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)

 Semester Exam Possible Solutions

Scoring

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 Choice

For each question, choose the one best 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.

The data are represented in a _?_.

a. line plot
b. stem-and-leaf plot
c. box-and-whisker plot
d. scatter plot
e. vertical plot
f. nice plot

2.

What portion of the bulbs tested lasted less than 400 hours? (See question 1.)

a. 6%
b. 20%
c. 24%
d. 80%

3.

Determine the 75th percentile of this data set. (See question 1.)

a. 420 hours
b. 480 hours
c. 490 hours
d. 630 hours

4.

 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.

5.

This visual representation shows test scores of 48 students in a science course. How many students scored at least 80 on the test?

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

6.

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? (See question 5 figure.)

a. a = b
b. b < a
c. a = 19
d. b = 13

7.

In 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.

8.

 For the plot here, what correlation coefficient is most likely? Assume that the axes scales are equal. a. 1.0 b. 0.67 c. 0.06 d. &endash;0.64 e. &endash;1.0

9.

The time that it takes to drive from Stevenson Hall to Eastland Mall at 9:00 am on a Saturday is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes.

Driving times ranging from 10 minutes to 12 minutes represent approximately what portion of all the driving times?

a. 5%
b. 13.5%
c. 27%
d. 34%
e. 68%
f. 95%

10.

[Refer to question 9.] Determine the driving time that will be exceeded by approximately 97.5% of all drivers making the trip from Stevenson Hall to Eastland Mall at 9:00 am on a Saturday.

a. 6 minutes
b. 8 minutes
c. 10 minutes
d. 14 minutes
e. 16 minutes
f. 18 minutes

11.

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

12.

State an appropriate estimate for the slope of a straight line that could be fit to these data. Note the axes scales. (See question 11.)

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

13.

A running 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.

14.

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 13.)

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

15.

 The spinner shown here is spun once. Each of the center angles measures 60û. Determine the probability that the pointer lands on 20 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.

16.

Here are the theoretical probabilities for an experiment whose sample space is {0,1,2,3,4,5}.

 x 0 1 2 3 4 5 p(x) 1/5 3/8 3/10 1/10 1/50 1/200

According to the theoretical probabilities, which value ought to occur most often?

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

17.

Determine P(x ³ 3). (See question 16.)

a. 0
b. 1/40
c. 1/10
d. 1/8
e. 1

18.

Determine the expected value of this experiment. (See question 16.)

a. 1.0
b. 1.38
c. 2.0
d. 2.5
e. 3.0

19.

Suppose the experiment was carried out 200 times and a histogram of the results was created. The histogram would most likely appear _?_.

a. bimodal
b. symmetric
c. negatively skewed
d. uniform
e. positively skewed

20.

Two numbers, a and b, are each created using a TI-83 random number generator. The random number generator creates numbers between 0 and 1. What is the probability that both a and b are less than 0.5 ?

a. 0
b. 0.05
c. 0.25
d. 0.5
e. 0.75
f. 1.0

Part II: Open Response

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

A.

A student group sells donuts at the mall. On recent Saturdays, they've been recording the number of donuts sold, along with the selling price. Here's the data:

A.1. On the grid provided, create a scatter plot for these data. Represent sale price in cents on the horizontal axis (x) and number of donuts sold on the vertical axis (y). Clearly indicate how you have scaled each axis.

A.2. Explain whether either the table of values or your scatter plot reveal a relationship between donut sale price and the number of donuts sold.

A.3. Least-squares linear regression is applied to the donut sales data set, resulting in the equation y = &endash;36.583x + 2121.10, where x represents donut sale price in cents and y represents the number of donuts sold. A correlation coefficient of r = &endash;0.9995 is computed with this least-squares linear regression equation.

 A.3.a. What is the slope of the regression equation? Describe its meaning in the context of this data set. Be specific. A.3.b. State two reasons you might question or doubt the meaningfulness of the y&endash;intercept of the regression equation. Be specific. A.3.c. Use the linear regression equation to predict the number of donuts sold when the sale price is 60 cents.

