bus math qrb 501 wk5 qu-2

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Business Math, Ch. 13

Retirement?

Don’t panic – learn the rules & start investing! 🙂

Class, one consideration you need to make when you have not yet started your retirement investing is the fact that you can start small and build gradually if you feel that you need to do that, but getting started is the part where many people fail.  Did you realize that if you’re investing in a 401K or similar account through your employer, the funds are taken out pretax, so your taxable income will be lowered?  That’s wonderful news because many young people who are just starting to fund their retirement are also single and need a way to reduce their taxable income so taxes don’t “eat them alive.”

 

Class, don’t panic when you are crunching the numbers!  Instead, but be mindful of what benefits your employer offers.  Do they match any portion of your contributions?  Do they offer an employee pension plan in addition to a 401K or other type of investment plan?  Where I work at my full-time job, we are fortunate enough to have an employer sponsored pension plan that is 100% funded by our company.  In addition, they also match a portion of our 401K contributions. Finally, an added bonus for employees who have been with the company 20 or more years and retire at age 55 or older is that they can have health insurance.  I have already invested my 20 years with the company.  In eight more years, I will be eligible to retire with health benefits – and as you know, that’s a huge plus!  Honestly, there have been many times in the past that I have been ready to find a new job, but when I crunch the numbers and realize how much added benefit these two benefits provide in the planning for my retirement, I close my mouth and continue on with my work. Innocent

 

For those of you who are thinking of changing jobs or who are in the process of changing jobs now, as you are reviewing future employers, be sure to ask about benefits that will help you build your retirement nest egg!  

 

Samir, in answer to your question, I look at retirement investing the same way I look at grades – anything is better than zero, so get started! You will find that once you get started, you can increase your investments at times that you will barely notice.  For instance, I increase my investment percentage when I receive raises. 

 

Class, check into the retirement benefits your employer offers!  What do they offer? 

 

Statistics Bottling Company Case Study

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Imagine you are a manager at a major bottling company. Customers have begun to complain that the bottles of the brand of soda produced in your company contain less than the advertised sixteen (16) ounces of product. Your boss wants to solve the problem at hand and has asked you to investigate. You have your employees pull thirty (30) bottles off the line at random from all the shifts at the bottling plant. You ask your employees to measure the amount of soda there is in each bottle. Note: Use the data set provided by your instructor to complete this assignment.

 

 

 

Write a two to three (2-3) page report in which you:

 

  1. Calculate the mean, median, and standard deviation for ounces in the bottles.
  2. Construct a 95% Confidence Interval for the ounces in the bottles.
  3. Conduct a hypothesis test to verify if the claim that a bottle contains less than sixteen (16) ounces is supported. Clearly state the logic of your test, the calculations, and the conclusion of your test.
  4. Provide the following discussion based on the conclusion of your test: 
    • If you conclude that there are less than sixteen (16) ounces in a bottle of soda, speculate on three (3) possible causes. Next, suggest the strategies to avoid the deficit in the future.

    Or

    • If you conclude that the claim of less soda per bottle is not supported or justified, provide a detailed explanation to your boss about the situation. Include your speculation on the reason(s) behind the claim, and recommend one (1) strategy geared toward mitigating this issue in the future.

 

  1. Use at least two (2) quality resources in this assignment. Note: Wikipedia and similar Websites do not qualify as quality resources. The body of the paper must have in-text citations that correspond to the references.

 

Your assignment must follow these formatting requirements:

 

  • Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; citations and references must follow APA or school-specific format. Check with your professor for any additional instructions.
  • Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the date. The cover page and the reference page are not included in the required assignment page length.

Bus Math wk 1 Qu-1

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Math is a language, so let’s translate it!

Check out Table 5-1 on page 162 in your Business Mathematics text.  It has a table that is helpful when you are “translating” a bunch of words into a mathematical expression or equation.  

