Illinois State University Mathematics Department

MAT 312: Probability and Statistics for Middle School Teachers

Dr. Roger Day (

Test #3: Possible Solutions

  • Part I: 20 Multiple Choice Questions (1 pts each)
  • Part II: 15 Open-Response Questions (2 pts each)
  • Total: 50 points
  • Impact on Course Grade: 20% of your Semester Grade

Criteria Used to Evaluate Part II Responses

Your responses to questions 21 through 35 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: {1,4,9,16,25,36,49,64,81,100}

P(1) = 0.05, P(4) = 0.02, P(9) = 0.13, P(16) = 0.05, P(25) = 0.10,

P(36) = 0.20, P(49) = 0.12, P(64) = 0.08, P(81) = 0.15, P(100) = 0.10

1. This experiment illustrates a uniform distribution.

a. True.
b. False.
c. It is impossible to determine whether this is a uniform distribution.

2. Let A be the outcome "25 occurs." What is the probability of the complement of A?

a. 0.10
b. 0.25
c. 0.50
d. 0.70
e. 0.90
f. 0.95

3. If a random variable x represents an outcome of this experiment, how many different values can the random variable x take on?

a. 0
b. 1
c. 4
d. 10
e. 16
f. 100

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

P(green) = 0.20, P(blue) = 0.15, P(white) = 0.25, P(yellow) = 0.10, P(gray) = 0.05

4. If "red" is the only other outcome that belongs in the sample space with {green,blue,white,yellow,gray} for this experiment, determine P(red).

a. 0.10
b. 0.15
c. 0.20
d. 0.25
e. 0.30
f. P(red) cannot be determined from this information.

5. Let F be the event that the outcome is a color spelled with more than four letters. Determine P(F).

a. 0.20
b. 0.25
c. 0.45
d. 0.55
e. 0.75
f. P(F) cannot be determined from this information.

6. Which statement below is most correct regarding the following suggested probability distributions?

Distribution I
Distribution II
Distribution III
Distribution IV
a. Distribution I is the only valid probability distribution.
b. Distribution II is the only valid probability distribution.
c. Distribution III is the only valid probability distribution.
d. Distribution IV is the only valid probability distribution.
e. None of these are valid probability distributions.
f. More than one of these are valid probability distributions.

7. People are lined up at a food stand during the Sugar Creek Arts Festival. The food stand offers exactly and only five different beverage choices and each person in line selects exactly one beverage. How many people must be in line to assure that at least one beverage option is chosen by at least 10 people?

a. 10 people
b. 11 people
c. 45 people
d. 46 people
e. 50 people
f. 51 people

For questions 8 and 9, suppose that the heights of women undergraduates at ISU have a mean of 144 cm with a standard deviation of 9 cm. Suppose that the shape of this heights distribution is a normal distribution (symmetric and mound-shaped).

8. What portion of the heights are between 144 cm and 162 cm?

a. approximately 34%
b. approximately 47.5%
c. approximately 50%
d.approximately 68%

9. What height represents the approximate 50th percentile of all heights?

a. 135 cm
b. 144 cm
c. 153 cm
d. 162 cm

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

a. -0.52
b. 0.37
c. 0.52
d. 0.99

The visual representation shown here represents test scores of 48 students in a science course. Use it for questions 11 and 12.

11. How many students scored at least 80 on the test?

a. 12
b. 20
c. 40
d. 50

12. In the plot above, if a represents the number of students who scored in the second quartile and b represents the number of students who scored in the third quartile, which statement is most correct?

a. a = b
b. b < a
c. a = 15
d. b = 14

13. For a distribution that is negatively skewed, which statement is most likely to be true?

a. The mean and median will be equal.
b. The mean will be greater than the median.
c. The mean will be less than the median.
d. The mean will equal 0.

14. The plot here shows the distribution of heights of residents in a Rockford nursing home. The median height lies in which measurement class?

a. 50-55 inches
b. 55-60 inches
c. 60-65 inches
d. 65-70 inches
e. None of these measurement classes contain the median height.

15. The visual representation shown here helps describe the relationship between mathematics placement test scores and writing test scores for an incoming class of students. The plot provides information about the _?_ of that relationship.

a. center, spread, and shape
b. direction, shape, and location
c. location, value, and shape
d. shape, strength, and direction
e. source, direction, and value

16. State an appropriate estimate for the slope of a spaghetti line that could be fit to these data. Note the axes scales. (See question 15.)

a. &endash;0.75
b. &endash;0.35
c. 0
d. 1.2
e. 4.0
f. 8.54

17. A dance club on a college campus has 20 members, each of whom is eligible to serve on the club's 4-member executive committee. In how many ways can the executive committee be selected from the membership?

a. 4845
b. 116,280
c. 160,000
d. None of these are correct.

18. If the chair of the executive committee is to be elected from among the 20 members, in how many ways could the 4-member executive committee be formed, assuming that the elected chair is to be one of the four members? (See question 17.)

a. 4845
b. 116,280
c. 160,000
d. None of these are correct.

19. The spinner shown here is spun once. Each of the center angles measures 60. Determine the probability that the pointer lands on 10 or 30, given that it lands on a multiple of 10.

a. 1/6
b. 1/3
c. 1/2
d. 2/3
e. None of these are correct.

20. A box-and-whiskers plot provides a visual _?_ of a data set.

a. description
b. display
c. representation
d. summary

Part II: Open Response

Complete each question and write your response in the space provided. Please include descriptive comments to help me understand your solution strategy.

