Key | The 2^nd Hourly | Math 1107 | Summer Term 2010

Protocol

You will use only the following resources: Your individual calculator; individual tool-sheet (one (1) 8.5 by 11 inch sheet), writing utensils, blank paper (provided by me) and 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 six cases. Using only one side of the blank sheets provided, present your work. Write on one side only of the sheets provided, and present your work only on these sheets. Do not share 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, Your Toolsheet. Then hand all of this in to me.

Sign and Acknowledge: I agree to follow this protocol.

________________________________________________________________________

Name (PRINTED) Signature Date

Case One | Descriptive Statistics | Lewy Body Disease

Dementia with Lewy bodies, the second most frequent cause of degenerative dementia in elderly adults, is a neurodegenerative disorder associated with abnormal structures (Lewy bodies) found in certain areas of the brain. Symptoms can range from traditional Parkinsonian effects, such as loss of spontaneous movement (bradykinesia), rigidity (muscles feel stiff and resist movement), tremor, and shuffling gait, to effects similar to those of Alzheimer's disease, such as acute confusion, loss of memory, and loss of, or fluctuating cognition.

Visual hallucinations may be one of the first symptoms noted, and patients may suffer from other psychiatric disturbances such as delusions and depression. Onset of the disorder usually occurs in older

adults, although younger people can be affected as well. The disease can progress slowly. Over time, the tissues in two parts of the brain (the temporal and frontal lobes) shrink. This shrinking is called atrophy. Symptoms such as behavioral changes, speech difficulty, and impaired intellect occur gradually, but continue to get worse. Suppose that we have a random sample of patients with Lewy body disease, diagnosed pre-mortem and confirmed post-mortem. The time from initial diagnosis of dementia to death for each sample patient (in years) is given below.

1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 15, 15, 15, 16, 17, 19, 20, 21, 22, 25

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

(p75 – p25), (p50 – p25). Show complete detail and work for full credit. Follow case study solutions and sample hourly keys in presenting your solutions.

Numbers

N P0 P25 P50 P75 P100 RANGE41 RANGE31 RANGE21

50 1 6 7.5 12 25 19 6 1.5

There are 50 Lewy body patients in the sample.

The patient in the sample with the shortest survival time lived 1 year past diagnosis.

Approximately 25% of the patients in the sample survived 6 years or less after diagnosis.

Approximately 50% of the patients in the sample survived 7.5 years or less after diagnosis.

Approximately 75% of the patients in the sample survived 12 years or less after diagnosis.

The patient in the sample with the longest survival time lived 25 years past diagnosis.

RANGE41 = P100 - P25 = 25 - 6 = 19

Approximately 75% of the patients in the sample survived between 6 and 25 years after diagnosis with dementia with Lewy bodies. The largest possible difference in survival time between any pair of patients in this upper three-quarter sample is 19 years.

RANGE31 = P75 - P25 = 12 - 6 = 6

Approximately 50% of the patients in the sample survived between 6 and 12 years after diagnosis with dementia with Lewy bodies. The largest possible difference in survival time between any pair of patients in this middle half- sample is 6 years.

RANGE21 = P50 - P25 = 7.5 - 6 = 1.5

Approximately 25% of the patients in the sample survived between 6 and 7.5 years after diagnosis with dementia with Lewy bodies. The largest possible difference in survival time between any pair of patients in this lower quarter - sample is 1.5 years.

Case Two | Summary Intervals | Lewy Body Disease

Using the context and data from Case One, let m denote the sample mean time from initial diagnosis of dementia to death for each sample patient (in years), and sd the sample standard deviation.

Compute and interpret the intervals m±2sd and m±3sd, using Tchebysheff’s Inequalities and the Empirical Rule. Show complete detail and work for full credit. Follow case study solutions and sample hourly keys in presenting your solutions.

