Section 1 – Applications of Unconstrained Optimization

1) A perfectly competitive pumpkin patch faces the following cost function:

TC(Q) = 14Q2 – 650Q + 15

And the patch sells pumpkins for $15.

What is the profit maximizing output?

Is the output you found a profit max? Show.

2) A local hammer manufacturer dominates the market in selling hammers. The demand function for the hammers is: P(Q)=135-.025Q. The total annual cost of producing the hammers is: C(Q)=10500+5Q.

First, what is the company’s revenue function?

? What is the company’s maximum revenue?

? What should the company charge for each hammer?

Now, What is the company’s maximum profit?

? What should the company charge for each hammer?

? How many hammers should be produced?

? Is this a profit max? Show.

Section 2 – Intro to Statistics

3) Find the number of total building permits for Colorado for each month in 2006. These can be found on the Census Bureau website. You may have to search around (be patient – it’s important to know how to navigate government websites since they hold so much data!) Put these in a spreadsheet or list them below.

Calculate the meanand median. I need to see these “by hand” meaning you can use excel to check your work but I need to see the calculation written out.

Calculate the varianceand standard deviation. Again, I need to see these “by hand” meaning you can use excel to check your work but I need to see the calculation written out.

What kind of observations can you make about the data?

Section 2 – Intro to Probability

4) You’ve noticed that with the class you are taking, you are unable to print or download files about 50 percent of the time. About 25 percent of the time you are unable to download. You find that about 15 percent of the time, you can’t print and you can’t download. What is the probabilitythat you won’t be able to print the next time you log on?

5) You love cooking shows and watch them obsessively. You have begun to notice that Paula Deen uses butter in 97 percent of her recipes. She also uses cream in 63 percent of her recipes. By your calculations, there is a 72 percent chance that she will use either butter or cream in a recipe. What is the chance she’ll use both?

6) Vegas experts estimate that the Patriots have a 78 percent chance of winning the Super Bowl given that they are able to steal the opposing team’s signals prior to the start of the game. There is only a 15 percent chance that the Patriots will be able to steal the signals now that they are being scrutinized. Assuming the Patriots make it to the Super Bowl, what is the chance that they will be able to win?

7) You have already decided that you will look for another job if you do not receive both a 7 percent raise and an extra week of vacation at your next review. Based on past experience, you estimate you have a 65 percent chance of receiving a 7 percent raise and 55 percent chance of receiving a week of vacation and about a 45 percent chance of receiving one or the other. What is the probability you’ll be sending out your resume in the near future?

8) The probability that the Pittsburgh Steelers win their division is 65 percent. The probability that the Steelers win the Super Bowl given that they win their division is ten percent. If they don’t win their division, the probability they win the Super Bowl is 2.5 percent. What is the probability that the Steelers win the Super Bowl?

9) Your company is embarking on a new marketing strategy for a widget and you have developed a new commercial as part of the plan. Initial focus group results show that there is a 27 percent chance that adult consumers will purchase your widget after they view the commercial. If they don’t see the commercial, there is a 4 percent probability they will purchase the widget anyway. You estimate the probability of adults seeing your commercial is about 13 percent. The adult population is about 200,000,000. How many people will purchase your widget?

If the widget sells for a quarter, what is your total revenue?

10) About 72 percent of MBA students at UW are female and 65 percent of MBA students at CSU are female. If you pick a school at random (50/50) and then choose an MBA student at random, what is the probability that the student you choose will be female?

11) The HR director at your firm knows that, on average, about 80 percent of new hires turn out to be good workers. After workers are hired, they are given a personality test. Past results show that about 75 percent of the good workers pass the test. About 50 percent of the bad workers pass the test too. If the firm starts giving the test prior to hiring and only hires those that pass, what is the likelihood that the new hires will be good workers?

This solution gives the step by step method for solving basic statistics problems.