Jan 9, 2017

# Probability Distribution — practice test 5B

20 cards
Problems with solutions in Elementary Statistics — Random Variables; Binomial Probability Distributions: Mean, Variance, and Standard Deviation for the Binomial Distribution; The Poisson Distribution.

Directions:

• Provide an appropriate response.
• 1. A game is said to be “fair” if the expected value for winnings is 0, that is, in the long run, the player can expect to win 0. Consider the following game. The game costs \$1 to play and the winning are \$5 for red, \$3 for blue, \$2 for yellow, and nothing for white. The following probabilities apply. What are your expected winnings? Does the game favor the player or the owner? The expected winnings are –\$0.46. The game is not fair, and it favors the owner of the game.

1. 2. Describe the Poisson distribution and give an example of a ransom variable with a Poisson distribution.

The Poisson distribution is a discrete probability distribution that applies to occurrences of some event over a specified. Examples will vary.

2. Identify the given random variable as being discrete or continuous.
• 3. The cost of a randomly selected orange.
1. Continuous
2. Discrete

• 4. The height of a randomly selected student.
1. Continuous
2. Discrete

3. Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.
• 5. Probability distribution.

1. 6. A police department reports that the probabilities 0, 1, 2, 3, and 4 car thefts will be reported in a given day are 0.150, 0.284, 0.270, 0.171, and 0.081, respectively.

Not a probability distribution. The sum of the P(x)’s is not 1, since 0.956 ≠ 1.000.

2. Find the mean of the given probability distribution.
• 7. The accompanying table shows the probability distribution for x, the number that shows up when a loaded die is rolled. 1. 3.50
2. 3.85
3. 3.98
4. 0.17

3. Solve the problem.
• 8. The random variable x is the number of houses sold by a realtor in a single month at the Sendsom’s Real Estate Office. Its probability distribution is as follows. Find the standard deviation for the probability distribution. 1. 4.45
2. 2.25
3. 6.86
4. 2.62

• 9. A contractor is considering a sale that promises a profit of \$31,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of \$13,000 with a probability of 0.3. What is the expected profit?
1. \$21,700
2. \$30,800
3. \$17,800
4. \$18,000

• 10. Assume that a researcher randomly selects 14 newborn babies and counts the number of girls selected, x. The probabilities corresponding to the 14 possible values of x are summarized in the given table. Answer the question using the table. Find the probability of selecting 2 or more girls.
1. 0.006
2. 0.001
3. 0.999
4. 0.994

• 11. Focus groups of 14 people are randomly selected to discuss products of the Famous Company. It is determined that the mean number (per group) who recognize the Famous brand name is 8.9, and the standard deviation is 0.69. Would it be unusual to randomly select 14 people and find that fewer than 12 recognize the Famous brand name?
1. Yes
2. No

5. Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
• 12. Rolling a single “loaded” die 61 times, keeping track of the numbers that are rolled.
1. Not binomial: the trials are not independent.
2. Procedure results in a binomial distribution.
3. Not binomial: there are too many trials.
4. Not binomial: there are more than two outcomes for each trial.

6. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.
• 13. n = 6, x = 3, p = 1/6
1. 0.0286
2. 0.0536
3. 0.0322
4. 0.0154

7. Find the indicated probability.
• 14. A company purchases shipments of machine components and uses this acceptance sampling plan: Randomly select and test 24 components and accept the whole batch id there are fewer than 3 defectives. If a particular shipment of thousands of components actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted?
1. 0.5553
2. 0.1799
3. 0.0550
4. 0.9307

8. Find the standard deviation, σ, for the binomial distribution which has the stated values of n and p. Round your answer to the nearest hundredth.
• 15. n = 2815, p = .63
1. 29.74
2. 28.89
3. 25.62
4. 23.21

9. Use the given values of n and p to find the minimum usual value μ – 2σ and the maximum usual value μ + 2σ.
• 16. n = 274, p = 0.273
1. Minimum: 89.551; maximum: 60.053
2. Minimum: 60.053; maximum: 89.551
3. Minimum: 64.373; maximum: 85.231
4. Minimum: 67.428; maximum: 82.176

10. Solve the problem.
• 17. The probability of winning a certain lottery is 1/77,822. For people who play 961 times, find the mean number of wins.
1. 0.000013
2. 0.0123
3. 81.0
4. 0.0010

• 18. On a multiple choice test with 22 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the number of correct answers.
1. 1.99
2. 2.031
3. 1.944
4. 1.984

11. Determine is the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than μ – 2σ or greater than μ + 2σ.
• 19. The Acme Candy Company claims that 60% of the jawbreakers it produces weigh more than .4 ounces. Suppose that 800 jawbreakers are selected at random from the production lines. Would it be unusual for this sample of 800 to contain 470 jawbreakers that weigh more than .4 ounces?
1. Yes
2. No

12. Use the Poisson Distribution to find the indicated probability.
• 20. The Columbia Power Company experiences power failures with a mean of μ = 0.120 per day. Find the probability that there are exactly two power failures in a particular day.
1. 0.085
2. 0.018
3. 0.036
4. 0.027