Key

The 3rd Hourly

Math 1107

Spring 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 | Confidence Interval: Population Mean | The Framingham Heart Study

The objective of the Framingham Heart Study was to identify the common factors or characteristics that contribute to Cardiovascular disease (CVD) by following its development over a long period of time (since 1948)  in a large group of participants who had not yet developed overt symptoms of CVD or suffered a heart attack or stroke. Blood pressure is a measurement of the force applied to the walls of the arteries as the heart pumps blood through the body. Blood pressure readings are measured in millimeters of mercury (mm Hg) and usually given as 2 numbers. The top number is the systolic blood pressure reading. It represents the maximum pressure exerted when the heart contracts.  The bottom number is the diastolic blood pressure reading. It represents the pressure in the arteries when the heart is at rest. A sample of FHS adult subjects yields the following readings: (top/bottom)

130/70, 175/75, 136/84, 124/84, 144/88, 154/90, 164/97, 210/120, 110/75, 166/108, 100/70, 172/110, 160/90, 122/84, 162/80, 155/85, 120/65, 128/84, 130/90, 210/110, 110/68, 160/106, 132/72, 120/80, 200/104, 165/105, 132/88, 134/84, 152/74, 138/70, 122/80, 155/90, 166/108, 120/80, 200/130, 121/85, 150/100, 135/75, 140/78, 142/85, 146/94, 185/90, 166/78, 193/106, 160/80, 140/80, 150/110, 140/84, 130/82, 130/80, 238/122, 128/72, 220/118, 165/95, 208/114, 126/80, 140/90, 166/104, 130/70, 131/88

 

Estimate the population mean systolic blood pressure with 95% confidence. That is, compute and discuss a 95% confidence interval for this population mean. Show your work. Fully discuss the results. This discussion must include a clear discussion of the population and the population mean, the family of samples, the family of intervals and the interpretation of the interval.

 

Numbers

 

n       m          sd         se      Z    lower95    upper95

60    150.967    29.6573    3.82875    2    143.309    158.624

 

n = 60

m » 150.967

sd » 29.6573

se = sd/sqrt(n) = sd/sqrt(60) » 29.6573/sqrt(60) » 3.82875

Z = 2 from the mean/proportion row: 2.00 0.022750 0.95450

 

lower95 = m – Z*se  » 150.967 – 2*(29.6573/sqrt(60)) » 150.967 – 2*3.82875 » 143.309  

upper95 = m + Z*se  » 150.967 + 2*(29.6573/sqrt(60)) » 150.967 + 2*3.82875 » 158.624

 

Report the interval as [143.3,158.6].

 

Interpretation

Our population is the population of Framingham Heart Study subjects and our population mean is the mean systolic blood pressure (SBP, mm Hg).

Our Family of Samples (FoS) consists of every possible random sample of 60 Framingham Heart Study subjects.

From each member sample of the FoS, we compute the sample mean (m) and standard deviation (sd) for FHS SBP, and then compute the interval

[m – 2.00*( sd/sqrt(n)), m + 2.00*( sd/sqrt(n))].

Computing this interval for each member sample of the FoS, we obtain a Family of Intervals (FoI), approximately 95% of which cover the true population mean systolic blood pressure for Framingham Heart Study subjects.

If our interval, [143.3,158.6] is among the approximate 95% super-majority of intervals that cover the population mean, then the true population mean systolic blood pressure for Framingham Heart Study subjects is between 143.3 and 158.6 mm Hg.

