1. Given the following linear programming model, answer the questions that follow. You are given the result of a computer program. The results are

Maximize 9 X₁ + 12 X₂ + 10 X₃

Subject to:

Machine Constraint:   3 X₁ + 4 X₂ + 3 X₃ < 160

Labor Constraint:        6 X₁ + 10 X₂ + 4 X₃ < 288

Materials Constraint: 2 X₁ + 2 X₂ + 7 X₃ < 200

Product 2 Constraint: X₁ < 16

OPTIMAL SOLUTION

Objective Function Value =         483.097

      Variable             Value             Reduced Costs  

   --------------     ---------------      ------------------

         X1                    16.000                   0.000

         X2                    10.839                   0.000

         X3                    20.903                   0.000

     Constraint        Slack/Surplus           Dual Prices   

   --------------     ---------------      ------------------

         1                      5.935                   0.000

         2                      0.000                   1.032

         3                      0.000                   0.839

         4                      0.000                   1.129

OBJECTIVE COEFFICIENT RANGES

    Variable       Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

      X1                  7.871              9.000   No Upper Limit

      X2                  2.857             12.000           14.059

      X3                  4.800             10.000           18.750

RIGHT HAND SIDE RANGES

  Constraint      Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

       1                154.065            160.000   No Upper Limit

       2                192.000            288.000          304.727

       3                 70.400            200.000          226.286

       4                  0.000             16.000           30.154

1. If the profit of Product 3 was changed to $20 and the profit of Product 1 was changed $20, would that change the solution?   Provide proof.

2. If the number of minutes of Machine time was decreased to 155 minutes and the amount of materials were decreased to 170 pounds, would this change the solution? Provide proof.

3. The Dual Price for Constraint 1 (Machine time) is 4.2. In terms of this problem what does that mean?

1. Given the following linear programming model, answer the questions that follow. You are given the result of a computer program. The results are

Maximize 9 X₁ + 12 X₂ + 10 X₃

Subject to:

Machine Constraint:   3 X₁ + 4 X₂ + 3 X₃ < 160

Labor Constraint:        6 X₁ + 10 X₂ + 4 X₃ < 288

Materials Constraint: 2 X₁ + 2 X₂ + 7 X₃ < 200

Product 2 Constraint: X₁ < 16

OPTIMAL SOLUTION

Objective Function Value =         483.097

      Variable             Value             Reduced Costs  

   --------------     ---------------      ------------------

         X1                    16.000                   0.000

         X2                    10.839                   0.000

         X3                    20.903                   0.000

     Constraint        Slack/Surplus           Dual Prices   

   --------------     ---------------      ------------------

         1                      5.935                   0.000

         2                      0.000                   1.032

         3                      0.000                   0.839

         4                      0.000                   1.129

OBJECTIVE COEFFICIENT RANGES

    Variable       Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

      X1                  7.871              9.000   No Upper Limit

      X2                  2.857             12.000           14.059

      X3                  4.800             10.000           18.750

RIGHT HAND SIDE RANGES

  Constraint      Lower Limit       Current Value     Upper Limit

------------   ---------------    --------------- ---------------

       1                154.065            160.000   No Upper Limit

       2                192.000            288.000          304.727

       3                 70.400            200.000          226.286

       4                  0.000             16.000           30.154

1. What is the optimal solution?

1. What is the profit?

2. If the profit of Product 3 was changed to $20 and the profit of Product 1 was changed $20, would that change the solution?   Provide proof.

3. If the number of minutes of Machine time was decreased to 155 minutes and the amount of materials were decreased to 170 pounds, would this change the solution? Provide proof.

4. The Dual Price for Constraint 1 (Machine time) is 4.2. In terms of this problem what does that mean?

2. (CFGs) Give CFGs that generate the following languages. Use as few variables as you can (and no more than requested). Unless specified otherwise, the alphabet is Σ = {0, 1}.

(c) L3 = {w | w represents a binary number that starts with 1 and is divisible by 3}. Use at most 4 variables.

(d) L4 = {x₁#x₂# · · · #x_k | k ≥ 1, each x_i ∈ {0, 1}^* , and x_i = x_j^R for some i != j}. Recall that w^R represents string w written backwards. Use at most 4 variables.

1. You have been given the following return information for two mutual funds (Papa and Mama), the market index, and the risk-free rate. (25 points)

1. What is the Sharpe ratio for both fund? (4 points)

2. What is the Treynor ratio for both fund? (4 points)

3. What is the Jensen’s alpha for both fund? (4 points)

4. What is the information ratio for both fund? (4 points)

5. What is R-squared for both fund? (4 points)

6. Which is the best choice for your portfolio? (5 points)

As schools moved to the remote online training model some faculty were requested to learn new online skills. The effectivity of the training was evaluated with a “before and after” test of the teacher’s evaluation of their skills. The scale was a 10-point Likert scale and the teacher was asked to give a “pre” before and “post” after the training. A small sample of 10 teachers is in your data files at the end of this test. Use an alpha of .05. Run the proper analysis of “before and after” and answer the following questions.

Give the following:

1- Mean pre-test.  

2- Variance pre-test   

3- Mean post-test   

4- Variance post-test

5- Number of observations   

6- Number of teachers surveyed in this test

7- tStat    

8- P(T<=t) one

9- Conclusion reached:

1. You are trying to value Resources Limited. Your projections for the next four years are based on the following assumptions

(Total 50 Marks)

FIN3013 August

•  Sales will be 600 million in Year 1.

•  Sales will grow at 7 percent in Years 2 and 3 and 9 percent in

  Year 4.

•  Operating profits (EBIT) will be 23 percent of sales in each year.

•  Interest expense will be 17 million in Year 1 and increase by 2 million every year until Year 4.

•  Income tax rate is 40 percent.

•  Earnings retention ratio will stay at 0.70.

•  The per-share dividend growth rate will be constant from Year 4 forward, and the final growth rate will be 300 basis points less than the growth rate from Year 3 to Year 4.

  The company has 10 million shares outstanding. The required return on the stock is estimated to be 13 percent.

1. (a) Estimate the value of the stock at the end of Year 4.

2. (b) Estimate the current value of the stock.

443

Consider the data set:

8 4 3 6 17 15 13 7 21 11 32 1 1.

1.What is the percentile rank of xi = 21?

2. What is the value corresponding to the 61th percentile, P61?

3. Find outliers, if exist.

2. For each of the following joint pmfs nd the value c.

a. p(x,y) = c(x + y), x = 1,2,3, y = 1,...,x

b. p(x,y) = c( 1/4)^x (1/3)^y, x = 0,1,2,..., y = 0,1,2,...

Outline one possible process for implementing a plan for environmental sustainability.

Please select no more than three significant phases in implementation.

using Process analysis.

1) one thesis statement

2) three-sentence outlines

3) 1 body paragraph

4) 1 conclusion