 

To illustrate how many ways we can say the same thing, use the table to say the following in words in 3 different ways (please try not to repeat what others have said):

25 – 2 = 23

 British Pound
    

1.0000
    

0.6069
    

0.8687
    

0.007088
    

0.6371
    

0.09323  

Canadian Dollar
    

1.6504
    

1.0000
    

1.4331
    

0.011691
    

1.0508
    

0.1538

Euro
    

1.1524
    

0.6990
    

1.0000
    

0.00816
    

0.7338
    

0.1074

Japanese Yen
    

141.2860   
    

85.7113  
    

122.6500   
    

1.0000
    

89.9832  
    

13.1678  

US Dollar
    

1.5706
    

0.9525
    

1.3638
    

0.011126
    

1.0000
    

0.1463

Chinese Yuan Reminbi
    

10.7352  
    

6.5106
    

9.3216
    

0.07604
    

6.8351
    

1.0000

Source: Currency Exchange Rates provided by OANDA, the currency site.

MA260 Statistical Analysis I – || SOLUTION

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MA260 Statistical Analysis I

Assignment 1:

Directions: Sources must be cited in APA format. Your response should be a minimum of one (1) single-spaced page to a maximum of two (2) pages in length
NOTE: Show your work in the problems.

1. Compute the mean and variance of the following discrete probability distribution.

x P(x)
2 .50
8 .30
10 .20

2. The Computer Systems Department has eight faculty, six of whom are tenured. Dr. Vonder, the chair, wants to establish a committee of three department faculty members to review the curriculum. If she selects the committee at random:

a. What is the probability all members of the committee are tenured?
b. What is the probability that at least one member is not tenured? (Hint: For this question, use the complement rule.)

3. New Process, Inc., a large mail-order supplier of women’s fashions, advertises same-day service on every order. Recently, the movement of orders has not gone as planned, and there were a large number of complaints. Bud Owens, director of customer service, has completely redone the method of order handling. The goal is to have fewer than five unfilled orders on hand at the end of 95% of the working days. Frequent checks of the unfilled orders follow a Poisson distribution with a mean of two orders. Has New Process, Inc. lived up to its internal goal? Cite evidence.

4. Recent information published by the U.S. Environmental Protection Agency indicates that Honda is the manufacturer of four of the top nine vehicles in terms of fuel economy.
a. Determine the probability distribution for the number of Hondas in a sample of three cars chosen from the top nine.
b. What is the likelihood that in the sample of three at least one Honda is included?

5. According to the “January theory,” if the stock market is up for the month of January, it will be up for the year. If it is down in January, it will be down for the year. According to an article in the Wall Street Journal, this theory held for 29 out of the last 34 years. Suppose there is no truth to this theory. What is the probability this could occur by chance?

Assignment 2:

MA260 Statistical Analysis I

Directions: Sources must be cited in APA format. Your response should be a minimum of one (1) single-spaced page to a maximum of two (2) pages in length;
NOTE: Show your work in the problems.

1. A recent article in the Myrtle Beach Sun Times reported that the mean labor cost to repair a color television is $90 with a standard deviation of $22. Monte’s TV Sales and Service completed repairs on two sets this morning. The labor cost for the first was $75 and it was $100 for the second. Compute z values for each and comment on your findings.

2. The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds.
a. What is the area between 415 pounds and the mean?
b. What is the area between the mean and 395 pounds?
c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?

3. The monthly sales of mufflers in the Richmond, VA area follow the normal distribution with a mean of 1200 and a standard deviation of 225. The manufacturer would like to establish inventory levels such that there is only a 5% chance of running out of stock. Where should the manufacturer set the inventory levels?

4. Research on new juvenile delinquents revealed that 38% of them committed another crime.
a. What is the probability that of the last 100 new juvenile delinquents put on probation, 30 or more will commit another crime?
b. What is the probability that 40 or fewer of the delinquents will commit another crime?
c. What is the probability that between 30 and 40 of the delinquents will commit another crime?

5. An Air Force study indicates that the probability of a disaster such as the January 28, 1986 explosion of the space shuttle Challenger was 1 in 35. The Challenger flight was the 25th mission.
a. How many disasters would you expect in the first 25 flights?
b. Use the normal approximation to estimate the probability of at least one disaster in 25 missions.

QSO 510: Module 8 Solution

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Notes: Before doing this assignment, do the practice problem posted under Apply and Discover. Word-process your answers within this document. Do not create a new file. Copy and paste the reports from Excel to this document. Show all steps used in arriving at the final answers. Incomplete solutions will receive partial credit.