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


21. Determine P(y = 2).


P(y = 2) = 0.20

22. What is P(y 1)?


P(y is greater than or equal to 1) = 0.70

23. Suppose event M is "y is 0, 1, or 2." What is the probability that the complement of M will occur?


P(complement of M) = 1 - P(M) = 1 - 0.90 = 0.10

Suppose the four outcomes in the table represent the distribution of responses to the following survey question, posed to shoppers at a regional shopping mall:

"How many times in the last 30 days have you washed your car?"

24. Jeremy claims that the outcome y = 1 and the outcome y = 2 are mutually exclusive outcomes. Do you agree or disagree with Jeremy? Explain your response.


Jeremy is correct. y = 1 and y = 2 are mutually exclusive because both outcomes cannot occur simultaneously.

25. Based on the distribution shown above, explain how you know that no survey respondent stated that he or she had washed his or her car 5 times in the last 30 days.


The probabilities in the table sum to 1, indicating there are no other possible outcomes.

Here is a portion of the menu board at a new snack shop called Pierre's Sandwich Corner. Use it to answer questions 26 through 30.

Pierre's Sandwich Corner
Bread Choices
Meat and Dairy Choices
Vegetable Choices
Roast Beef
Root Beer
Diet Cola

Whole Wheat

Bean Sprouts

Seven Grain

26. On Sundays, Pierre's offers a sandwich special with your choice of two meat/dairy and three vegetable ingredients on your choice of sandwich bread. How many different sandwich specials are possible? Do not account for any differerences in how the ingredients are placed within the sandwich.


6 * C(4,2) * C(5,3) = 360 choices


There are 6 bread types to choose from, two of four meat/dairy items to select, and three of five vegetable items to select.

27. If we know the last customer at Pierre's ordered a sandwich on Raisin Bread with either one or two meat/dairy items and either three or four vegetable items, how many different sandwiches could have been ordered? Do not account for any differerences in how the ingredients are placed within the sandwich.


C(4,1) * C(5,3) + C(4,1) * C(5,4) + C(4,2) * C(5,3) + C(4,2) * C(5,4) = 150 choices


The products above show the four item combinations the customer could have selected. We sum them because each is a possibility and the customer selected from only one of them.

28. How many people must line up at Pierre'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.


Four people. By the Pigeonhole Principle, the first three may each select a different beverage, but the four person would then be forced to suplicate one of the three.

29. If a combinatorics instructor wants a sandwich on Rye that contains every possible meat/dairy and vegetable choice, and that instructor does pay attention to how the items are layered on the sandwich, how many different sandwiches could the instructor order?


P(9,9) = 9!


There are a total of nine items to choose from and the instructor wants all nine items, so we count the permutations of those 9 items.

30. A choral director wants to buy sandwiches and feed a community choir, one sandwich to each choir member, each sandwich different from all the others. A sandwich can be made with any of the bread choices, and can contain anything from no meat/dairy items and no vegetable items (i.e., just the bread) to a sandwich with every possible meat/dairy and vegetable item on it. How many different such sandwiches could be made? Do not account for any differerences in how the ingredients are placed within the sandwich.


6 * 2^9 = 3072


Begin by considering the ingredients. There are nine items in all, including four meat/dairy and five veggie. For each of these items, we either include the ingredient or do not include the ingredient, that is, two options for each ingredient. By the multiplication principle, there are 2^9 ways to complete these options. Note that nine "NO" responses gives us the naked sandwich and nine "YES" responses gives us a sandwich with all nine ingredients on it.


We then multiply this result by 6 because for each of the 2^9 ingredient-selection options, we can have one of six different types of bread.

For questions 31 through 35, suppose that the Say-Good-Bye lottery requires players to correctly choose three numbers from the first 12 natural numbers (1, 2, 3, . . . , 12).

For example, Elmo might choose 6, 10, and 9. He can choose no number more than once. The order Elmo choose the numbers does matter. The choice 7, 12, 1 is different from 12, 7, 1. Elmo pays $3 to choose one arrangement of three numbers.

Each day, a computer program randomly chooses three numbers, and Elmo wins if his choice matches the computer's, in the correct order. When Elmo wins, State-O-Fun pays him $500.

31. How many different arrangements of three numbers can Elmo choose in the Say-Good-Bye lottery?


P(12,3) = 12 * 11 * 10 = 1320 arrangements

32. What is the probability Elmo will choose the winning arrangement of three numbers when he makes one choice as described above? Express your response as a common fraction.


P (wining on one ticket) = 1/1320

33. Let w be a number that represents Elmo's winnings with one ticket in the Say-Good-Bye lottery. Create a table to show the two possible values for w and the probability associated with each possible value. Express each probability as a common fraction.


34. Suppose the Say-Good-Bye lottery is changing the game to include the first 20 natural numbers (1, 2, 3, . . . , 20), with all other aspects of the game unchanged. Under these conditions, what is the probability Elmo will choose the winning arrangement of three numbers when he makes one choice as described here? Express your response as a common fraction.


P (wining on one ticket) = 1/[P(20,3)] = 1/6840

35. The State-O-Mind lottery requires players to correctly pick five numbers from among the first 49 natural numbers (1, 2, 3, . . . , 49), with no regard for the order the numbers are selected. Players pay $1 for each pick and the State-O-Mind lottery pays a winner $1,000,000 for a correct pick (matching all five chosen numbers). What is the probability Elmo will win this lottery game? Express your response as a common fraction.


P (wining on one ticket) = 1/[C(49,5)] = 1/1906884


Three unique individuals sit at a coffee bar every morning to drink coffee and read a local newspaper. There are eight identical stools in a line at the coffee bar. If the three people refuse to speak with each other and demand that at least one bar stool stands between that person and either of the other two seated coffee drinkers, how many possible sitting arrangements are there? Do not concern yourself with any other sitters than the three described here.

Keep Thinkin' !!!