Key | The 3rd Hourly | Math 1107 | Fall 2009

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 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.

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 | Hypothesis Test – Population Median | Glioblastoma Multiforme

 

Glioblastoma multiforme (GBM) is the highest grade glioma tumor and is the most malignant form of astrocytomas. These tumors originate in the brain. GBM tumors grow rapidly, invade nearby tissue and contain cells that are very malignant. GBM are among the most common and devastating primary brain tumors in adults.

 

Suppose that we have a random sample of GBM patients, with survival time (in months) listed below:

 

0, 1, 2, 2, 3 | 3, 3, 3, 4, 4 | 4, 4, 4, 5, 5 | 5, 5, 5, 5, 6 | 6, 6, 7, 7, 8 | 8, 8, 9, 10, 10 | 11, 11, 11, 12, 12

12, 13, 13, 13, 14 | 14, 15, 16, 17, 17| 18, 18, 19, 23, 24 | 24, 25, 27, 30, 36 | 38, 40, 58, 60, 61

 

Test the following: null (H0): The median GBM survival time is 20 months (h = 20) against the alternative (H1):

h < 20.  Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

 

Null Hypothesis: Median Survival Time = 20 Months

Alternative  Hypothesis: Median Survival Time < 20 Months (Guess is too Large)

Error Function: Number of Sample Patients Surviving Strictly Less Than 20 Months

 

0, 1, 2, 2, 3 | 3, 3, 3, 4, 4 | 4, 4, 4, 5, 5 | 5, 5, 5, 5, 6 | 6, 6, 7, 7, 8 | 8, 8, 9, 10, 10 | 11, 11, 11, 12, 12

12, 13, 13, 13, 14 | 14, 15, 16, 17, 17| 18, 18, 19, ||||20||||,  23, 24 | 24, 25, 27, 30, 36 | 38, 40, 58, 60, 61

 

sample error = Number of Sample Patients Surviving Strictly Less Than 20 Months = 48

n = sample size = 60

from 60 48 <0.00001, p < 0.00001 < .01 (p is strictly less than 1/100,000)

 

Our population consists of patients who have been diagnosed with and who have died with glioblastoma multiforme (GBM). Our null hypothesis is that the population median survival time for this population is 20 months.

 

Each member of the family of samples (FoS) is a single random sample of 60 patients who have been diagnosed with and who have died with glioblastoma multiforme (GBM). The FoS consists of all possible samples of this type.

 

From each member of the (FoS), compute an error as the number of sample GBM patients surviving strictly less than 20 months. Computing this error for each member of the FoS forms a family of errors (FoE).

 

If the true population median survival time for GBM patients is 20 months, then fewer than .001% of  member samples from the FoS yield errors as bas as or worse than our error. The sample presents highly significant evidence against the null hypothesis.

 

Case Two | Confidence Interval: Population Proportion | Glioblastoma Multiforme

 

Using the Glioblastoma multiforme (GBM) data from Case One, consider the proportion of GBM patients who survive 18 months or longer. Compute and interpret a 97% confidence interval for this population proportion. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

 

0, 1, 2, 2, 3 | 3, 3, 3, 4, 4 | 4, 4, 4, 5, 5 | 5, 5, 5, 5, 6 | 6, 6, 7, 7, 8 | 8, 8, 9, 10, 10 | 11, 11, 11, 12, 12

12, 13, 13, 13, 14 | 14, 15, 16, 17, 17| 18, 18, 19, 23, 24 | 24, 25, 27, 30, 36 | 38, 40, 58, 60, 61

 

event = GBM patient survives 18 months or longer

e = event count = 15

n = sample size = 60

p = sample proportion of GBM patients surviving 18 months or longer = 15/60 = .25

sdp = square root of (p*(1 – p)/n ) = sqrt( (.25*.75)/60) )  = 0.055902   

from 2.20 0.013903 0.97219, Z=2.20

lower97 = p –  (Z*sdp) = .25 –  (2.2*0.055902)  = 0.12702

upper97 = p +  (Z*sdp) = .25 +  (2.2*0.055902)  = 0.37298

  

Our population consists of patients who have been diagnosed with and who have died with glioblastoma multiforme (GBM). Our population proportion is the population proportion of GBM patients surviving 18 or more months.

