)] – repeating these calculations for each member sample yields a Family of Intervals (FoI), approximately 95% of which cover the population proportion of Sea Weasel kits that yield 150 or more Sea Weasels. If our single computed interval resides in this 95% supermajority, then between 18.1 and 42.6% of Sea Weasel kits yield 150 or more Sea Weasels. 

 

Case Four | Hypothesis Test – Categorical Goodness-of-Fit | Traumatic Brain Injury

 

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. That is,

 

Augmented Glasgow Coma Scale Categories

Mild  = 13, 14, 15

Moderate = 9, 10, 11, 12

Severe/Coma =  3, 4, 5, 6, 7, 8

Pre-admission Death/PAD/DOA = 0

 

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 with TBI, with GCS at initial treatment and diagnosis listed below:

 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8,

9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12,

13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15

 

Our null hypothesis is that TBI case outcomes are 20% Pre-admission Deaths, 50% Severe, 20% Moderate and 10% Mild. Test this Hypothesis. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

 

Numbers

 

Pre-admission Death (PAD)

 

ObservedPAD = 14

ExpectedPAD = n*PPAD= 71*.20 = 14.2

ErrorPAD = (ObservedPAD – ExpectedPAD)2/ExpectedPAD = (14 –14.2)2/14.2 » 0.00282

 

Severe (GCS in 3, 4, 5, 6, 7, 8)

 

ObservedSevere = 28

ExpectedSevere = n*P = 71*.50 = 35.5

ErrorSevere = (ObservedSevere – ExpectedSevere)2/ExpectedSevere = (28 –35.5)2/35.5 » 1.58451

 

Moderate (GCS in 9, 10, 11, 12)

 

ObservedModerate = 18

ExpectedModerate = n*PModerate = 71*.20 = 14.2

ErrorModerate = (ObservedModerate – ExpectedModerate)2/ExpectedModerate = (18 –14.2)2/14.2 » 1.01690

 

Mild (GCS in 13, 14, 15)

 

ObservedMild = 11

ExpectedMild = n*PMild = 71*.10 = 7.1

ErrorMild = (ObservedMild – ExpectedMild)2/ExpectedMild = (11 –7.1)2/7.1 » 2.14225

 

Total Error = ErrorPAD + ErrorSevere + ErrorModerate + ErrorMild » 0.00282 + 1.58451 + 1.01690 + 2.14225 » 4.74648 over 4 categories.

 

From rows: 4 4.6416 0.200 and 4 4.9566 0.175, .175 £ p £ .200.

 

Each member of the Family of Samples(FoS)  is a random sample of 71 Traumatic Brain Injury patients – the FoS consists of all possible samples of this type.

 

From each member sample of the FoS, compute the following items at each level of severity:

 

Pre-admission Death (PAD)

ObservedPAD

ExpectedPAD = n*PPAD= 71*.20 = 14.2

ErrorPAD = (ObservedPAD – ExpectedPAD)2/ExpectedPAD

 

Severe (GCS in 3, 4, 5, 6, 7, 8)

 

ObservedSevere

ExpectedSevere = n*PSevere = 71*.50 = 35.5

ErrorSevere = (ObservedSevere – ExpectedSevere)2/ExpectedSevere

 

Moderate (GCS in 9, 10, 11, 12)

 

ObservedModerate

ExpectedModerate = n*PModerate = 71*.20 = 14.2

ErrorModerate = (ObservedModerate – ExpectedModerate)2/ExpectedModerate

 

Mild (GCS in 13, 14, 15)

 

ObservedMild

ExpectedMild = n*PMild = 71*.10 = 7.1

ErrorMild = (ObservedMild – ExpectedMild)2/ExpectedMild

 

Then compute Total Error = ErrorPAD + ErrorSevere + ErrorModerate + ErrorMild

Repeating these calculations for each member sample of the FoS yields a Family of Errors (FoE).

If the population proportions for TBI Severity are PPAD=.20,  PSevere =.50, PModerate = .20 and PMild = .10, then between 17.5% and 20% of the member samples of the FoS yield errors as severe or more extreme than our single computed error. Our sample does not seem to 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. Categories/Goodness of Fit 

Categories ERROR  p-value

3 0.0000 1.000                                  3 0.2107 0.900                                   3 0.4463 0.800                                   3 0.7133 0.700                                   3 1.0217 0.600                                   3 1.3863 0.500                                   3 1.5970 0.450                                   3 1.8326 0.400                                   3 2.0996 0.350                                   3 2.4079 0.300                                   3 2.7726 0.250                                   3 3.2189 0.200                                   3 4.6052 0.100                                   3 4.8159 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