1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players take turns. When it is a player's turn, she can remove any number of matches from either pile. Each player is required to remove some number of matches if either pile has matches remaining, and can only remove matches from one pile at a time. Whichever player removes the last match wins the game. Winning gives a player a payoff equal to 1, and losing gives a player a payoff equal to 0. The initial configuration of the piles has one match in one of the piles, and two in the other one. a) Write down the game tree for this sequential game. [3 points] b) What is the Nash equilibrium? [3 points]

1. A racing team owner wants to attempt to qualify his car for a major auto race. The owner believes that it will take a mean qualifying speed of over 223 mph to qualify for the race. During the two days of testing prior to qualifying, the team conducted 10 practice qualifying runs. The mean speed of these qualifying runs was 224.5 mph, with a standard deviation of .75 mph. Based on this information, does the owner have reason to believe that his car will qualify for the race, with at least 95% confidence. Assume all normality conditions apply. Solve using the p-value approach.

1. State the null hypothesis

2. State the alternative hypothesis

3. State the significance level.

4. Perform the calculations.

Race One Motors is an Indonesian car manufacturer. At its largest manufacturing? facility, in? Jakarta, the company produces subcomponents at a rate of 295 per? day, and it uses these subcomponents at a rate of 12,100 per year? (of 250 working? days). Holding costs are ?$3 per item per? year, and ordering costs are ?$31 per order.

a. what is the economic production quantity? (round to two decimal places)

b. how many production runs per year will be made? (round to two decimal places)

c. what will be the maximum inventory level? (round to two decimal places)

d. what percentage of time will the facility be producing components? (enter your response as a percentage rounded to two decimal places)

e. what is the annual cost of ordering and holding inventory? (round to two decimal places)

Design and construct a computer program in one of the approved languages (C, C++, C#, Java, Pascal, Python, etc.) that will illustrate the use of a fourth-order
explicit Runge-Kutta method of your own design. In other words, you will first have to solve the Runge-Kutta equations of condition for the coefficients
of a fourth-order Runge-Kutta method.

Then, you will use these coefficients in a computer program to solve the ordinary differential equation below.
Be sure to follow the documentation and programming style policies of the Computer Science Department.

The initial value problem to be solved is the following:

                          x'(t) = 3 x² cos(5 t)

subject to the initial condition:  x(0) = 1.0   

Obtain a numerical solution to this problem over the range from t=0.0 to t=2.0

for seven different values of the stepsize,

h=0.1, 0.05 , 0.025 , 0.0125 , 0.00625 , 0.003125 , and 0.0015625 .

In other words, make seven runs with 20, 40, 80, 160, 320, 640, and 1280 steps, respectively.
For each run, print out the value of h, then a table of t and x, and then the error at t=2. You may use the following
very precise value for your "true answer" in order to compute the error at t=2:   0.753913186469598763502963347.

In 2012, the mean number of runs scored by both teams in a Major League Baseball game was 8.62. Following are the numbers of runs scored in sample of 24 games in 2013. A test of whether the mean number of runs in 2013 is less than it was in 2012 will be conducted, using significance level 0.05, by answering the questions below..

a) What are the observational units?

b) What is the variable collected?

c) Summarize the data with the mean, SD, quartiles, the median, the min and max.

d) Produce a boxplot for the data. Comment on whether you think the data is normally distributed based on boxplot.

e) Write out the appropriate null and alternative hypotheses.

f) Find the test statistic, showing your work.

g) What is the p-value for the test?

h) Conduct the test to make your conclusion by clearly comparing the p-value to the significance level of the test.

i) State your conclusion in the context.