Numbers

N M SD lower2 upper2 lower3 upper3

50 9.02 5.71604 -2.41207 20.4521 -8.12811 26.1681

lower2 = m – 2*sd = 9.02 - 2*5.71604 = -2.41207 [0]

upper2 = m + 2*sd = 9.02 + 2*5.71604 = 20.4521

lower3 = m – 3*sd = 9.02 - 3*5.71604 = -8.12811 [0]

upper3 = m + 3*sd = 9.02 + 3*5.71604 = 26.1681

There are 50 Lewy body patients in the sample.

At least 75% of the patients in the sample survived between 0 and 20.4 years or less after diagnosis.

At least 89% of the patients in the sample survived between 0 and 26.1 years or less after diagnosis.

If the dementia with Lewy bodies survival times cluster symmetrically around a central value, becoming rare as the distance from the center increases, then:

Approximately 95% of the patients in the sample survived between 0 and 20.4 years or less after diagnosis.

Approximately 100% of the patients in the sample survived between 0 and 26.1 years or less after diagnosis.

Case Three | 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 at injury (in years) of the case is determined. The sample ages at injury are listed below:

4, 5, 5, 6, 6, 7, 7, 8, 9, 12, 12, 13, 14, 15, 15, 16, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 23, 25, 27, 27, 30, 30, 30, 31, 32, 32, 33, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 41, 41, 41, 42, 42, 45, 47, 50, 52, 60, 63, 65, 70, 70, 71, 71, 71, 71, 72, 72,  72, 73, 73, 74, 74, 75, 75, 76, 76, 76, 77, 79, 80, 81, 89, 90, 91

Compute and interpret a 97% confidence interval for the population mean age at injury of TBI patients. Show complete detail and work for full credit. Follow case study solutions and sample hourly keys in presenting your solutions.

Numbers

From row 2.20 0.013903 0.97219, Z=2.2.

N M SD se z lower97 upper97

87 41.4713 25.6161 2.74633 2.2 35.4293 47.5132

se = sd/sqrt(n) = 25.6161/sqrt(87) ≈ 2.74633;

lower97 = m - (z*se) = 41.4713 - (2.2*2.74633) ≈ 35.4293

upper97 = m + (z*se) = 41.4713 + (2.2*2.74633) ≈ 47.5132

Our population consists of patients with Traumatic Brain Injury (TBI). Our population mean is the population mean age at injury.

Each member of the family of samples (FoS) is a single random sample of 87 TBI patients. The FoS consists of all possible samples of this type.

From each member of the (FoS), compute:

m = sample mean age at injury

sd = sample standard deviation for the sample mean age at injury

se = sample standard error = sd/sqrt(87)

From row 2.20 0.013903 0.97219, Z=2.2.

and then compute se = sd/sqrt(87) and then the interval as:

[lower97 = m - (2.2*se), upper97 = m + (2.2*se)].

Computing this interval for each member of the FoS forms a family of intervals (FoI).

Approximately 97% of the FoI captures the true population mean age at injury for Traumatic Brain Injury

If our interval resides in this 97% supermajority, then the population mean age at injury for Traumatic Brain Injury is 35.4 and 47.5 years.

Case Four | Confidence Interval, Proportion | Traumatic Brain Injury (TBI) and the Extended Glasgow Outcome Scale (GOS-E)

The extended Glasgow Outcome Scale (GOS-E) was developed to address the limitations of the original Glasgow Outcome Scale (GOS), including the use of broad categories that are insensitive to change and difficulties with reliability due to lack of a structured interview format. The GOS-E extends the original 5 GOS categories to 8. The 8 categories are: Dead, Vegetative State, Lower Severe Disability, Upper Severe Disability, Lower Moderate Disability, Upper Moderate Disability, Lower Good Recovery, and Upper Good Recovery. A structured interview has been provided to improve reliability of rating. The extended Glasgow outcome scale (GOS-E):

1 (Dead),

2 (Vegetative state),

3 (Lower severe disability: completely dependent on others)

4 (Upper severe disability: dependent on others for some activities),

5 (Lower moderate disability: unable to return to work or participate in social activities)

