Key | The Comprehensive Final Examination | Math 1107 | Spring 2011 | 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 toolsheets, 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 | Probability Computation Rules

 

Here is our slot machine – on each trial, it produces a 6-color sequence, using the table below:

 

Sequence*

Probability

BBBBBB

.10

BBGGBG

.10

RRYYGG

.15

GBBGGY

.10

RGBYYR

.25

BBYYGR

.10

YBGRYY

.20

Total

1.00

* B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as 1st to 6th , from left to right: (1st 2nd 3rd 4th 5th6th )

Compute the following probabilities – if a rule is specified, you must use that rule:

 

Pr{ Blue Shows | Green Shows } – This is a conditional probability.

 

Sequence*

Probability

BBGGBG

.10

RRYYGG

.15

GBBGGY

.10

RGBYYR

.25

BBYYGR

.10

YBGRYY

.20

Total

0.90

 

Pr{ Green Shows } =

Pr{One of BBGGBG, RRYYGG, GBBGGY, RGBYYR, BBYYGR

or YBGRYY Shows} = Pr{BBGGBG}+Pr{RRYYGG}+Pr{GBBGGY}+Pr{RGBYYR}+Pr{BBYYGR}+Pr{YBGRYY} =

 .10 + .15 + .10 + .25 + .10 + .20 = 0.90

 

Sequence*

Probability

BBGGBG

.10

GBBGGY

.10

RGBYYR

.25

BBYYGR

.10

YBGRYY

.20

Total

0.75

 

Pr{ Blue and Green Show } =

Pr{One of BBGGBG, GBBGGY, RGBYYR, BBYYGR

or YBGRYY Shows} = Pr{BBGGBG}+ Pr{GBBGGY}+Pr{RGBYYR}+Pr{BBYYGR}+Pr{YBGRYY} =

 .10 + .10 + .25 + .10 + .20 = 0.75

 

Pr{ Blue Shows | Green Shows } = Pr{ Blue and Green Show }/ Pr{ Green Shows } = .75/.90

 

Pr{ Red and Blue Show } – Use the additive rule.

 

Sequence*

Probability

RGBYYR

.25

BBYYGR

.10

YBGRYY

.20

Total

.55

 

Pr{ Red and Blue Show } =

Pr{One of RGBYYR, BBYYGR or YBGRYY Shows} = Pr{RGBYYR}+Pr{BBYYGR}+Pr{YBGRYY} =

.25 + .10 + .20  =  0.55

 

Pr{ Yellow Shows } – Use the complementary rule. 

 

Other Event = “Yellow Does Not Show”

 

Sequence*

Probability

BBBBBB

.10

BBGGBG

.10

Total

.20

 

Pr{Yellow does not Show} = Pr{ BBBBBB } + Pr{ BBGGBG } = .10 + .10 = 0.20

Pr{Yellow Shows} = 1 – Pr{Yellow does not Show} = 1 –0.20 = 0.80

 

Optional Check:

 

Sequence*

Probability

RRYYGG

.15

GBBGGY

.10

RGBYYR

.25

BBYYGR

.10

YBGRYY

.20

Total

.80

 

Pr{Yellow Shows} = Pr{One of RRYYGG, GBBGGY, RGBYYR, BBYYGR or YBGRYY Shows} = Pr{RRYYGG} + Pr{GBBGGY} + Pr{RGBYYR} + Pr{BBYYGR} + Pr{YBGRYY } =

.15 + .10 + .25+ .10 + .20 = .80

 

Show full work and detail for full credit.

