Illinois State University Mathematics Department
MAT 312: Probability and Statistics for Middle School Teachers
Dr. Roger Day (day@ilstu.edu)
Semester Exam
Possible Solutions
Scoring
 Part I: 30 Multiple Choice Questions (30 points)
 Part II: 14 OpenResponse Questions (30 points)
 Bonus: 5 points possible
 Total: 60 points + 5 Bonus Points
 Impact on Course Grade: 30% of your Semester Grade
Criteria Used to Evaluate Part II Responses
Your responses to these questions 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 ChoiceFor each question, choose the most correct response and circle that letter at the appropriate spot on the answer sheet.
1. 
The manufacturer of a new type of light bulb wants to show that the new bulbs outlast those of a major competitor. The manufacturer tested 30 bulbs and recorded the life span of each. Here are the data. Use these data for questions 1 through 3. The data are represented in a _?_.


2. 
What portion of the bulbs tested lasted at least 500 hours?


3. 
Determine the 25th percentile of this data set.


4. 


5. 
The following visual representation shows test scores of 48 students in a science course. How many students scored 50 or less on the test?


6. 
In a distribution that is positively skewed, which statement is most likely to be true?


7. 


8. 
The time that it takes to drive from the Interstate Center in Bloomington to the Peoria Civic Center on a Saturday is normally distributed with a mean of 54 minutes and a standard deviation of 7 minutes. Driving times of no more than 54 minutes represent approximately what portion of all the driving times for this situation?


9. 
Which visual representation preserves the values in a data set?




10. 
the colors of the background pages in a family scrapbook d. nominal data 

11. 
the size of each family scrapbook, categorized as small, medium, large, or supersrcap b. ordinal data 

12. 
the size of each photo in a family scrapbook, expressed in square inches c. ratio data


13. 
Suppose that the distribution of the life spans of a certain dog breed has a symmetric, moundshaped (normal) distribution with a mean of 9.5 years and a standard deviation of 3.5 years. Within what bounds do we expect approximately 2.5% of the life spans of such dogs to fall?


Here are the theoretical probabilities for an experiment whose sample space is {0,10,20,30,40,50}. Use this for questions 14 through 16.


14. 
Determine P(x is less than or equal to 30).


15. 
Determine the expected value of this experiment.


16. 
Suppose the experiment was carried out 1000 times and a histogram of the results was created. The histogram would most likely appear _?_.


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 
For questions 24 and 25, consider the following probability distribution for some experiment.


25. 
Let F be the event "5 does not occur." What is the probability of the complement of F?


26. 
Which statement below is most correct regarding the following distributions?


27. 
Which of the following is the least justifiable criterion for positioning a spaghetti line on a scatter plot to represent the relationship between two variables?


28. 
A data set of 125 ordered pairs relates age of a car, in years (a), to its resale value, in dollars (R). For example, (3, 9250) represents a 3yearold car with a resale value of $9250. Suppose that a medianmedian line is calculated for these data and is represented by the equation R = 1256a + 11952. In this equation, the number 1256 represents _?_.


29. 


30. 

Part II: Open Response
Complete each question and write your response in the space provided.
A. 
On the wall at a local pizzeria is a rectangular dart board, similar to the one shown below. The board is composed of four rectangles centered upon each other. Here are the dimensions of the rectangle (Note: Figure not drawn to scale.):
For $5 a customer can try to win a pizza by throwing a dart at the board. Any dart landing in Rectangle #4 earns a familysize pizza ($18 value). If a dart sticks in Rectangle #3 (and not within #4) the customer gets a large pizza ($12 value). For a dart sticking in Rectangle #2 (and not within #3 nor within #4) the customer gets a medium pizza ($8 value). A dart on any other portion of the board (in Rectangle #1 and not within Rectangle #2 nor #3 nor #4) wins no prize. 31. Suppose a dart hits the board at some random point. What is the probability of winning a medium pizza? Some Useful Information: Areas of Rectangles
To get a medium pizza, dart must be in #2 but not in #3. This region has area 80 square inches (14060). We compare this region to the area of entire board, just the area of Rectangle #1, 240 square inches. This generates a probability of winning a medium pizza as 80/240 or 1/3. 32. Let w represent a random variable that represents the prize values possible on a toss of a dart, assuming that darts always hit the board at some random location. Create a table to show the probability distribution for the random variable w. To generate the
desired probabilities, we proceed as illustrated in question
31 above. 33. Calculate the expected value of w. Explain what this represents for a pizzeria customer. E(w) = $0*(5/12) + $8*(1/3) + $12*(7/40) + $18*(3/40) = $6.12 (rounded) This indicates that a $5 investment yields a $6.12 outcome, or a "profit" of $1.12. Note the assumption, however, that every thrown dart hits the board in a random location. You might be a better thrower, or, alternatively, you may not! 
2,3,2: 