Problem

Print Media Advertising (PMA) has been given a contract to market Buzz Cola via newspaper ads in a major southern newspaper. Full-page ads in the weekday editions (Monday through Saturday) cost $2000, whereas on Sunday a full-page ad costs $8000. Daily circulation of newspaper is 30,000 on weekdays and 80,000 on Sundays.

PMA has been given a $40,000 advertising budget for the month of August. The experienced advertising executives at PMA feel that both weekday and Sunday newspaper ads are important; hence they wish to run the equivalent of at least eight weekday and at least two Sunday ads during August. (Assume that a fractional ad would simply mean that a smaller ad is placed on one of the days; that is, 3.5 ads would mean three full-page ads and one half-page ad. Also, assume that smaller ads reduce exposure and costs proportionately.) This August has 26 weekdays and 5 Sundays.

The objective is to determine the optimal placement of ads by PMA in the newspaper during August so as to maximize the cumulative total exposure (as measured by circulation) for the month of August.

a)      Formulate the linear programming model for the problem.

b)      Use the Graphical method to find the optimal solution. Show all steps.

c)       Use Excel Solver to find the optimal solution. Copy and paste your spreadsheet and the Answer report in its entirety from Excel. Remember to not delete/modify any part of the Answer Report.

Here is how the spreadsheet was set up for Excel Solver. The spreadsheet shows 10 weekdays and 10 for Sundays.  These numbers were selected arbitrarily. Any quantities can be entered weekdays and Sundays since Excel Solver would find the optimal solution anyway.

             

 

 

Re-Post to Math Quiz 3 (1-20)

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1. Add and simplify: 3/8 + 13/8

a. 
b. 
c. 
d. 
 
2. Subtract and write a mixed numeral for the answer: 12 3/5 – 8 7/8

a. 
b. 
c. 
d. 
 
3. Multiply and write a mixed numeral for the answer: 5 2/3 x 3/4

a. 
b. 
c. 
d. 
 
4. Find the LCM of 18 and 30.

a. 
b. 
c. 
d. 
 
5. Convert to a mixed numeral: 17/8

a. 
b. 
c. 
d. 
 
6. Convert to a mixed numeral: 13/2

a. 
b. 
c. 
d. 
 
7. Find the LCM of 6, 27, and 45.

a. 
b. 
c. 
d. 
 
8. Convert to fraction notation: 5 1/2

a. 
b. 
c. 
d. 
 
9. Add and write a mixed numeral for the answer: 9 1/5 + 4 4/5

a. 
b. 
c. 
d. 
 
10. Subtract and simplify: 5/4 – 3/4

a. 
b. 
c. 
d. 
 
11. Divide and write a mixed numeral for the answer: 12 ÂĽ Ă· 4 2/3

a. 
b. 
c. 
d. 
 
12. Subtract and write a mixed numeral for the answer: 18 – 4 3/4

a. 
b. 
c. 
d. 
 
13. A stack of books is 31 ½ in. high. Each book is 1 ½ in. thick. How many books are in the stack?

a. 
b. 
c. 
d. 
 
14. Solve for p. 1/3 + p = 4

a. 
b. 
c. 
d. 
 
15. Multiply and write a mixed numeral for the answer: 6 x 4 2/3

a. 
b. 
c. 
d. 
 
16. George and Veronica both get haircuts on the same day. George gets his haircut every 4 weeks and Veronica gets hers cut every 6 weeks. When will they next have haircuts the same week?

a. 
b. 
c. 
d. 
 
17. Add and write a mixed numeral for the answer: 3 ÂĽ + 2 2/5

a. 
b. 
c. 
d. 
 
18. Convert to fraction notation: 5 4/7

a. 
b. 
c. 
d. 
 
19. Find the LCM of 8, 15, and 20.

a. 
b. 
c. 
d. 
 
20. A postcard is 4 ÂĽ in. long and 3 Âľ in. wide. What is the area of the postcard?

a. 
b. 
c. 
d. 