2, 2, 6: 10 points total

B.

On the wall at a local pizzeria is a square dart board, each side 10" long. For \$1 a customer can try to win a pizza by throwing a dart at the board.

The board contains three smaller squares whose centers are at the center of the board. Any dart landing in the innermost square, a square with side length 1", earns a large pizza (\$10 value). If a dart sticks in the first layer outside the innermost square, part of a square of side length 3", the customer gets a medium pizza (\$5 value). For a dart sticking in the next layer, part of a square with side length 5", the customer gets a small pizza (\$2 value). A dart on any other portion of the board wins no prize.

B.4. Suppose a dart hits the board at some random point. What is the probability of winning a medium-size pizza?

B.5. If customers played this game many many times, and we assume that darts always hit the board at some random location, what is the expected gain or loss per play, from a customer's standpoint?

B.6. Assume again that darts always hit the board at some random location. What is the net gain the pizzeria can expect if 1000 rounds of this game are played some weekend? Take into account only the cost to play and the value of the prizes.

3,3,4: 10 points total

C.

C.7. In the far-off world of Balbion III in the Mostarth Galaxy, each year every family in the city of Krameth is given a pet. There are three species of pets randomly distributed to the Kramethian families. A family receives a Quark with probability 0.2, a Rorst with probability 0.3, and a Swimp with probability 0.5.

Determine the probability that in a three-year sequence, a family gets:

C.7.a. two Rorsts and a Quark

C.7.b. three pets all of the same species

C.7.c. at least two Swimps

3,3,4: 10 points total

D.

Suppose we know that the distribution of the waiting times (in minutes) for drivers boarding a ferry boat on Lake Erie is mound-shaped and symmetrical, that is, the waiting times are normally distributed. The mean waiting time is 16 minutes and the standard deviation is 4 minutes.

D.8. What portion of all drivers will wait 16 minutes or less?

D.9. What is probability that a driver will wait 20 minutes or more?

D.10. Based on the information given, we know that approximately 2.5% of all drivers wait at least x minutes. What value of x makes this true?

3, 3, 4: 10 points total

E.

This problem requires you to design and carry out a simulation. The situation is first described to you and then several questions are asked related to the simulation.

My nephew Seth noticed that Kellogg's cereals offered a set of 3 cartoon characters in its current cereal selections. One cartoon character is in each specially marked box of cereal and the cartoon characters are equally distributed among the cereal boxes currently coming off the production line. Seth wondered how many boxes of cereal he'd have to purchase to get the entire set of cartoon characters.

Design and carry out a simulation to address Seth's question. Assume that one trial of your simulation will determine the number of boxes of cereal he must purchase to get a complete set of 3 cartoon characters.

E.11. Describe the model you will use to simulate this situation. In your description:

E.11.a. Indicate how you will generate random outcomes.
E.11.b. Specify the decisions you will make based on the random outcomes.
E.11.c. Justify that your model accurately represents the situation.

E.12. Show the details of one trial of your simulation. Include:

E.12.a. a list of the random outcomes you generated for one trial,
E.12.b. the decisions you made based on the random outcomes, and
E.12.c. the number of cereal boxes required to get a complete set of cartoon characters, based on this single trial.

E.13. Carry out at least 10 trials of this simulation. Use the results to answer Seth's original question: How many boxes of cereal will he have to purchase to get the entire set of cartoon characters?

3, 3, 4: 10 points total

BONUS!

Assume that 2% of the population is on drugs. A test is 98% accurate in indicating whether or not a person is on drugs. This means that people on drugs will test positive* 98% of the time and people not on drugs will test negative* 98% of the time.

Determine the probability that a person is on drugs give that the person's test result is positive. Provide clear and specific evidence to support your response.

*A positive test means the test results indicate the person is on drugs; a negative test means the test results indicate the person is not on drugs.