You collect the following information on a sample of 100 adults:

• Y=LOTTERY= percentage of income the person spends on lottery tickets (in %)
• EDUC= amount of education the individual has (in years)
• AGE = age of the individual (in years),
• CHILDREN = number of children the person has (in number of children)
• INC1000= annual income of the individual (in 1,000s of dollars)

The data set can be found in Mod9-1Data. Run the multiple regression in Minitab. Assume a level of significance of 5%.

null hypothesis for the test on the slope/coefficient on AGE-

alternative hypothesis=

computed test statistic=

table test statistic=

p-value=

statistical conclusion=

Predicted percentage of income spent of lottery tickets for a person with 12 years of education; 20 years old; 0 children; and an income of $25,000.=

null hypothesis for valid regression test=

alternative hypothesis for valid regression=

computed test statistic for the useful regression test=

table test statistic for the valid regression test=

p-value for the valid regression test=

statistical conclusion for the valid regression test=

Part II: Evaluate Disaster Risk in Supply Chain

Suzy Jones is trying to decide whether to use one or two suppliers for the motors than go into the chain saws that her company produces. She wants to use local suppliers because her firm runs a JIT operation. Her factory is located in a coastal town that is prone to hurricanes. She estimates that the probability in any year of a "super-event" that might shut down all suppliers at the same time for at least two weeks is 5%. Such a total shutdown would cost the company approximately $100,000. She estimates the "unique-event" risk for any of the suppliers to be 10%. Assuming that the marginal cost of managing an additional supplier is $12,000 per year, should Suzy use one or two suppliers?

Trace (show the steps) the outer loops and draw the final array. In your documentation, use a chart with columns titled: outer loop, inner loop, counter, a, b. Assume the array is six (6) rows, seven (7) columns and filled with zeros (0s) when started.

counter = 1; for(int a = 1; a < 6; a++) { for(int b = 5; b >= 0; b--) { if(b %a == 0) { arr[a][b] = counter; counter++; } else { arr[b][a] = counter; counter--; } } }

Two 50 ?? long, thin parallel straight wires (grey) are connected at their ends by metal springs. The mass of each thin wire is 1.0 ?. The upper wire is connected to the ceiling by (non-conducting) stiff rods. Each spring has an equilibrium length of 5.0 ?? and a spring constant of ? = 0.50 ?/?. A steady current ? runs clockwise through the wire-spring loop as indicated by the arrow. At equilibrium, the lower rod hangs at a level 6.0 ?? below the upper wire. Find the magnitude of the current. You may ignore the magnetic fields generated by the springs, and you may approximate the magnetic fields generated by the wires as those from long, straight wires

After running the experiment with the pivot, comment out the line, update the pivot selection to use the median of the first, middle, and last items, and run the experiment.

What line needs to be commented out and how would I update the pivot selection to use the median?

Example.java

package sorting;

import java.io.IOException;
import java.util.Random;
import sorters.QuickSort;

public class Example {

   public static void main(String args[]) throws IOException {
       int n = 100;   // adjust this to the number of items to sort
       int runs = 11;
      
       
       partB(n, runs);

   public static void partB(int n, int runs) {
       int [] data = new int[n];
       QuickSort quicksort = new QuickSort();
      
       labels(runs);
       for (int i = 0; i < runs; i++) {
           randomArray(data);
           quicksort.sort(data);
       }
       System.out.println();
      
   }
  
   public static void labels(int runs) {
       for(int i = 0; i < runs; i++) {
           String label = "Run " + i;
           System.out.printf("%12s ", label);
       }
       System.out.println();      
   }
  
   public static void randomArray(int [] data) {
       Random rand = new Random();      
       for(int j = 0; j < data.length; j++) {
           data[j] = rand.nextInt();
       }  
   }
  
}

QuickSort.java

package sorters;

// note that this class can only sort primitive ints correctly.

public class QuickSort {
  
   private int[] items;
  
   public void sort(int[] items) {
       this.items = items;
      
       long start = System.nanoTime();
       quicksort(0, items.length-1);
       long finish = System.nanoTime();
       //System.out.println(Arrays.toString(items));
       System.out.printf("%12s ", finish-start);
   }
  
   private int partition(int left, int right) {
       int i = left;
       int j = right;
        int temp;
        int pivot = (int) items[(left + right) / 2];
      
      
        while (i <= j) {
           while((int) items[i] < pivot)
               i++;
            while((int) items[j] > pivot)
                j--;
            if(i <= j) {
                temp = items[i];
                items[i] = items[j];
                items[j] = temp;
                i++;
                j--;
            }
        }
        return i;
   }

   private void quicksort(int left, int right) {

       int index = partition(left, right);

        if(left < index - 1)
           quicksort(left, index-1);
        if(index < right)
            quicksort(index, right);
   }
}