Question 1: Test-preparation organizations like Kaplan, Princeton Review, etc. often advertise their services by claiming that students gain an average of 100 or more points on the Scholastic Achievement Test (SAT). Do you think that taking one of those c

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Question 1:

Test-preparation organizations like Kaplan, Princeton Review, etc. often advertise their services by claiming that students gain an average of 100 or more points on the Scholastic Achievement Test (SAT). Do you think that taking one of those classes would give a test taker 100 extra points? Why might an average of 100 points be a biased estimate?

Question 2:

A few random results for you to ponder…do the results justify the conclusions? Why or why not?  Pick one or two statements and give a reason why the conclusions don’t necessarily follow from the statement

 

1.      In the NFL, teams win more often when they score 13 points than when they score 14. Thus, scoring points is bad.

2.      Often when people use regression analysis to estimate the effect of police officers or police spending on crime, they find that cities with larger police forces/budgets have higher crime rates. Therefore, police cause crime.

3.      As ice cream sales increase, so to do drowning deaths. Thus, ice cream causes people to drown.

4.      Studies find that students who drink more tend to have lower grades. Therefore, drinking leads to poor student performance.

5.      Over the past 300 years, there has been a decrease in the number of pirates on the high seas, along with an increase in average global temperatures. Thus, the reduction in piracy has led to global warming.

6.      Did you know there is a health benefit to winning an Oscar? Doctors at Harvard Medical School say that a study of actors and actresses shows that winners live, on average, for four years more than losers. And winning directors live longer than non-winners.

7.      Children who come further down in the birth order have, on average, lower IQs than those born earlier in the birth order (e.g. First born children vs. 5th born children). Therefore birth order determines intelligence 

 

Question 3:

Consider the following:

There is a game called Under-or-Over Seven.  A pair of fair dice is rolled once and the resulting sum determines whether the player wins or loses his or her bet.  For example, the player can bet $1 that the sum will be under 7 — that is 2, 3, 4, 5, or 6.  For this bet, the player wins $1 if the result is under 7 and loses $1 if the outcome equals to or is greater than 7.  Similarly, the player can bet $1 that the sum will be over 7  — that is 8, 9, 10, 11, or 12.   Here the player wins $1 if the result is over 7, but loses $1 if the outcome is 7 or under.  A third method of play is to bet $1 on the outcome 7.  For this bet, the player wins $4 if the result of the roll is 7 and loses $1 otherwise.

a) Construct the probability distribution representing the different outcomes that are possible for a $1 bet on under 7.

b)  Construct the probability distribution representing the different outcomes that are possible for a $1 bet on over 7.

c)  Construct the probability distribution representing the different outcomes that are possible for a $1 bet on 7.

 

d)  Show that the expected long run profit (or loss) to the player is the same, no matter which method of play is used.

MAT540 Week 2 Homework

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MAT540

Week 2 Homework

Chapter 12

8.    A local real estate investor in Orlando is considering three alternative investments: a motel, a restaurant, or a theater. Profits from the motel or restaurant will be affected by the availability of gasoline and the number of tourists; profits from the theater will be relatively stable under any conditions. The following payoff table shows the profit or loss that could result from each investment:

 

Gasoline Availability

Investment

Shortage

Stable Supply

Surplus

Motel

$-8,000

$15,000

$20,000

Restaurant

    2,000

    8,000

    6,000

Theater

    6,000

    6,000

    5,000

 

Determine the best investment, using the following decision criteria.

a.       Maximax

b.      Maximin

c.       Minimax regret

d.      Hurwicz (α = 0.4)

e.      Equal likelihood

16. A concessions manager at the Tech versus A&M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates that the following profits will result from each decision, given each set of weather conditions:

 

Weather Conditions

Decision

Rain

Overcast

Sunshine

 

.30

.15

              .55

Sun visors

$-500

$-200

$1,500

Umbrellas

2,000

0

-900

 

a.       Compute the expected value for each decision and select the best one.

b.      Develop the opportunity loss table and compute the expected opportunity loss for each decision.

24. In Problem 13 the Place-Plus real estate development firm has hired an economist to assign a probability to each direction interest rates may take over the next 5 years. The economist has determined that there is a .50 probability that interest rates will decline, a .40 probability that rates will remain stable, and a .10 probability that rates will increase.

a.       Using expected value, determine the best project.

b.      Determine the expected value of perfect information.

