Solve the following linear programming problem. You must use the dual. First write down the dual maximization LP problem, solve that, then state the solution to the original minimization problem.

(a) Minimize w = 4y₁ + 5y₂ + 7y₃

Subject to: y₁ + y₂ + y₃ ≥ 18

2y₁ + y₂ + 2y₃ ≥ 20

y₁ + 2y₂ + 3y₃ ≥ 25

y₁, y₂, y₃ ≥ 0

(b) Making use of shadow costs, if the 2^nd original constraint changed to

2y₁ + y₂ + 2y₃ ≥ 24, now what will the minimum of w be? Explain clearly.

(c) Making use of shadow costs, if the 1^st original constraint changed to

y₁ + y₂ + y₃ ≥ 21, now what will the minimum of w be? Explain clearly.

E. Martz tells us that chi-square analysis can be used for more than just practical applications:

Chi-square analysis compares the counts of two categorical variables to tell you if a relationship exists between the variables or not. You can apply chi-square analysis to answer important questions about factors in everyday life, and even about events like elections... or Halloween. If you are a character in a slasher film, is there a connection between your gender and your dying in some horrible manner?

As explained in the above passage, you can use chi-square analysis to compare a wide range of topics.

To complete the Discussion activity, please do the following:

• Propose a life event or situation and two categorical variables for it. Complete a chi-square analysis of this event or situation and these variables, and share your results.
• Do you agree with your findings? Why or why not?
• What other factors, if any, should be considered?
• In your peer responses, explain whether you agree with other’s posts on their chi-square analyses. Why or why not? Fully explain your answers.

Use Worksheet 9.2. Ben West, a 35-year-old computer programmer, earns $57,000 a year. His monthly take-home pay is $3,000. His wife, Ashley, works part-time at their children's elementary school but receives no benefits. Under state law, Ashley's employer contributes to a workers' compensation insurance fund that would provide $2,050 per month for six months if Ben were disabled and unable to work.

1. Use Worksheet 9.2 to calculate Ben's disability insurance needs assuming that he won't qualify for Medicare under his Social Security benefits. Round your answer to the nearest whole.
   $
2. Based on your answer in part a, what would you advise Ben about his need for additional disability income insurance? Discuss the type and size of disability income insurance coverage he should consider, including possible provisions he might want to include. What other factors should he take into account if he decides to purchase a policy?
   

CRIMINAL JUSTICE: The Death Penalty is a hard area to talk about since many people have strong feelings on the issue. For this discussion board, state the arguments for the death penalty AND against the death penalty. Then you can state your feelings on the death penalty and why.

Answer the questions below using the appropriate statistical technique. For questions involving the use of hypothesis testing, you must:

1. State the null and research hypotheses

2. Provide the Z(critical), T(critical), or χ 2 (critical) score corresponding to the α threshold for your test

3. Provide your test statistic

4. Provide your decision about statistical significance

An advantage that often comes with a basic knowledge of statistics is a change in salary. To see whether this was the case for Tulane University graduates, you took a random sample of 57 students who completed a statistics class and asked about their starting salaries (in thousands) after graduation. The sample had a mean of 53.3 with a standard deviation of 3.72 (i.e., x = 53.3 and s = 3.72). A call to the Office of the Registrar indicates that the average starting salary value for all Tulane students is 47.1. Do students who take statistics courses earn an equal salary compared to Tulane students generally? Use α = 0.001.

Answer the questions below using the appropriate statistical technique. For questions involving the use of hypothesis testing, you must:

1. State the null and research hypotheses

2. Provide the Z(critical), T(critical), or χ 2 (critical) score corresponding to the α threshold for your test

3. Provide your test statistic

4. Provide your decision about statistical significance

An advantage that often comes with a basic knowledge of statistics is a change in salary. To see whether this was the case for Tulane University graduates, you took a random sample of 57 students who completed a statistics class and asked about their starting salaries (in thousands) after graduation. The sample had a mean of 53.3 with a standard deviation of 3.72 (i.e., x = 53.3 and s = 3.72). A call to the Office of the Registrar indicates that the average starting salary value for all Tulane students is 47.1. Do students who take statistics courses earn an equal salary compared to Tulane students generally? Use α = 0.001.

2. A jar contains 5 balls, 4 of which are blue and one red.

(a) If you draw balls one at a time and replace them, what is the expected draw at which

you will see the red ball?

(b) If you draw balls one at a time, but without replacing them, what is the expected time

to see the red ball?

3. A jar contains 1 red ball and an unknown number of blue balls. You make 20 draws with

replacement from the jar. What is a maximum likelihood estimator of the number of blue

balls in the jar?

2. Economic growth. Two key sources of economic growth have been due to increases in resources and increases in productivity.
a. Please discuss 5 factors that might contribute to falling U.S. GDP growth rates into the future,
b. Explain in detail, what we could do to avoid each of the 5 factors causing our GDP growth rates to fall.
c. Please discuss 5 factors that might contribute to rising U.S. GDP growth rates into the future,
d. Explain in detail, what we could do to cause each of the 5 factors to happen and cause our GDP growth rate to increase. .
e. Do a bit of research into our ability to get more resources and the cost associated with getting the
hardest to recover bits of resources.
f. What could prevent us from “Engineering our way out of the problem”?

The Red Cross has a cafeteria in its headquarters that serves lunch to its employees five days a week. The cafeteria typically serves 500 lunches per day, 5 days per week, although it has the capacity to serve at least twice as many before it would have to invest in new equipment or hire new employees. The cost for the equipment and cafeteria employees is $180,000 per year. The cafeteria’s food costs for each meal served are $1.00, and the cafeteria charges $2.50 per meal. An elementary school next door has heard about the cafeteria and offers to send 200 students a day to the cafeteria and pay the Red Cross $1.50 for each student meal. Should the Red Cross take this offer? Why or why not? (6 points)

A basket contains 100 balls.40 are red,45 are orange and 15 are yellow.Three balls will be drawn out one at a time at random with replacement.Match the probabilities.

(a) P(all three draws are red)

(b) P(all three draws are orange)

(c) P(at least one draw is red)

(d) P(at least one draw is orange)

2.Refer to the previous question about the balls in the basket.Instead of drawing out three balls one at a time with repalcement, suppose we selected balls one at a time at random without replacement until all the yellow balls were removed from the basket. Y=the number of draws necessary.What are the possible values of Y.

(a)[15,16,17,18...] (b)[15,1617,18...100] (c)[0,1,2,3...15] (d)[1,2,3,...85]