A round-robin tournament involving n plays is modeled with digraph D where, for every two distinct vertices (players) u and v, either (u,v) is an edge (player u defeats player v) or (v,u) is an edge (player v defeats play u). Prove that if D is acyclic, i.e., no directed cycles, then there always exists a player who has defeated everyone (out-degree is n – 1) and a player who has lost to everyone (in-degree is n – 1).

(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0.

(4) Prove that for any field F, F is a vector space over F.

(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over R (with respect to the usual addition and scalar multiplication of polynomials).

For a comparison of two study guides for a mathematics course, 14 student volunteers were found. They were randomly assigned, 7 to study guide A, and 7 to study guide B. Following a two-day period of independent study, the students were examined on the material. The students received the following scores (out of 100):

Study Guide A: 95 96 97 80 92 97 95
Study Guide B: 72 73 80 69 78 74 73

Use a permutation test to test the null hypothesis that the distributions of scores are the same for each study guide, against the two-sided alternative that the distributions are different.

1. How many possible ways are there to randomly allocate the 14 students into two groups of 7 students?  [2 pt(s)]

2. Compute the difference in sample means, A(bar) - B(bar) as the test statistic.  [1 pt(s)]

3. How many arrangements of the data would lead to an absolute value of the computed test statistic as great or greater than the absolute value of the test statistic you calculated in the previous question? Note that you don't have to list all possible arrangements in order to answer this. Examine the data carefully and use the trick that was shown in class (also on pages 8-9 of the typed notes) for the permutation problem.  [5 pt(s)]

4. Suppose that after examining all the possible arrangements, it is found that the absolute value of the difference in means for 16 of the arrangements (including the real one that you observed) are greater than or equal to the absolute value of the computed test statistic. What is the two-sided p-value for the permutation test? Use at least 5 digits to the right of the decimal.  [2 pt(s)]

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.

X is an independent standard uniform random variable X ∼ Uniform(0, 1)

Y is an independent standard uniform random variable Y ∼ Uniform(0, 1)

U = min(X, Y )

V = max(X, Y )

Find the correlation coefficient of V and U , ρ(U, V) = Correlation(U, V).

What is Cramer's V for each of the following values for the chi-square test for independence? (Round your answers to two decimal places.)

(a) X2 = 3.95, n = 50, dfsmaller = 1 V =

(b) X2 = 8.81, n = 110, dfsmaller = 2 V =

(c) X2 = 11.62, n = 180, dfsmaller = 3 V =

Directions: Solve the question 1st, then place the answer in the appropriate blank.

In Drosophila, cinnabar eye color (c), vestigal wings (v) and plexus wing veins (p) are all autosomal recessive mutations linked on chromosome 2. A true-breeding cinnabar, vestigal winged female was crossed to a true-breeding plexus vein male. The wild type F1 females were mated to a plexus, vestigal winged male with cinnabar eyes and the following progeny were obtained.  

Please solve the problem and complete the following fill-in the blank questions.

a. What is the gene order on chromosome 2?  

b. What is the distance between cinnabar eyes and vestigal wings?  

c. What is the distance between vestigal wings and plexus veins?  

d. What is the distance between cinnabar eyes and plexus veins?  

e. What is the interference in this cross?  

1. In a person without Type 2 diabetes and with a 10-year ASCVD risk determined to be 15% with two other risk factors, what do the 2018 cholesterol guidelines suggest?
   1. The person may be withheld statin therapy because his 10-year risk is <20%
   2. The person should be considered for moderate intensity statin therapy
   3. The person is recommended for high intensity statin therapy
   4. None of the above
2. Which of the following is NOT true regarding the Pooled Cohort Risk Score (also called the ASCVD Risk Estimator)?
   1. It is not appropriate for use in persons with pre-existing myocardial infarction
   2. It predicts the lifetime risk of ASCVD in persons aged 20-59 years
   3. It includes the prediction of total cardiovascular disease including heart failure
   4. It allows for the prediction of ASCVD in persons with diabetes
   5. All of the above are true

Read the beginning of the article “Control of Cardiovascular Risk Factors in Patients With Diabetes and Hypertension at Urban Academic Medical Centers” in Diabetes Care, Volume 25, Number 4, April 2002 (http://care.diabetesjournals.org/content/25/4/718.long). For now, you need only to read up to the “Data Analysis” section on p. 719 (top of second column). Answer the questions below regarding the article. Note: The “cohort” in the article is the sample.

a) What was the objective of the study?

b) Was this a retrospective study or a prospective study? (Circle one)

c) What was the target population?

d) How many patients were in the sample?

e) How were the data collected (e.g., survey, experiment, chart review,…)?

f) What percentage of subjects in the study met the combined ADA goal for BP, LCL cholesterol, and HbA_1c?