Case Two | Confidence Interval: Population Proportion | Prenatal Care

 

A random sample of Year 2000 Georgia resident live births are checked for prenatal care status, in the following categories:

 

Prenatal Care Status

Number in Sample

Prenatal Care Began 1st Trimester (Months 1-3 of Pregnancy)

410

Prenatal Care Began 2nd Trimester (Months 4-6 of Pregnancy)

52

Prenatal Care Began 3rd Trimester (Months 7-9 of Pregnancy)

12

No Prenatal Care

6

Prenatal Care Status Unknown

20

Total

500

 

Compute and interpret a 99% confidence interval for the population proportion of year 2000 Georgia resident live births with prenatal care beginning in the 1st or 2nd trimesters. That is, compute and discuss a 99% confidence interval for this population proportion. Show your work. 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

 

n = 500

event = “Prenatal Care Begins in 1st or 2nd Trimester”

nevent = 410 + 52 = 462

pevent =nevent/n = 462/500 = .924

sdp = sqrt(pevent *(1 – pevent )/ n) » sqrt(.924*(1 – .924 )/ 500) » .01185

Z = 2.60 from row: 2.60 0.004661 0.99068

lower99 = pevent – Z*sdp » .924 – 2.6*.01185  » .893187

upper99 = pevent + Z*sdp » .924 + 2.6*.01185  » .954812

    

Report the interval as [.893, .954].

 

Interpretation

Our population is the population of Georgia resident live births occurring in the year 2000. We seek to estimated the population proportion of year 2000 Georgia resident live births where prenatal began in the 1st or 2nd trimester.

Our Family of Samples (FoS) consists of every possible random sample of 500 year 2000 Georgia resident live births.

From each member sample of the FoS, we compute the sample proportion p of live births where prenatal care began in the 1st or 2nd trimester, sdp = sqrt(p*(1–p)/n) and then compute the interval

[p – (2.60*sdp),  p + (2.60*sdp)].

Computing this interval for each member sample of the FoS, we obtain a Family of Intervals (FoI), approximately 99% of which cover the true population proportion of year 2000 Georgia resident live births where prenatal care began in the 1st or 2nd trimester.

If our interval, [.893, .954] is among the approximate 99% super-majority of intervals that cover the population proportion, then between 89.3% and 95.4% of year 2000 Georgia resident live births had prenatal care beginning in the 1st or 2nd trimester.

Case Three | Hypothesis Test: Population Median | The Framingham Heart Study

 

Using the Framingham Heart Study sample from Case One, test the following: null (H0): The median diastolic blood pressure (DBP) is 80 (h = 80) against the alternative (H1): h > 80.  Show your work. Fully discuss the results. This discussion must include a clear discussion of the population and the population median, the family of samples, the family of errors and the interpretation of the p-value.

 

Numbers

The data from Case One…The bottom number is the diastolic blood pressure reading. It represents the pressure in the arteries when the heart is at rest. A sample of FHS adult subjects yields the following readings: (top/bottom)

130/70, 175/75, 136/84, 124/84, 144/88, 154/90, 164/97, 210/120, 110/75, 166/108, 100/70, 172/110, 160/90, 122/84, 162/80, 155/85, 120/65, 128/84, 130/90, 210/110, 110/68, 160/106, 132/72, 120/80, 200/104, 165/105, 132/88, 134/84, 152/74, 138/70, 122/80, 155/90, 166/108, 120/80, 200/130, 121/85, 150/100, 135/75, 140/78, 142/85, 146/94, 185/90, 166/78, 193/106, 160/80, 140/80, 150/110, 140/84, 130/82, 130/80, 238/122, 128/72, 220/118, 165/95, 208/114, 126/80, 140/90, 166/104, 130/70, 131/88

 

n = 60

 

Error Form: “Guess is too small” – Count strictly above the guess:

 

136/84, 124/84, 144/88, 154/90, 164/97, 210/120, 166/108, 172/110, 160/90, 122/84, 155/85, 128/84, 130/90, 210/110, 160/106, 200/104, 165/105, 132/88, 134/84, 155/90, 166/108, 200/130, 121/85, 150/100, 142/85, 146/94, 185/90, 193/106, 150/110, 140/84, 130/82, 238/122, 220/118, 165/95, 208/114, 140/90, 166/104, 131/88

 

error = #FHS subjects in the sample whose diastolic blood pressure strictly exceeds 80 = 38

 

From the row: 60 38 0.01367,  p-value » 1.367%

 

Interpretation

Our population is the population of Framingham Heart Study subjects.

Our Family of Samples (FoS) consists of every possible random sample of 60 Framingham Heart Study subjects.

