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Practice Test 7 AP Statistics Name:
Directions: Work on these sheets.
Part 1: Multiple Choice. Circle the letter corresponding to the best answer.
1. In a population of students, the number of calculators owned is a random variable X with P (X = 0) = 0.2, P (X = 1) = 0.6, and P (X = 2) = 0.2. The mean of this probability distribution is (a) 0. (b) 2. (c) 1. (d) 0.5. (e) The answer cannot be computed from the information given. 2. Refer to the previous problem. The variance of this probability distribution is (a) 1. (b) 0.63. (c) 0.5. (d) 0.4. (e) The answer cannot be computed from the information given. 3. The number of calories in a one-ounce serving of a certain breakfast cereal is a random variable with mean 110. The number of calories in a full cup of whole milk is a random variable with mean 140. For breakfast you eat one ounce of the cereal with 1/2 cup of whole milk. Let Z be the random variable that represents the total number of calories in this breakfast. The mean of Z is (a) 110. (b) 140. (c) 180. (d) 250. (e) 195. 4. The weight of reports produced in a certain department has a Normal distribution with mean 60 g and standard deviation 12 g. What is the probability that the next report will weigh less than 45 g? (a) 0. (b) 0. (c) 0. (d) 0. (e) The answer cannot be computed from the information given. 5. A business evaluates a proposed venture as follows. It stands to make a profit of $10,000 with probability 3/20, to make a profit of $5000 with probability 9/20, to break even with probability 1/4, and to lose $5000 with probability 3/20. The expected profit in dollars is (a) 1500. (b) 0. (c) 3000. (d) 3250. (e) –1500.
6. A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5000 profit. If it is very cold that day, she expects to suffer a $12,000 loss. Based upon historical records, the weather office has estimated the chances of a warm day to be 0.60; the chances of a cool day to be 0.25. What is the producer’s expected profit? (a) $ (b) $13, (c) $15, (d) $13, (e) $11, 7. A random variable X has a probability distribution as follows: X 0 1 2 3_ P (X) 2k 3k 13k 2k
Then the probability that P (X < 2.0) is equal to (a) 0.90. (b) 0.25. (c) 0.65. (d) 0.15. (e) 1.00.
8. Cans of soft drinks cost $ 0.30 in a certain vending machine. What is the expected value and variance of daily revenue (Y) from the machine, if X, the number of cans sold per day has E(X) = 125, and Var(X) = 50? (a) E(Y) = 37.5, Var(Y) = 50 (b) E(Y) = 37.5, Var(Y) = 4. (c) E(Y) = 37.5, Var(Y) = 15 (d) E(Y) = 37.5, Var(Y) = 30 (e) E(Y) = 125, Var(Y) = 4. 9. Let the random variable X represent the amount of money Dan makes doing lawn care in a randomly selected week in the summer. Assume that X is Normal with mean $240 and standard deviation $60. The probability is approximately 0.6 that, in a randomly selected week, Dan will make less than (a) $ (b) $ (c) $ (d) $ (e) The answer cannot be determined from the information given.
Questions 10, 11 and 12 refer to the following information. X is a random variable that takes the value A between X = 0 and X = 2 and is zero everywhere else. X defines a uniform density curve.
10. The probability that X is between 0.5 and 1.5 is (a) 1/3. (b) 1/2. (c) 3/4. (d) 1. (e) 1/
Part 2: Free Response Answer completely, but be concise. Write sequentially and show all steps.
16. The probability that 0, 1, 2, 3, or 4 people will seek treatment for the flu during any given hour at an emergency room is shown in the following distribution.
X 0 1 2 3 4_ P(X) 0.12 0.25 0.33 0.24 0.
(a) What does the random variable count or measure?
(b) Calculate the mean of X, and interpret this value in context.
(c) What are the variance and standard deviation of X?
17. If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If the person gets a 7, he wins $5. The cost to play the game is $3. Find the expectation of the game.
18. Picard Partners is planning a major investment. The amount of profit X is uncertain but a probabilistic estimate gives the following distribution (in millions of dollars):
Profit 1 1.5 2 4 10 Probability 0.1 0.2 0.4 0.2 0.
(a) Find the mean profit μ X and the standard deviation of the profit.
(b) Picard Partners owes its source of capital a fee of $200,000 plus 10% of the profits X. So the firm actually retains Y = 0.9X – 0.2 from the investment. Find the mean and standard deviation of Y.
19. A study of the weights of the brains of Swedish men found that the weight X was a random variable with mean 1400 grams and standard deviation 20 grams. Find numbers a and b such that Y = a + b X has mean 0 and standard deviation 1. 20. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Let X and Y be the lengths of two randomly chosen pregnancies. What is P (X + Y) > 600?