6 (Upper moderate disability: return to work at reduced capacity, reduced participation in social activities)

7 (Lower good recovery: good recovery with minor social or mental deficits)

8 (Upper good recovery)

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. TBI includes: 1) the head being struck, 2) the head striking an object, and 3) the brain undergoing an acceleration/deceleration movement (i.e., whiplash) without direct external trauma to the head. Consider a random sample of patients surviving with TBI, with GOS-E at diagnosis + six months, listed below:

2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8

Define the event “Patient Presents Severe (GOS-E at 3 or 4) Disability via Assessment by GOS-E at Six Months after Injury” Estimate the population proportion for this event with 95% confidence. That is, compute and discuss a 95% confidence interval for this population proportion. Provide concise and complete details and discussion as demonstrated in the case study summaries.

Numbers

n=60

2, 2, 2, 2, 2 |, 3, 3, 3, 3, 3 |, 3, 3, 3, 3, 4 |, 4, 4, 4, 4, 4|, 4, 4, 4, 4, 4|, 4, 4, 4, 5, 5|, 5, 5, 5, 5, 5|, 5, 5, 5, 5, 5|, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7|, 7, 7, 7, 7, 8|, 8, 8, 8, 8, 8

event = “Patient Presents Severe (GOS-E at 3 or 4) Disability via Assessment by GOS-E at Six Months after Injury”

e = sample event count = 23

p = e / n = 23/60 = 0.38333

1 – p = 1 – (23/60) = 37/60 = 0.61667

sdp = sqrt(p*(1-p)/n) = sqrt((23/60)*(37/60)/60) = 0.062768

from 2.00 0.02275 0.95450, z=2.00

lower95 = p – z*sdp = 0.38333 – (2* 0.062768) = 0.25780

upper95 = p + z*sdp = 0.38333 + (2* 0.062768) = 0.50887

Our population consists of subjects surviving six or more months after acquiring a Traumatic Brain Injury. Our population proportion is the population proportion of subjects surviving six or monthsw after acquiring TBI whose extended Glasgow Outcome Scale score is severe (GOSE =3 or 4).

Each member of the family of samples (FoS) is a single random sample of 60 subjects surviving six or more months after acquiring a Traumatic Brain Injury. The FoS consists of all possible samples of this type.

From each member of the (FoS), compute:

e = sample count of subjects surviving six or monthsw after acquiring TBI whose extended Glasgow Outcome Scale score is severe (GOSE =3 or 4).

p = sample proportion of subjects surviving six or monthsw after acquiring TBI whose extended Glasgow Outcome Scale score is severe (GOSE =3 or 4) = e/60

sdp = square root of (p*(1 – p)/n )

from 2.00 0.02275 0.95450, z=2.00

and then compute the interval as: lower95 = p – z*sdp, upper95 = p + z*sdp.

Computing this interval for each member of the FoS forms a family of intervals (FoI).

Approximately 95% of the FoI captures the true population proportion of Our population proportion is the population proportion of subjects surviving six or monthsw after acquiring TBI whose extended Glasgow Outcome Scale score is severe (GOSE =3 or 4). If our interval resides in this 95% supermajority, then between 25.8% and 50.8% of Our population proportion is the population proportion of subjects surviving six or monthsw after acquiring TBI whose extended Glasgow Outcome Scale score is severe (GOSE =3 or 4).

Case Five | Hypothesis Test, Categorical Goodness of Fit | Traumatic Brain Injury (TBI) and the Extended Glasgow Outcome Scale (GOS-E)

Using the data and context of Case Four, consider these categories:

Dead/PVS: 1 (Dead), 2 (Vegetative state)

Severe: 3 (Lower severe disability), 4 (Upper severe disability)

Moderate: 5 (Lower moderate disability), 6 (Upper moderate disability)

Good: 7 (Lower good recovery), 8 (Upper good recovery)

Our null hypothesis is that TBI case outcomes are 15% Dead/PVS, 40% Severe, 25% Moderate and 20% Good. Test this Hypothesis. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