From each member sample of the FoS, we compute Error = Number of sample subjects whose diastolic blood pressure strictly exceeds 80 mm Hg. Computing this error for each member sample of the FoS, we obtain a Family of Errors (FoE).

If the true population diastolic blood pressure for Framingham Heart Study subjects is 80 mm Hg, then approximately 1.367% of the Family of Samples yield errors as bad as or worse than our single error. The sample presents significant evidence against the null hypothesis.

Case Four | Hypothesis Test: Categorical Goodness of Fit | Prenatal Care

 

A random sample of Year 2000 Georgia resident live births are checked for prenatal care status, in the following categories – consider the portion of the sample reporting prenatal care:

 

Prenatal Care Status

Number in Sample

Prenatal Care Began 1st Trimester (Months 1-3 of Pregnancy)

410

Prenatal Care Began 2nd Trimester (Months 4-6 of Pregnancy)

52

Prenatal Care Began 3rd Trimester (Months 7-9 of Pregnancy)

12

Test the hypothesis that:

Pr{ Prenatal Care Began 1st  Trimester (Months 1-3 of Pregnancy)} = .70

Pr{ Prenatal Care Began 2nd Trimester (Months 4-6 of Pregnancy)} = .20

Pr{ Prenatal Care Began 3rd Trimester (Months 7-9 of Pregnancy)} = .10 .

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. Show all work and detail for full credit.

Numbers

 

Prenatal Care Status

n

P

E=nP

Error

PNC T1

410

0.7

331.8

18.4305003

PNC T2

52

0.2

94.8

19.32320675

PNC T3

12

0.1

47.4

26.43797468

Total Sample with PNC

474

1

474

64.19168174

 

n=474

 

Prenatal Starts 1st Trimester

Observed=410

Expected = n*PT1= 474*.70 » 331.8

Error = (Observed – Expected)2/Expected » (410 – 331.8)2/331.8 » 18.4305003

 

Prenatal Starts 2nd Trimester

Observed=52

Expected = n*PT2= 474*.20 » 94.8

Error = (Observed – Expected)2/Expected » (52 – 94.8)2/94.8 » 19.32320675

 

Prenatal Starts 3rd Trimester

Observed=12

Expected = n*PT3= 474*.10 » 47.4

Error = (Observed – Expected)2/Expected » (12 – 47.4)2/47.4 » 26.43797468

 

Total Error » 18.4305003 + 19.32320675 + 26.43797468 » 64.19168174 over three categories.

 

From row: 3 9.2103 0.010, p-value < .01, since our error exceeds 9.2103. 

 

Interpretation

Our population is the population of Year 2000 Georgia resident live births. Our categories are based on those who reported receiving Prenatal Care and include: (T1)Prenatal Care Began 1st Trimester (Months 1-3 of Pregnancy), (T2)Prenatal Care Began 2nd Trimester (Months 4-6 of Pregnancy) and (T3)Prenatal Care Began 3rd Trimester (Months 7-9 of Pregnancy). Our null hypothesis is that the categories are distributed as: 70% T1, 20% T2  and 10% T3.

Our Family of Samples (FoS) consists of every possible random sample of 474 Year 2000 Georgia resident live births. Under the null hypothesis, within each member of the FoS, we expect approximately:

ET1 = N*PT1 = 474*.70 » 331.8

ET2  = N*PT2  = 474*.20 » 94.8

ET3 = N*PT3  = 474*.10 » 47.4

 

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

 

ET1 = N*PT1 = 474*.70 » 331.8

ErrorT1 = (OT1 ─ ET1)2/ ET1

 

ET2  = N*PT2  = 474*.20 » 94.8

ErrorT2  = (OT2  ─ ET2 )2/ ET2  

 

ET3 = N*PT3  = 474*.10 » 47.4

ErrorT3 = (OT3 ─ ET3)2/ ET3

 

Then add the individual errors for the total error as Total Error = ErrorT1 + ErrorT2  + ErrorT3

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

If the prenatal care categories are distributed as: 70% T1, 20% T2 and 10% T3, then fewer 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