Reference Problem 13:  Place-Plus, a real estate development firm, is considering several alternative development projects.  These include building and leasing an office park, purchasing a parcel of land and building an office building to rent, buying and leasing a warehouse, building a strip mall, and building and selling condominiums. The financial success of these projects depends on interest rate movement in the next 5 years. The various development projects and their 5-year financial return (in $1,000,000s) given that interest rates will decline, remain stable, or increase, are shown in the following payoff table:

 

 

Interest Rate

 

Project

Decline

Stable

Increase

Office park

$0.5

$1.7

$4.5

Office building

  1.5

  1.9

  2.5

Warehouse

  1.7

  1.4

  1.0

Mall

  0.7

  2.4

  3.6

Condominiums

  3.2

  1.5

  0.6

                                               

32. The director of career advising at Orange Community College wants to use decision analysis to provide information to help students decide which 2-year degree program they should pursue. The director has set up the following payoff table for six of the most popular and successful degree programs at OCC that shows the estimated 5-year gross income ($) from each degree for four future economic conditions:

 

Economic Conditions

Degree Program

Recession

Average

Good

Robust

Graphic design

145,000

175,000

220,000

260,000

Nursing

150,000

180,000

205,000

215,000

Real estate

115,000

165,000

220,000

320,000

Medical technology

130,000

180,000

210,000

280,000

Culinary technology

115,000

145,000

235,000

305,000

Computer information technology

125,000

150,000

190,000

250,000

 

Determine the best degree program in terms of projected income, using the following decision criteria:

a.       Maximax

b.      Maximin

c.       Equal likelihood

d.      Hurwicz (α = 0.50)

36. Construct a decision tree for the decision situation described in Problem 25 and indicate the best decision.

Reference Problem 25:  Fenton and Farrah Friendly, husband-and-wife car dealers, are soon going to open a new dealership. They have three offers: from a foreign compact car company, from a U.S. producer of full-sized cars, and from a truck company. The success of each type of dealership will depend on how much gasoline is going to be available during the next few years. The profit from each type of dealership, given the availability of gas, is shown in the following payoff table:

 

Gasoline Availability

Dealership

Shortage

Surplus

 

.6

.4

Compact cars

$ 300,000

$150,000

Full-sized cars

-100,000

600,000

Trucks

120,000

170,000

 

 

Decision Tree diagram to complete:

Steven collected data from 20 college students on their emotional responses to classical music. Students listened to two 30-second segments from “The Collection from the Best of Classical Music.” After listening to a segment, the students rated it on a sc

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Problem 1)

Steven collected data from 20 college students on their emotional responses to classical music. Students listened to two 30-second segments from “The Collection from the Best of Classical Music.” After listening to a segment, the students rated it on a scale from 1 to 10, with 1 indicating that it “made them very sad” to 10 indicating that it “made them very happy.” Steve computes the total scores from each student and created a variable called “hapsad.” Steve then conducts a one-sample t-test on the data, knowing that there is an established mean for the publication of others that have taken this test of 6. The following is the scores:

5.0 5.0
10.0 3.0
13.0 13.0
7.0 5.0
5.0 15.0
14.0 18.0
8.0 12.0
10.0 7.0
3.0 15.0
4.0 3.0
a) Conduct a one-sample t-test. What is the t-test score? What is the mean? Was the test significant? If it was significant at what P-value level was it significant?
b) What is your null and alternative hypothesis? Given the results did you reject or fail to reject the null and why? 
(Use instructions on page 349 of your textbook, under Hypothesis Tests with the t Distribution to conduct SPSS or Excel analysis).

Problem 2)

Billie wishes to test the hypothesis that overweight individuals tend to eat faster than normal-weight individuals. To test this hypothesis, she has two assistants sit in a McDonald’s restaurant and identify individuals who order the Big Mac special for lunch. The Big Mackers as they become known are then classified by the assistants as overweight, normal weight, or neither overweight nor normal weight. The assistants identify 10 overweight and 10 normal weight Big Mackers. The assistants record the amount of time it takes them to eat the Big Mac special.