 

Each member of the family of samples (FoS) is a single random sample of 60 patients who have been diagnosed with and who have died with glioblastoma multiforme (GBM). The FoS consists of all possible samples of this type.

 

From each member of the (FoS), compute: 

 

e = sample count of GBM patients surviving 18 months or longer

p = sample proportion of GBM patients surviving 18 months or longer = e/60

sdp = square root of (p*(1 – p)/n )  

from 2.20 0.013903 0.97219, Z=2.20

 

and then compute the interval as: lower97  = p –  (2.20*sdp), upper97 = p +  (2.20*sdp).

 

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

 

Approximately 97% of the FoI captures the true population proportion of GBM patients survivning 18 months or longer. If our interval resides in this 97% supermajority, then between 12.7% and 37.2% of GBM patients survive 18 or more months past diagnosis.

 

Case Three | Confidence Interval: Population Mean | Gestational Age

 

Gestational age is the time spent between conception and birth, usually measured in weeks. In general, infants born after 36 or fewer weeks of gestation are defined as premature, and may face significant challenges in health and development. Infants born after 37-40 weeks of gestation are generally viewed as full term, and those born after 41 or more weeks of gestation are generally viewed as post term. Suppose that a random sample of 2007 US resident live born infants yields the following gestational ages (in weeks):

 

31, 32, 33, 34, 34, 34, 35, 35, 36, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 41, 41, 41, 41, 42, 42

 

Estimate the population mean gestational age for 2007 US resident live born infants with 96% confidence. That is, compute and discuss a 96% confidence interval for this population mean. Provide concise and complete details and discussion as demonstrated in the case study summaries.

 

n       m             sd               se             Z     lower96    upper96

60    38.0167    2.32519    0.30018    2.1    37.3863    38.6470

 

from 2.10 0.017864 0.96427, Z = 2.1;

se = sd/sqrt(n) = 2.32519/sqrt(60) = 0.30018;

 

lower96 = m - (z*se) = 38.0167 - (2.1*0.30018) = 37.3863

upper96 = m + (z*se) = 38.0167 + (2.1*0.30018) = 38.6470

 

Our population consists of US resident live births occurring in 2007. Our population mean is the population mean gestational age in weeks.

 

Each member of the family of samples (FoS) is a single random sample of 60 year 2007 US resident live born infants.. The FoS consists of all possible samples of this type.

 

From each member of the (FoS), compute: 

 

m = sample mean gestational age in weeks

sd = sample standard deviation for the sample mean gestational age in weeks

se = sample standard error = sd/sqrt(60)

from 2.10 0.017864 0.96427, Z = 2.1;

 

and then compute the interval as: lower96 = m - (2.1*se), upper96 = m + (2.1*se).

 

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

 

Approximately 96% of the FoI captures the true population mean gestational age for year 2007 US resident live born infants. If our interval resides in this 96% supermajority, then the population mean gestational age for year 2007 US resident live born infants is between 37.4 and 38.6 weeks.

 

Case Four | Hypothesis Test: Categorical Goodness of Fit | Gestational Age

 

Gestational age is the time spent between conception and birth, usually measured in weeks. In general, infants born after 36 or fewer weeks of gestation are defined as premature, and may face significant challenges in health and development. Infants born after 37-40 weeks of gestation are generally viewed as full term, and those born after 41 or more weeks of gestation are generally viewed as post term. Using the data and context from Case Three, test the null hypothesis that the 2007 US resident live births are are distributed as 15% Premature, 82% Full Term and 3% Post Term. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries. Fully discuss the testing procedure and results. This discussion must include a clear discussion of the population and the null hypothesis, the family of samples, the family of errors and the interpretation of the p-value.