 

 

 

Case Two | Descriptive Statistics | C-reactive Protein

Inflammation is the first response of the immune system to infection or irritation. Inflammation is characterized by redness, heat, swelling, pain and dysfunction of the organs involved. The C-reactive protein (CRP) owes its name to the ability of this protein to precipitate pneumococcal C-polysaccharide in the presence of calcium. It was first discovered in 1930 by Tillet and Frances. CRP is a direct measure of the level of inflammation in the body. CRP Elevations in CRP are seen in infectious, inflammatory, and malignant diseases but also with pregnancy, and trauma. Consider a random sample of adult US residents. From each sample subject we obtain serum, and determine the serum CRP level as milligrams CRP per deciliter serum (mg/dL). Here is the CRP data:

0.1, 0.15, 0.20, 0.3, 0.35, 0.40, 0.47, 0.49, 0.52, 0.57, 0.63, 0.65, 0.70, 0.72, 0.73, 0.75, 0.77, 0.80, 0.83, 0.90, 0.92, 0.93, 0.94, 0.95, 0.98, 1.03, 1.10, 1.15, 1.20, 1.25, 2.5, 3.4, 5.6, 5.4, 5.5, 5.8, 6.0, 7.5, 8.9, 9.5, 11.1, 11.5, 12, 13, 15, 20, 25, 27, 30, 35

Compute and interpret the following statistics: sample size, p00, p25, p50, p75, p100, (p100-p50), (p75-p25) and (p50-p00). Show your work, and discuss completely for full credit.

Numbers

n    p00    p25     p50     p75    p100      R42     R31     R20

50    0.1    0.7    1.005    7.5     35     33.995    6.8    0.905

 

R42 = p100 – p50  = 35 – 1.005 = 33.995

R31 = p75 – p25 = 7.5 – 0.7 = 6.8

R20 = p50 –p00  = 1.005 – 0.1 = 0.905     

 

Interpretation

 

There are 50 US Resident Adults in our sample.

The adult in our sample with the lowest CRP level has CRP at 0.1 mg/dL.

Approximately 25% of the adults in our sample have CRP levels at 0.7 mg/dL or less.

Approximately 50% of the adults in our sample have CRP levels at 1.005 mg/dL or less.

Approximately 75% of the adults in our sample have CRP levels at 7.5 mg/dL or less.

The adult in our sample with the highest CRP level has CRP at 35 mg/dL.

Approximately 50% of the adults in our sample have CRP levels between 1.005 and 35 mg/dL. The largest difference in CRP between any pair of adults in this upper half sample is 33.995 mg/dL. 

Approximately 50% of the adults in our sample have CRP levels between 0.7 and 7.5 mg/dL. The largest difference in CRP between any pair of adults in this middle half sample is 6.8 mg/dL.

Approximately 50% of the adults in our sample have CRP levels between 0.1 and 1.005 mg/dL. The largest difference in CRP between any pair of adults in this lower half sample is 0.905 mg/dL.

Case Three | Confidence Interval Proportion | C-reactive Protein

Using the context and data from case two, consider the proportion of adult US residents present CRP levels at 10 or higher with 95% confidence interval for this population proportion. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries.

Event = “CRP at 10 mg/dL or more”

0.1, 0.15, 0.20, 0.3, 0.35, 0.40, 0.47, 0.49, 0.52, 0.57, 0.63, 0.65, 0.70, 0.72, 0.73, 0.75, 0.77, 0.80, 0.83, 0.90, 0.92, 0.93, 0.94, 0.95, 0.98, 1.03, 1.10, 1.15, 1.20, 1.25, 2.5, 3.4, 5.6, 5.4, 5.5, 5.8, 6.0, 7.5, 8.9, 9.5, 11.1, 11.5, 12, 13, 15 | 20, 25, 27, 30, 35

Numbers

 

n = 50

e = sample event count = 10

p = sample event proportion = e/n = 10/50 = .20

1 – p =  1 – .20 = .80

standard error proportion = sdp = sqrt(p*(1 –p)/n) = sqrt(.2*.8/50) ≈ sqrt(0.0032) ≈ 0.056569

From 2.00  0.022750  0.95450, Z≈2.00

lower95 = p – (Z*sdp) ≈ .20 – (2*0.056569) ≈ 0.086862915          

upper95 = p + (Z*sdp) ≈ .20 + (2*0.056569) ≈ 0.313137085

 

Interpretation

We estimate the population proportion of US resident adults whose serum C Reactive Protein (CRP) level is at 10 mg/dL or greater.