Numbers

2, 2, 2, 2, 2, (Dead/Vegetative @ 5)

3, 3, 3, 3, 3|, 3, 3, 3, 3, 4|, 4, 4, 4, 4, 4|, 4, 4, 4, 4, 4|, 4, 4, 4, (Severe @ 23)

5, 5, 5, 5, 5|, 5, 5, 5, 5, 5|, 5, 5, 5, 5, 6|, 6, 6, 6, (Moderate @ 18)

7, 7, 7, 7, 7|, 7, 7, 7, 8, 8|, 8, 8, 8, 8 (Good @ 14)

n = 5 + 23 + 18 + 14 = 60

Dead/Vegetative

E_{Dead/Vegetative} = .15*60 = 9

O_{Dead/Vegetative} = 5

Error_{Dead/Vegetative} = (O_{Dead/Vegetative} – E_{Dead/Vegetative} )²/ E_{Dead/Vegetative} = (5 – 9 )²/ 9 = 16/9

Severe

E_Severe = .40*60 = 24

O_Severe = 23

Error_Severe = (O_Severe – E_Severe )²/ E_Severe = (23 – 24 )²/ 24 = 1/24

Moderate

E_Moderate = .25*60 = 15

O_Moderate = 18

Error_Moderate = (O_Moderate – E_Moderate )²/ E_Moderate = (18 – 15 )²/159 = 9/15

Good

E_Good = .20*60 = 12

O_Good = 14

Error_Good = (O_Good – E_Good )²/ E_Good = (14 – 12 )²/ 12 = 4/12

Error = Error_{Dead/Vegetative} + Error_Severe + Error_Moderate + Error_Good =

1.777777778 + 0.041666667 + 0.6 + 0.333333333 =

2.752777778 over 4 categories

From 4 2.6430 0.450 and 4 2.9462 0.400 , .40 < p < .45

Our population consists of subjects surviving six or more months after acquiring a Traumatic Brain Injury.

Each member of the family of samples (FoS) is a single random sample of 60 subjects surviving six or more months after acquiring a Traumatic Brain Injury. The FoS consists of all possible samples of this type.

From each member of the (FoS), compute:

Dead/Vegetative: GOS-E = 1 or 2

E_{Dead/Vegetative} = .15*60 = 9

O_{Dead/Vegetative}

Error_{Dead/Vegetative} = (O_{Dead/Vegetative} – E_{Dead/Vegetative} )²/ E_{Dead/Vegetative}

Severe: GOS-E = 3 or 4

E_Severe = .40*60 = 24

O_Severe

Error_Severe = (O_Severe – E_Severe )²/ E_Severe

Moderate: GOS-E = 5 or 6

E_Moderate = .25*60 = 15

O_Moderate

Error_Moderate = (O_Moderate – E_Moderate )²/ E_Moderate

Good: GOS-E = 7 or 8

E_Good = .20*60 = 12

O_Good

Error_Good = (O_Good – E_Good )²/ E_Good

and finally:

Error = Error_{Dead/Vegetative} + Error_Severe + Error_Moderate + Error_Good

Repeating these calculations for each member of the family of samples yields a family of errors. If the true population proportions for the Extended Glasgow Outcome Scale are 15% Dead/Vegetative, 40% Severe, 25% Moderate and 20% Good, then between 40% and 45% of the members of the family of samples yield errors as bad as or worse than our error. Our sample does not appear to present statistically significant evidence against the null hypothesis.

Case Six | Hypothesis Test, Median | Traumatic Brain Injury (TBI) and Glasgow Coma Scale (GCS)

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,

GCS=Eye+Verbal+Motor.

Glasgow Coma Scale Scoring: Add scores across Eye, Motor, Verbal.

Eye opening: 4. Spontaneous. Indicates arousal, not necessarily awareness; 3. To speech. When spoken to – not necessarily the command to open eyes; 2. To pain. Applied to limbs, not face where grimacing can cause closure and 1. None.