B. 
34. The prom supervisor at a local high school asked for volunteers for next year's prom committee. There were seven 10thgrade and five 9thgrade volunteers. The advisor only needed three from each of the two classes. How many 6person prom committees could be formed under these conditions? Explain your response. C(7,3)*C(5,3) = 350 different committees Arrangement within the committee is not a factor, so choose 3 of the 7 10th graders and 3 of the 5 9th graders. 35. The design of a quilt for a newborn baby is shown here, composed of three parts: an inset, a body, and a border. Suppose each of the three parts of the quilt is to be a solid color, with each part a different color. If there are only seven colors to choose from for the quilt, how many different quilts could be made? Explain your response. P(7,3) = 210 different quilts can be made. Here, arrangement matters, so we select 1 of 7 colors for the border, 1 of the remaining 6 for the body, and 1 of the remaining 5 for the inset. 36. Junior Samples was left in charge of the hatcheck room at a recent square dance. Against one wall in the hatcheck room was a rectangular array of numbered cubicles into which hats could be placed. A big sign hung over the cubicles. Here's what it said:
During the night, Junior kept a tally sheet showing the time on the clock and the number of hats that were in check at that time. A portion of the tally sheet is shown below. After the dance, Wally Woundhouse, Junior's supervisor, sauntered over to the hatcheck room and surveyed Junior's tally sheet. He looked at the 9:05 entry and said, "Junior, if your figures are accurate, at precisely 9:05 I know that you violated the hatcheck policy! You may have violated it some other time not shown here, but I can assure you that you did at 9:05!" Based on Wally's statement, how many cubicles were there in the hatcheck room? Explain how you know. Because the only violation noted was at 9:05, it must be the case, by the pigeonhole principle, that 150 hats was the maximum allowed and therefore that there are 50 cubicles in the hatcheck area.

2,2,3: 

C. 
Use the following sets for questions 37 through 40:
37. A letter is to be chosen from Set I or from Set II. How many choices are there? Explain. 9 + 8  2 = 15 choices By the addition principle, we can choose one of 9 from the first set or one of 8 from the second set. However, there are two elements common to the two sets, so we must subtract these. 38. Twoletter "words" (meaningful or otherwise) are to be created using a letter from Set I as the first letter of the word and a letter from Set III as the second letter of the word. How many different twoletter words can be created in this manner? Explain your response. 9*6 = 54 words Using the multplication principle, there are 9 ways to fill the first position and 6 ways to fill the second position. 39. Threeletter sets are to be created using the letters from Set I with no repetition allowed. For example, {a,c,g} can be created, but not {a,a,a} nor {a,a,g}. Note, also, that the set {a,c,g} is equivalent to the set {g,c,a}. How many unique threeletter sets can be created? Explain your response. C(9,3) = 84 sets With no repetition allowed and order being nonsignificant, we grab 3 of the 9 letters from Set I. 40. How many unique new sets can the created from the letters in Set IV, under the following conditions? Explain your response.

2,2,2,2: 

D. 

2,2,2,2: 

Juanita is a political satirist. She claims to know enough jokes today so that she could tell a different set of three jokes in her warmup act, every night of the year, for at least 40 years. What is the minimum number of jokes she must know? NOTE: The set of jokes {A,B,C} is considered one set of jokes, no matter what order Juanita tells the three jokes. There are 365 days per year during nonleapyear years. Through 40 years, this amounts to 14600 days. During this time, there can be, at most, 11 leap years. So, all tolled we must account for 14611 days. Juanita, therefore, must have a pool of jokes from which she can grab 3 at a time and never repeat the same set of 3 during 14611 days of joke telling. This reduces to determining a value for j such that C(j,3) is greater than or equal to 14611. By guess and test, we get that and that C(46,3) = 15180. This indicates that 46 jokes is the minimum number in Juanita's collection. 