 

Prematurity (< 37 weeks): 31, 32, 33, 34, 34 | 34, 35, 35, 36 (9)

 

Full Term (37-40 weeks): 37, 37, 37, 37, 37| 37, 37, 37, 37, 37| 37, 37, 37, 38, 38| 38, 38, 38, 38, 38|

38, 38, 38, 38, 38| 38, 39, 39, 39, 39| 39, 39, 40, 40, 40| 40, 40, 40, 40, 40| 40, 40, 40, 40, 40| (45)

 

Post Term (41+ weeks): 41, 41, 41, 41, 42 | 42 (6)

 

Total = 9 + 45 + 6 = 54 + 6 = 60

 

Expected Counts from the Null Hypothesis for n=60

 

ePremature = 60*PPremature = 60*.15 = 9

eFull = 60*PFull = 60*.82 = 49.2

ePost = 60*PPost = 60*.03 = 1.8

 

Error Calculations

 

errorPremature = (nPremature – ePremature )2/ ePremature = (9 – 9 )2/ 9 = 0

errorFull = (nFull – eFull )2/ eFull = (45 – 49.2 )2/ 49.2 = 0.358536

errorPost = (nPost – ePost )2/ ePost = (6 – 1.8 )2/1.8 = 9.80000

errorTotal = errorPremature +  errorFull +  errorPost = 0 + 0.358536 + 0.000813 +  9.8 = 10.158536

 

From 3 9.2103 0.010, p < 0.01 – the p-value is less than 1%.

  

Interpretation

Our population consists of US resident live births occurring in 2007. Our categories are based on gestational age: Prematurity(<37 weeks), Full Term(37-41 weeks) and Post Term (42 or more weeks). Our null hypothesis is that the categories are distributed as: 15% Prematurity, 82% Full Term and 3% Post Term.

Our Family of Samples (FoS) consists of every possible random sample of 60 US resident live births occurring in 2007. Under the null hypothesis, within each member of the FoS, we expect approximately:

ePremature = 60*PPremature = 60*.15 = 9

eFull = 60*PFull = 60*.82 = 49.2

ePost = 60*PPost = 60*.03 = 1.8

 

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

 

errorPremature = (nPremature – ePremature )2/ ePremature

errorFull = (nFull – eFull )2/ eFull

errorPost = (nPost – ePost )2/ ePost

 

Then add the individual errors for the total error as errorTotal = errorPremature +  errorFull +  errorPost.

 

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

If the gestational age categories for US resident live births occurring in 2007 are distributed as 15% Prematurity, 82% Full Term and 3% Post Term, 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 highly significant evidence against the null hypothesis.

Work all four (4) cases.

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

60       0     1.00000                                     60       1     1.00000                                     60       2     1.00000                                     60       3     1.00000                                     60       4     1.00000                                     60       5     1.00000                                     60       6     1.00000                                     60       7     1.00000                                     60       8     1.00000                                     60       9     1.00000                                     60      10     1.00000                                     60      11     1.00000                                     60      12     1.00000                                     60      13     0.99999                                     60      14     0.99998                                     60      15     0.99993                                     60      16     0.99980                                     60      17     0.99947                                     60      18     0.99866                                     60      19     0.99689                                     60      20     0.99326                                    

n         error          base p-value

60      21     0.98633                                     60      22     0.97405                                     60      23     0.95377                                     60      24     0.92250                                     60      25     0.87747                                     60      26     0.81685                                     60      27     0.74052                                     60      28     0.65056                                     60      29     0.55129                                     60      30     0.44871                                     60      31     0.34944                                     60      32     0.25948                                     60      33     0.18315                                     60      34     0.12253                                     60      35     0.07750                                     60      36     0.04623                                     60      37     0.02595                                     60      38     0.01367                                     60      39     0.00674                                     60      40     0.00311                                      

n         error          base p-value

60      41     0.00134                                     60      42     0.00053                                     60      43     0.00020                                     60      44     0.00007                                     60      45     0.00002                                     60      46     0.00001                                     60      47     <0.00001                                     60      48     <0.00001                                     60      49     <0.00001
                                 60      50     <0.00001                                   60      51     <0.00001                                   60      52     <0.00001                                   60      53     <0.00001                                   60      54     <0.00001                                   60      55     <0.00001                                   60      56     <0.00001                                   60      57     <0.00001                                   60      58     <0.00001                                        60      59     <0.00001                                            60      60     <0.00001        

   Table 3. 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