Each member of the family of samples is a single random sample of 50 adults. The family of samples consists of possible samples of this type.

From each member sample, compute:

e = sample event count (serum CRP is 10 mg/dL or greater), p = sample event proportion = e/n, standard error proportion = sdp = sqrt(p*(1 –p)/n), and the interval as [lower95 = p – (Z*sdp), upper95 = p + (Z*sdp)]. Doing this for each member of the family of samples yields a family of intervals, approximately 95% of which capture the population proportion of US resident adults whose serum C Reactive Protein (CRP) level is at 10 mg/dL or greater. If our interval is in this 95% supermajority, then between 8.68% and 31.31% of US Resident adults have serum CRP levels of 10 mg/dL or greater.

Case Four | Median Test | Green Lynx Spiders

The green lynx spider, Peucetia viridans, is a conspicuous, large, bright green spider found on many kinds of shrub-like plants throughout the southern United States and is the largest North American lynx spider. Although it is common throughout Florida and aggressively attacks its insect prey, it very seldom bites humans. Lynx spiders get their name from the way that they sometimes pounce on their prey in a catlike fashion.

 

A random sample of green lynx spiders (male) yields the following body lengths (excluding legs), in millimeters per spider: 

4.2, 4.7, 5.5, 5.7, 7.8 | 8.5, 9.8, 10.2, 10.9, 11.2 | 12.0, 12.1, 12.2, 12.2, 12.3 | 12.5, 12.5, 12.6, 12.6, 12.7 

12.8, 12.9, 13.1, 13.2, 13.5 | 13.6, 14.2, 15.8, 16.4, 20.2

 

Test the following: null (H0): The median body length for male Green Lynx spiders is 14 mm (h = 14) against the alternative (H1): h ≠ 14. Completely discuss and interpret your test results, as indicated in class and case study summaries. Show all work and detail for full credit.

4.2, 4.7, 5.5, 5.7, 7.8 | 8.5, 9.8, 10.2, 10.9, 11.2 | 12.0, 12.1, 12.2, 12.2, 12.3 | 12.5, 12.5, 12.6, 12.6, 12.7 

12.8, 12.9, 13.1, 13.2, 13.5 | 13.6, 14.2, 15.8, 16.4, 20.2

n = 30

Sample Count Below 14: 26

Sample Count Above 14: 4

Sample Error for Two-Sided Test = Max(26, 4) = 26

From the row 30 26 0.00003, base p-value = 0.00003

p-value for two-sided test = 2*0.00003 = 0.00006

 

Interpretation

 

Our test concerns the median Green Lynx body length (male).

 

Each member of the family of samples is a single random sample of 30 Green Lynx spiders (male). The family of samples consists of all possible samples of this type.

 

From each member of the family of samples, compute the sample error as the maximum of the number of spiders in the sample whose body lengths are strictly less than 14 millimeters, and the number of spiders in the sample whose spider legths are strictly larger than 14 millimeters. Computing this error for each member sample yields a family of errors.

 

If the true population Green Lynx spider body length (male) is 14 millimeters, then approximately 0.0006% of the samples yield errors equal to or more extreme than our sample. The sample presents highly significant evidence against the null hypothesis.

 

Table: 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   0.146860   0.70628

      1.10   0.135670   0.72867

1.15  0.125070  0.74986

1.20  0.115070  0.76986

1.25  0.105650  0.78870

1.30  0.096800  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  0.0034670 0.99307

2.75  0.0029798 0.99404

2.80  0.0025551 0.99489

2.85  0.0021860 0.99563

2.90  0.0018658 0.99627

2.95  0.0015889 0.99682

3.00  0.0013499 0.99730

 

 

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

Table: Medians

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