Background: You are a clinical psychologist who is interested in examining polysubstance use among a populations of legal staff in local law firms. As part of a research project you have collected data from 32 employees serving in various positions in the law firms of a large metropolitan area.

Data: The following table provides information on Role (1 = intern, 2 = paralegal, 3 = lawyer) Sex (1 = female, 2 = male), Alcohol Use (scale; 0 = does not drink, 10 = heavy drinker), and Marijuana Use (scale; 1 = does not use, 10 = heavy user) of the 32 firm employees in your sample. Your task is to enter this data into an SPSS data file such that each column represents a variable and each row represents an individual person. Please make sure that you treat Role and Sex as nominal variables and Alcohol Use and Marijuana Use as scale variables. After you complete data entry please double check all values. Any error that you make at this stage will carry over to procedures that you perform later. You should have 32 rows in the data file each representing a sample participant.

Be sure to copy and paste relevant SPSS syntax as part of your answers to any item requiring analysis, so I know how you came to the answers you did.

Q1. Compute a z-score on Alcohol Use for participants 6 and 29.

2. Prove that |[0, 1]| = |(0, 1)|

Two firms, 1 and 2, are engaged in Bertrand price competition. There are10possible buyers, each of whom is willing to pay up to $4, and no more, for an item the firms sell .Firms 1 and 2 have identical unit costs of $2. However, each firm has a capacity of 8units, so that it cannot satisfy the whole market by itself (it can only satisfy 8 possible buyers, at most).The firms simultaneously announce prices, p1 and p2 respectively and the prices are publicly known. If p1=p2<=4; the first five buyers buy from firm 1 and the remaining 5from firm 2. Then firm 1's profit becomes (p1-2)*5 and similarly for firm 2.If p1< p2<=4; the first eight buyers buy from firm 1 and the remaining 2 from firm 2.If p2< p1<=4; then only the first two buyers buy from firm 1 and the remaining 8 from firm 2. If a firm charges a price above $4, it does not get any buyers. The prices p1;,p2 are restricted to be $0,$1, $2, $3, $4..

Find all the Nash equilibria in pure strategies for this game.

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true?

4. Prove that the following is not true by finding a counterexample.

“The sum of any 3 consecutive integers is even"

5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd

6.  Show an informal Direct Proof for “The sum of 2 even integers is even.”

Recursive Definitions

7.  The Fibonacci Sequence is defined as follows:                             

            F(1) = 1

            F(2) = 1

            F(n) = F(n-1) + F(n-2) for n>2.

     The first 10 numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

      Find F(13) = ______________ and F(14) =______________

8.  Prove the following property of the Fibonacci numbers directly from the definition.                                                                                                          

            F(n+3) = 2F(n+1) + F(n-3)+ 2F(n-2)

9.  Find the 3^rd, 4^th and 5^th values in the sequence.

            D(1) = 0

            D(n) = 2D(n-1) + 2 for n>1.

            D(2) = _______________________________________________

            D(3) = _______________________________________________

            D(4) = _______________________________________________

10.  Find the 3^rd, 4^th and 5^th values in the sequence.

            D(1) = 1

            D(2) = 4

            D(n) = 2D(n-1) + D(n-2) for n>2.

            D(3) = _______________________________________________

D(4) = _______________________________________________

            D(5) = _______________________________________________

11.  Using the SelectionSort algorithm on p 169 in your textbook, simulate the execution of the algorithm on the following list L; write the list after every exchange.  

L:    9, 2, 4, 8, 6

_______________

_______________

_______________

_______________

The next three questions will refer to the following game table:

Q1. In the space below, answer the following questions:

• Does this game contain any strictly dominated strategies? (You do not need to list them right now.)
• If so, perform IESDS until none are left. List the order in which you eliminated strategies.

Q2. In the space below, answer the following questions:

• Does this game contain any non-rationalizable strategies? (You do not need to list them right now.)
• If so, perform IENBR until none are left. List the order in which you eliminated strategies.

Q3. In the space below, answer the following questions:

• Did either IESDS or IENBR, by themselves, reveal the Nash equilibrium of the game?
• What are the Nash equilibria of the game? (If they were not revealed by IESDS or IENBR, use any method you like to find them.)

y=c(18.90, 17, 20, 18.25, 20.07, 11.2, 22.12, 21.47, 34.70, 30.40, 16.50, 36.50, 21.50, 19.70, 20.30, 17.80, 14.39, 14.89, 17.80, 16.41, 23.54, 21.47, 16.59, 31.90, 29.40, 13.27, 23.90, 19.73, 13.90, 13.27, 13.77, 16.50)

x1=c(350, 350, 250, 351, 225, 440, 231, 262, 89.7, 96.9, 350, 85.3, 171, 258, 140, 302, 500, 440, 350, 318, 231, 360, 400, 96.9, 140, 460, 133.6, 318, 351, 351, 360, 350)

x2=c(4, 4, 1, 2, 1, 4, 2, 2, 2, 2, 4, 2, 2, 1, 2, 2, 4, 4, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4)

The le hmw7 prob4.R contains data on the gasoline mileage performance of 32 dierent au-
tomobiles.
(a) Fit a simple linear regression model relating gasoline mileage (y) to engine displace-
ment (x1) and carburetor (x2).
(b) Construct and interpret the partial regression plots for each predictor.
(c) Compute the studentized residuals and the R-student residuals for this model. What
information is conveyed by these scaled residuals?

Brothers and sisters: Thirty students in a first-grade class were asked how many siblings they have. Following are the results.

Java question: Write a method printCombinationR(int[] array, int r) that generates and prints all possible combinations of r elements in array of size n,. For example, if input array is {1, 2, 3, 4} and r is 2, then output should be {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4} and {3, 4}.

I kind of know how to write this with DFS with four or five formal parameters, however I don't know how to solve it after the formal parameters are restricted by only using (int[] array, int r).

CHOOSE THE CORRECT ANSWER AND EXPLAIN FOR EACH OPTION WHY YOU DID NOT CHOOSE THAT OPTION.

Arrange the following events in steroid function in chronological order. Number each statement

from 1-5.

_____Level of estradiol increases

_____Decrease in the thickening of uterine lining

_____Ovulation triggered by luteinizing hormone

_____If no ovum is fertilized, progesterone and estradiol production stops

_____Bleeding due to sloughed off uterine lining

a. 3, 2, 4, 1, 5

b. 3, 1, 4, 2, 5

c. 1, 4, 3, 2, 5

d. 1, 4, 2, 3, 5

Arrange the following events in lipid transport in chronological order. Number each statement from 1-5.

____ Cell use the cholesterol as part of the membrane

____ apoB-100 binds to LDL-receptor in coated pits

____ VLDL contains triglycerides and cholesteryl esters

____ Endocytosis of LDL into the cell

____Removal of all VLDL molecules except for the cholesteryl core

a. 5, 3, 1, 4, 2

b. 5, 4, 3, 2, 1

c. 3, 1, 4, 5, 2      

d. 3, 5, 2, 1, 4

Which of the following pairs would form hydrogen bonds?

A. Gly-Ala    B. Ser-Asp   C.Val-Ile      D. Phe-Thr

Which peptide sequence has an aromatic residue?

A. L-C-D-V-G-T

B. I-K-R-Y-N-E

C. M-P-A-Q-S-D

D. T-C-S-L-Q-G

Given a simple project illustrated in the following table

The required project completion time is 12. Assume that "Activity 1" actually finished at 3 weeks, and "Activity 2" actually finished at 2 weeks. Given the actual project progress, what is the LF (lastest time) of "Activity 4"?