# MATHEMATICS

10. Customers arrive at a Vineyard Vines at an average of 15 per hour (0.25/min).

What is the probability that the manager must wait at least 5 minutes for the first customer? A) 0.2865 C) 0.6836 B) 0.7135 D) 0.1232

Use the following to answer questions 11–14: Consider the following probability histogram for a discrete random variable X: number of hot dogs Capt Jim eats at a cookout. 11. This probability histogram corresponds to which of the following distributions for X?

A) Value of X 1 2 3 4 5 Probability 0.06 0.25 0.38 0.25 0.06 B) Value of X 1 2 3 4 5

Probability 0.10 0.25 0.30 0.20 0.15 C) Value of X 1 2 3 4 5 Probability 0.10 0.25 0.30 0.25 0.10 D) None of the above.

12. What is the P(X = 3)?

A) 0 C) 0.25 B) 0.20 D) 0.30

13. What is P(X < 3)?

A) 0.10 C) 0.35 B) 0.25 D) 0.65

14. What is P(X ≤ 3)?

A) 0.10 C) 0.35 B) 0.25 D) 0.65

15. Central Limit Theorem only applies when sampling from Normal populations.

A) True B) False

Use the following to answer questions 16–18: The Department of Animal Regulations released information on pet ownership for the population consisting of all households in a particular county. Let the random variable X = the number of licensed dogs per household. The distribution for the random variable X is given below:

Value of X 0 1 2 3 4 5 Probability 0.52 0.22 0.13 0.03 0.01

16. The probability for X = 3 is missing. What is it?

A) 0.07 C) 0.1 B) 0.09 D) 0.0

17. What is the probability that a randomly selected household from this community owns at least one

licensed dog? A) 0.22 C) 0.48 B) 0.26 D) 0.52

18. What is the average number of licensed dogs per household in this county?

A) 0 dogs C) 1 dog B) 0.92 dogs D) 1.22 dogs

Use the following to answer questions 19–21: In a large city, 72% of the people are known to own a cell phone, 38% are known to own a pager, and 29% own both a cell phone and a pager. 19. What proportion of people in this large city own either a cell phone or a pager?

A) 0.29 C) 0.81 B) 0.67 D) 1.1

20. What is the probability that a randomly selected person from this city owns a pager, given that the

person owns a cell phone? A) 0.266 C) 0.403 B) 0.38 D) 0.528

21. Are the events “owns a pager” and “owns a cell phone” independent?

A) Yes. B) No, because P(owns a pager) and P(owns a cell phone) are not equal. C) No, because P(owns a pager) and P(owns a pager|owns a cell phone) are not equal. D) Cannot be determined.

Use the following to answer questions 22–23: Chocolate bars produced by a certain machine are labeled 8.0 oz. The distribution of the actual weights of these chocolate bars is claimed to be Normal with a mean of 8.1 oz and a standard deviation of 0.1 oz. 22. A quality control manager initially plans to take a simple random sample of size n from the production

line. If he were to double his sample size (to 2n), by what factor would the standard deviation of the

sampling distribution of X change?

A) 1/2 C) 2

B) 21 D) 2 23. If the quality control manager takes a simple random sample of ten chocolate bars from the

production line, what is the probability that the sample mean weight of the 10 sampled chocolate bars will be less than 8.0 oz? A) 0 C) 0.0316 B) 0.00078 D) 0.1587

24. A sample of size n is selected at random from a population that has mean µ and standard deviation

σ . The sample mean x will be determined from the observations in the sample. Which of the following statements about the sample mean, x , is (are) TRUE?

A) The mean of x is the same as the population mean, i.e., µ .

B) The variance of x is σ 2

n .

C) The standard deviation of x decreases as the sample size grows larger. D) All of the above are true. E) Only A and B are true.

25. From the central limit theorem, we know that if we draw a SRS from any population the sampling

distribution of the sample mean will be EXACTLY Normal. A) True B) False

26. The average batch of chocolate at Schakolad Chocolate Factory will be ready to serve in 1 hour.

What is the chance the next batch will be ready in less than 50 min? A) 0.3681 C) 0.6836 B) 0.5654 D) 0.1232

27. The following histogram shows the distribution of 1000 sample

observations from a population with mean µ = 4 and variance 2σ = 8: Suppose a simple random sample of 100 observations is to be

selected from the population and the sample average, x , calculated. Which of the following statements about the

distribution of x is (are) FALSE?

A) The distribution of x will have a mean of 4.

B) The distribution x will be approximately Normal. C) Because the distribution shown in the histogram above is

clearly skewed to the right, the shape of the distribution of x will also show skewness to the right.

D) Even though the distribution of the population variable appears to be skewed to the right, the

distribution of x will be approximately symmetric around µ = 4.

E) The standard deviation of the distribution of x will be 0.283. Use the following to answer questions 28–30: In a test of extrasensory perception (ESP), the experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle, a wavy line, or a square. An experimenter looks at each of 100 cards in turn, and the subject tries to read the experimenter’s mind and name the shape on each. A subject who is just guessing has probability 0.25 of guessing correctly on each card. 28. What is the probability that the subject gets more than 30 correct if the subject does not have ESP

and is just guessing? (Use the continuity correction.) A) Less than 0.0001 B) 0.1038 C) 0.25 D) 0.31

29. What is the probability of the subject obtaining his/her first correct guess on the 4th question?

A) 0.1055 C) 0.4219 B) 0.8945 D) 0.5781

30. What is the probability of the subject obtaining his/her first correct guess within the first 4 questions?

A) 0.3164 C) 0.6836 B) 0.8945 D) 0.5781

Extra Credit 1. What is the probability it will take exactly 6 rolls of two fair dice to make a 7?

A) 0.9330 C) 0.0804 B) 0.0670 D) 0.9196

2. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals

8?

A) (0.6) 8 (0.4)

2

B)

10!

8! (0.6)

8 (0.4)

2

C) 45(0.6) 8 (0.4)

2

D) 45(0.6) 2 (0.4)

8

E) None of the above.

3. A college basketball player makes 6 5

of her free throws. Assume free throws are independent. What is the probability that she makes exactly three of her next four free throws?

A) ( ) ( )1

6

53

6 14

C) ( ) ( )3

6

51

6 14

B) ( ) ( )1

6

53

6 1

D) ( ) ( )3

6

51

6 1

4. A batch of 100 chocolate chip cookies contains 10 burnt cookies. Five cookies are chosen at random,

without replacement. Find the probability that the sample contains at least one burnt cookie. A) 0.3349 C) 0.4162 B) 0.6606 D) 0.5839

5. The exponential distribution is ______. A) symmetric B) bell-shaped C) All of the above D) None of the above

6. Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one

person. For example, nationally 75.5% of the people drive to work alone. a) If you choose 12 vehicles driving to work at random, what is the probability that more than half

(that is, 7 or more) carry just one person? b) If you choose 80 vehicles at random, what is the probability that more than half (that is, 41 or

more) carry just one person?

Chs 6 – 8