Motor response: 6. Obeys commands. Exclude grasp reflex or postural adjustments; 5. Localises. Other limb moves to site of nailbed pressure 4. Withdraws. Normal flexion of elbow or knee to local painful stimulus; 3. Abnormal flexion. Slow withdrawal with pronation of wrist, adduction of shoulder 2. Extensor response. Extension of elbow with pronation and adduction and 1. No movement.

Verbal responses: 5. Orientated. Knows who, where, when; year, season, month; 4. Confused conversation. Attends & responds but answers muddled/wrong; 3. Inappropriate words. Intelligible words but mostly expletives or random; 2. Incomprehensible speech. Moans and groans only – no words and 1. None

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. A patient with mild traumatic brain injury is a person who has had a traumatically induced physiological disruption of brain function, as manifested by a least one of the following: Any period of loss of consciousness; Any loss of memory for events immediately before or after the accident; Any alteration in mental state at the time of the accident (eg, feeling dazed, disoriented, or confused); Any focal neurological deficit(s) that may or may not be transient; but where the severity of the injury does not exceed the following: posttraumatic amnesia (PTA) not greater than 24 hours, after 30 minutes, an initial Glasgow Coma Scale (GCS) of 13-15; and Any loss of consciousness of approximately 30 minutes or less. TBI includes: 1) the head being struck, 2) the head striking an object, and 3) the brain undergoing an acceleration/deceleration movement (ie, whiplash) without direct external trauma to the head.

Consider a random sample of patients with TBI, with GCS at initial treatment and diagnosis listed below:

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

Test the following: null (H₀): The median GCS at initial treatment and diagnosis is (h = 6) against the alternative (H₁): h > 6. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

Work all six (6) cases.

The alternative hypothesis is (H₁): h > 6 – “Guess (6) is too small.” The required error rule is then “Count above the Guess.”

n = 30

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

error = “sample count strictly above 6” = 20

From 30 20 0.04937, p = 0.04937

Our population consists of subjects with Traumatic Brain Injury. Our population median is the median Glasgow Coma Score. Our null hypothesis is that the population median Glasgow Coma Score at initial diagnosis and treatment is 6.

Each member of the family of samples (FoS) is a single random sample of 60 subjects with Traumatic Brain Injury. The FoS consists of all possible samples of this type.

From each member of the (FoS), compute the error as “number of sample subjects whose initial GCS is strictly greater than 6.”

Computing this error for each member of the family of samples yields a family of errors. If the true population median initial GCS for subjects with TBI is 6, then approximately 4.937% of the member samples of the family of samples yield errors as bad as or worse than our sample. This sample presents statistically 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 0.08850 0.82298 1.40 0.08075 0.83849 1.45 0.07352 0.85294 1.50 0.06680 0.86639 1.55 0.06057 0.87886 1.60 0.05479 0.89040 1.65 0.04947 0.90106 1.70 0.04456 0.91087 1.75 0.04005 0.91988 1.80 0.03593 0.92814 1.85 0.03215 0.93569 1.90 0.02871 0.94257 1.95 0.02558 0.94882 2.00 0.02275 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 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

Categories ERROR p-value 6 0.0000 1.000 6 1.6103 0.900 6 2.3425 0.800 6 2.9999 0.700 6 3.6555 0.600 6 4.3515 0.500 6 4.7278 0.450 6 5.1319 0.400 6 5.5731 0.350 6 6.0644 0.300 6 6.6257 0.250 6 7.2893 0.200 6 7.6763 0.175 6 8.1152 0.150 6 8.6248 0.125 6 9.2364 0.100 6 9.5211 0.090 6 9.8366 0.080 6 10.1910 0.070 6 10.5962 0.060 6 11.0705 0.050 6 11.6443 0.040 6 12.3746 0.030 6 13.3882 0.020 6 15.0863 0.010

Table 3. 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 “< 0.00001” 30 28 “< 0.00001” 30 29 “< 0.00001” 30 30 “< 0.00001”