1.0 585.0
1.0 540.0
1.0 660.0
1.0 571.0
1.0 584.0
1.0 653.0
1.0 574.0
1.0 569.0
1.0 619.0
1.0 535.0
2.0 697.0
2.0 782.0
2.0 587.0
2.0 675.0
2.0 635.0
2.0 672.0
2.0 606.0
2.0 789.0
2.0 806.0
2.0 600.0

a) Compute an independent-samples t-test on these data. Report the t-value and the p values. Were the results significant? (Do the same thing you did for the t-test above, only this time when you go to compare means, click on independent samples t-test. When you enter group variable into grouping variable area, it will ask you to define the variables. Click define groups and place the number 1 into 1 and the number 2 into 2).
b) What is the difference between the mean of the two groups? What is the difference in the standard deviation?
c) What is the null and alternative hypothesis? Do the data results lead you to reject or fail to reject the null hypothesis? 
d) What do the results tell you?

Problem 3)
Lilly collects data on a sample of 40 high school students to evaluate whether the proportion of female high school students who take advanced math courses in high school varies depending upon whether they have been raised primarily by their father or by both their mother and their father. Two variables are found below in the data file: math (0 = no advanced math and 1 = some advanced math) and Parent (1= primarily father and 2 = father and mother). 

Parent Math
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
1.0 0.0
2.0 0.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 1.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0
2.0 0.0

a) Conduct a crosstabs analysis to examine the proportion of female high school students who take advanced math courses is different for different levels of the parent variable.
b) What percent female students took advanced math class
c) What percent of female students did not take advanced math class when females were raised by just their father?
d) What are the Chi-square results? What are the expected and the observed results that were found? Are they results of the Chi-Square significant? What do the results mean? 
e) What were your null and alternative hypotheses? Did the results lead you to reject or fail to reject the null and why? 

Problem 4

This problem will introduce the learner into a technique called Analysis of Variance. For this course we will only conduct a simple One-Way ANOVA and touch briefly on the important elements of this technique. The One-Way ANOVA is an extension of the independent –t test that can only look at two independent sample means. We can use the One-Way ANOVA to look at three or more independent sample means. Use the following data to conduct a One-Way ANOVA:

Scores Group
1 1
2 1
3 1
2 2
3 2
4 2
4 3
5 3
6 3

Notice the group (grouping) variable, which is the independent variable or factor is made up of three different groups. The scores are the dependent variable. 

Use the instructions for conduction an ANOVA on page 366 of the text for SPSS or Excel. 

a) What is the F-score; Are the results significant, and if so, at what level (P-value)?
b) If the results are significant to the following: Click analyze, then click Compare Means, and then select one-way ANOVA like you did previously. Now click Post Hoc. In this area check Tukey. If there is a significant result, we really do not know where it is. Is it between group 1 and 2, 1 and 3, or 2 and 3? Post hoc tests let us isolate where the level of significance was. So if the results come back significant, conduct the post hoc test as I mentioned above and explain where the results were significant.
c) What do the results obtained from the test mean? 

FINITE MATH HELP

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  1. True or false. If all the coefficients a1, a2, …, an in the objective function P = a1x1 + a2x2 + … + anxn are nonpositive, then the only solution of the problem is x1 = x2 = … = xn and P = 0.
  2. True or false. The pivot column of a simplex tableau identifies the variable whose value is to be decreased in order to increase the value of the objective function (or at least keep it unchanged).
  3. True or false. The ratio associated with the pivot row tells us by how much the variable associated with the pivot column can be increased while the corresponding point still lies in the feasible set.
  4. True or false. At any iteration of the simplex procedure, if it is not possible to compute the ratios or the ratios are negative, then one can conclude that the linear programming problem has no solution.
  5. True or false. If the last row to the left of the vertical line of the final simplex tableau has a zero in a column that is not a unit column, then the linear programming problem has infinitely many solutions.
    1. True or false. Suppose you are given a linear programming problem satisfying the conditions:
    2. The objective function is to be minimized. 
  • All the variables involved are nonnegative, and 
  • Each linear constraint may be written so that the expression involving the variables is greater than or equal to a negative constant.
  1. Then the problem can be solved using the simplex method to maximize the objective function P = –C.
  2. True or false. The objective function of the primal problem can attain an optimal value that is different from the optimal value attained by the dual problem