Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Spring 1999 9:35 - 10:50 am TR STV 350A Dr. Roger Day (day@ilstu.edu)

 Test #3 Possible Solutions

Scoring
• Part I: 10 Multiple Choice Questions (2 pts each)
• Part II: 15 Open-Response Questions (2 pts each)
• Total: 50 points
• Impact on Course Grade: 15% of your Semester Grade

Criteria Used to Evaluate Part II Responses

Your responses to questions 11 through 25 will be evaluated for correct and accurate numerical solutions, appropriate and adequate explanations where required or indicated, and overall clarity of your response.

Part I: Multiple Choice

For each question, choose the one best response and circle that letter at the appropriate spot on the answer sheet.

For questions 1 through 3, consider the following probability distribution for some experiment.

Sample space: {0,2,4,6}

P(0) = 0.1, P(2) = 0.2, P(4) = 0.3, P(6) = 0.4

1. {0,2,4,6} is a uniform sample space.

a. True.
b. False.

2. Let A be the event "a 4 or a 6 occurs." What is the probability of ~A, the complement of A?

a. 0.1
b. 0.2
c. 0.3
d. 0.4
e. 0.6
f. 0.7

3. What is the expected value for this situation? That is, what is the average outcome we can expect for this experiment if the experiment were to be repeated many many times?

a. 0
b. 1
c. 2
d. 3
e. 4
f. 5

For questions 4 and 5, consider the following probabilities for some experiment.

P(7) = 0.10, P(8) = 0.20, P(9) = 0.20, P(10) = 0.15, P(11) = 0.15

4. If 12 is the only other value that belongs in the sample space with {7,8,9,10,11} for this experiment, determine P(12).

a. 0.10
b. 0.15
c. 0.20
d. 0.25

5. Let A be the event that the outcome is less than 9. Determine P(A).

a. 0.10
b. 0.20
c. 0.30
d. 0.50

6. Based on your knowledge of probability distributions, which one of the following is a valid distribution?

7. The bookstore offers to give away seven different T-shirts, each with a different college logo. Only one size shirt is available for any design. Each student in line for a free shirt chooses one T-shirt. How many people must be in line to be sure that at least one of the T-shirt designs is chosen by at least four people?

a. 4 people
b. 8 people
c. 11 people
d. 12 people
e. 16 people
f. 22 people

Homes in an exclusive Bayview neighborhood have a mean sales price of \$225,000 with a standard deviation of \$45,000. Suppose that the shape of the home-sales distribution for Bayview is a normal distribution (symmetric and mound-shaped).

8. What portion of the Bayview homes have sales prices between \$180,000 and \$315,000?

a. approximately 70%
b. approximately 81.5%
c. approximately 85%
d.approximately 95%

9. What price represents the approximate 84th percentile of all prices?

a. \$225,000
b. \$270,000
c. \$315,000
d. \$360,000

10. For the plot here, what correlation coefficient is most likely? Assume the vertical and horizontal axes are equally scaled.

a. 0.95
b. 0.37
c. -0.06
d. -0.64
e. -1.0

Part II: Open Response

Complete each question and write your response in the space provided. Please include descriptive comments as necessary.

The table here shows the probability distribution for an experiment with five outcomes. Use it to answer questions 11 through 15.

11. Determine P(k = 3).

12. What is P(k ³ 2)?

13. Suppose event E is "k is 0, 1, or 2." What is the probability that the complement of E will occur?

Suppose the five outcomes in the table represent the distribution of responses to the following survey question, posed to mid-level executives of a local corporation:

"How many cars does your immediate family own?"

14. From this situation, identify a pair of events that are mutually exclusive and explain why the two events are mutually exclusive.

15. What is the average number of cars owns by those responding to the survey question? Explain.

Here is a portion of the menu board at a new snack shop called Pascal's Popcorn Parlor. Use it to answer questions 16 through 20.

16. On Friday nights, Pascal's offers a double-flavored popcorn special. Last week, the special was Beefy Cheese and Tomato popcorn. Without regard for the taste of any such special, how many different double flavored specials are possible?

17. If we know the last customer at Pascal's ordered one item, either a flavored popcorn (exactly one flavor) or a beverage, how many different orders could have been placed?

18. How many people must line up at Pascal's to be sure that at least one of the beverage choices is selected by more than one person? Assume that everyone in line orders exactly one beverage.

19. Pascal creates daily specials and advertises them under the Specials portion of the menu board shown above. If the two spots on the Specials menu are to show one popcorn topping and one beverage, how many ways can Pascal fill out the Specials menu board?

20. Pascal's longtime friend Pierre Simon de Laplace entered the snack shop one Saturday afternoon and placed the most bizarre order Pascal had ever heard. Laplace asked for a bag of every possible popcorn-flavoring combination that Pascal could make, from a bag of no-topping plain popcorn to a bag with all eight toppings on the popcorn. How many bags did Pascal prepare for his friend?

The State-O-Fun lottery is based on participants correctly choosing two numbers from the first 15 natural numbers (1, 2, 3, . . . , 15).

For example, I might choose 12 and 7. I can choose no number twice, so 8 and 8 is not an allowable choice. The order I choose the numbers does not matter. The choice 7 and 12 is the same as 12 and 7. I pay \$1 to choose two numbers.

Each day, a computer program randomly chooses two numbers, and I win if my choice matches the computer's. When I win, State-O-Fun pays me \$50.

21. How many different pairs of numbers can be chosen in this lottery?

22. What is the probability I will choose the winning pair of numbers when I select one pair as described above?

23. Let w be a number that represents my net gain when buying one ticket in the State-O-Fun lottery. Create a table to show the two possible values for w and the probability associated with each possible value. Note that one value of w will be negative!

24. Compute my average net gain (expected value) using the information in (23).

25. If 100,000 people play the State-O-Fun lottery, how much money can State-O-Fun expect to have left after paying out prizes from the money collected from players? Do not consider any expenses of running the lottery.

 BONUS! Assume for this problem that the probability of the birth of a girl or a boy is 1/2. Suppose we know that a family has exactly two children. What is the probability that they are both boys, given that at least one of them is a boy?