Third Hourly

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 | Traumatic Brain Injury

Traumatic Brain Injury (TBI) involves the injury of the brain when it involves sudden or intense physical force resulting in the presence of Concussion, Skull Fracture, or Bleeding and Tissue Damage (Contusions, Lacerations, Hemorrhaging) involving the brain. A random sample of TBI cases is acquired, and the age (in years) at injury of each case is determined. The ages at injury for the sample cases are listed below:

4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 20, 21, 21, 22, 23, 25, 27, 27, 30, 30, 30, 31, 32, 32, 33, 35, 35, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 45, 47, 50, 52, 60,63, 65, 70, 70, 71, 71, 72, 72, 73, 73, 74, 74, 75, 75, 75, 76, 76, 77, 79, 80, 81, 89, 95

Compute and interpret a 97% confidence interval for the population mean age at injury for TBI cases. 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.

n m sd Z lower97 upper97

78 40.5128 25.6043 2.2 34.1348 46.8909

lower97 = m – ( Z*sd / sqrt(n) ) ≈ 40.5128 – ( 2.2*25.6043/sqrt(78) ) ≈ 34.1348

upper97 = m + ( Z*sd / sqrt(n) ) ≈ 40.5128 + ( 2.2*25.6043/sqrt(78) ) ≈ 46.8909

Our population is the population of people with traumatic brain injury(TBI), and our population mean is the population mean age at injury for TBI.

Each member of the Family of Samples is a single random sample of 78 people with TBI, from whom age at injury is determined.

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 age at injury for traumatic brain injury is between 34.1 and 46.9 years.

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 | Traumatic Brain Injury

Using the context and data of Case One, consider the proportion of TBI cases who acquire the injury at age 25 years or younger. Compute and interpret a 92% confidence interval for the population proportion of TBI cases who acquire the injury at age 25 years or younger. Compute and interpret a 92% confidence interval for this population proportion. Show your work. Completely discuss and interpret your results, as indicated in class and case study summaries.

Key

n e p sdp Z lower92 upper92

78 28 0.35897 0.054315 1.8 0.26121 0.45674

sample size n = 78

event=“TBI Case Acquiring TBI at Age 25 Years or Younger”

event count = e = 28

p = e /n = 28/78 ≈ 0.35897

1 – p = 1 – (28/78) = 50/78 ≈ 0.64103

sdp = sqrt(p*(1 – p)/n) = sqrt( ( 28/78 )*( 50/78 ) / 78 ) ≈ 0.054315

from 1.80 0.035930 0.92814, Z ≈ 1.80

lower92 = p – ( 1.8*sdp ) ≈ 0.35897 – ( 1.8* 0.054315 ) ≈ 0.26121

upper92 = p + ( 1.8*sdp ) ≈ 0.35897 + ( 1.8* 0.054315 ) ≈ 0.45674

Our population is the population of people with traumatic brain injury(TBI), and our population proportion is the population proportion of TBI cases acquiring their TBI at age 25 years or younger.

Each member of the Family of Samples is a single random sample of 78 people with TBI, from whom age at injury is determined.

From each member sample, compute: the sample number e of people with TBI acquiring their TBI at age 25 years or younger, the sample proportion p = e / 78 of people with TBI who acquire their TBI at age 25 years or younger, the standard error for proportion sdp = sqrt( p*(1 – p)/78 ). 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 26.1% and 45.7% of the population of TBI cases acquire their TBIs at age 25 years or younger.

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 among the three severity categories. 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)

e_SevereTBI = (1/3)*50 ≈ 16.6667

e_ModerateTBI = (1/3)*50 ≈ 16.6667

e_MildTBI = (1/3)*50 ≈ 16.6667

Severe

Observed n_SevereTBI = 11

Expected e_SevereTBI = (50/3)

Error_SevereTBI = (Observed_SevereTBI –Expected_SevereTBI)²/Expected_SevereTBI = (11 – (50/3) )²/ (50/3) ≈ 1.92667

Moderate

Observed n_ModerateTBI = 12

Expected e_ModerateTBI = (50/3)

Error_ModerateTBI = (Observed_ModerateTBI –Expected_ModerateTBI)²/Expected_ModerateTBI = (12 – (50/3) )²/ (50/3) ≈ 1.30667

Mild

Observed n_MildTBI = 27

Expected e_MildTBI = (50/3)

Error_MildTBI = (Observed_MildTBI –Expected_MildTBI)²/Expected_MildTBI = (27 – (50/3) )²/ (50/3) ≈ 6.40667

Total Error = Error_SevereTBI + Error_ModerateTBI + Error_MildTBI = Error_SevereTBI ≈ 1.92667 + 1.30667 + 6.40667 ≈ 9.6400

From 3 9.2103 0.010, p < .01

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 equally..

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:

e_SevereTBI = (1/3)*50 ≈ 16.6667

e_ModerateTBI = (1/3)*50 ≈ 16.6667

e_MildTBI = (1/3)*50 ≈ 16.6667

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

Severe

Observed n_SevereTBI

Expected e_SevereTBI = (50/3)

Error_SevereTBI = (Observed_SevereTBI –Expected_SevereTBI)²/Expected_SevereTBI

Moderate

Observed n_ModerateTBI

Expected e_ModerateTBI = (50/3)

Error_ModerateTBI = (Observed_ModerateTBI –Expected_ModerateTBI)²/Expected_ModerateTBI

Mild

Observed n_MildTBI

Expected e_MildTBI = (50/3)

Error_MildTBI = (Observed_MildTBI –Expected_MildTBI)²/Expected_MildTBI

Then add the individual errors for the total error as

Total Error = Error_SevereTBI + Error_ModerateTBI + Error_MildTBI = Error_SevereTBI

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

If the TBI severity levels for TBI survivors 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 equally likely, then less than 1% of the member samples of the Family of Samples yields errors as large as or larger than that of our single sample. Our sample presents 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