Name: __________________________ Date: _____________
1. For each of the following scenarios, determine the appropriate distribution for the random variable X.
A) A fair die is rolled seven times. Let X = the number of times we see an even number.
B) A card is selected at random from a standard deck of shuffled cards. The color of the card is determined and the card is returned to the deck. The cards are shuffled again. This selection procedure is repeated sixty times. Let X = the proportion of times the selected card is red.
C) The percentage of female students at a large university is known to be 46%. A simple random sample of 100 students is to be taken. Let X = the number of male students in the sample.
D) On any given Saturday during college football season, there are roughly 70 games being played. At each game, a fair coin is flipped to determine which team gets to kick off first. Let X = the proportion of these coins that land heads.
2. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable X.
A) X = the number of phone calls received in a one-hour period.
B) A hand of 5 cards will be dealt from a standard deck of 52 cards that has been thoroughly shuffled. Let X = the number of hearts in the hand of 5 cards.
C) Five random digits are to be randomly selected from your table of random digits (Table B). Let X = the number of sevens in this set of 5 random digits.
D) A set of random digits are to be randomly selected from your table of random digits (Table B). Let X = the number of random digits selected until we get 5 sevens.
3. For which of the following does the random variable X have a binomial distribution? A) X is the number of pastrami sandwiches sold at a deli in a month.
B) X is the number of speeding tickets given out at a randomly picked location in a city during a calendar year.
C) X is the number of defects found in 100 meters of fiber optic cable.
D) X is the number of people in a random sample of size 50 from a large population that have type-AB blood.
4. A production process, when functioning as it should, will still produce 2% defective items. A random sample of 10 items is to be selected from the 1000 items produced in a particular production run. Let X be the count of the number of defective items found in the random sample. What can be said about the variable X?
A) We can use a Normal distribution with mean 20 and standard deviation 4.43 as an approximation for the distribution of X.
B) X is approximately Normal with µ = 10 and σ = 0.44.
C) X has an approximate binomial distribution with parameters 1000 and 0.01. D) X has an approximate binomial distribution with mean 0.2 and standard deviation
0.443.
E) Without additional information we are unable to determine if X is approximately Normally distributed or if it has a binomial distribution.
5. Let X be a random variable, which has a binomial distribution with mean µ = 8 and standard deviation σ = 2.19. The parameters n and p for this binomial distribution are respectively A) n = 16, p = 0.5. B) n = 13.3, p = 0.6. C) n = 10, p = 0.8. D) n = 20, p = 0.6. E) n = 20, p = 0.4.
6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8?
A) (0.6) (0.4)8 2
B) 10!(0.6) (0.4)8 2
8!
C) 45(0.6) (0.4)8 2
D) 45(0.6) (0.4)2 8
E) None of the above.
Use the following to answer questions 7-8:
A coin is about to be tossed multiple times. Assume the coin is fair, i.e., the probability of heads and the probability of tails are both 0.5.
7. If the coin is tossed six times, what is the probability that less than ⅓ of the tosses are heads? A) 0.0049 B) 0.094 C) 0.109 D) 0.344
8. If the coin is tossed 60 times, what is the probability that less than1
3of the tosses are
heads? A) 0.0049 B) 0.094 C) 0.109 D) 0.344
Use the following to answer questions 9-10:
It is claimed that 55% of marriages in the state of California end in divorce within the first 15 years. A large study was started 15 years ago and has been tracking hundreds of marriages in the state of California.
9. Suppose ten marriages are randomly selected. What is the probability that less than two of them ended in a divorce?
A) 0.0021 B) 0.0045 C) 0.0130 D) 0.0274
10. Suppose 100 marriages are randomly selected. What is the probability that less than 20 of them ended in a divorce?
A) Less than 0.0001 B) 0.0055
C) 0.0130 D) 0.0229
11. How many voters need to be sampled to guarantee that the standard deviation σ is no ˆp more than 0.025? A) 249 B) 250 C) 399 D) 400
12. Suppose that the polling organization takes a simple random sample of 500 voters. What is the probability that the sample proportion will be greater than 0.5, causing the polling organization to predict the result of the upcoming election incorrectly?
A) 0 B) 0.185 C) 0.212 D) 0.5
13. A college basketball player makes 5
6 of his free throws. Assume free throws are
independent. What is the probability that he makes exactly three of his next four free throws? A)
( ) ( )
1 3 5 1 6 6 4 B)( ) ( )
1 3 5 1 6 6 C)( ) ( )
1 1 5 3 6 6 4 D)( ) ( )
1 1 5 3 6 614. A fair die is rolled 12 times. Let X = the number of times an even number occurs on the 12 rolls. What is the appropriate distribution for the random variable X?
A) A binomial distribution with a mean of 2.
B) A binomial distribution with a standard deviation of 3. C) A binomial distribution with a mean of 0.5.
D) A binomial distribution with a mean of 6.
Use the following to answer questions 15-17:
15. At the end of a game, his team is losing by two points. He is fouled attempting a three-point shot and is awarded three free throws. Assuming each free throw is independent, what is the probability that he makes at least two of the free throws?
A) 0.384 B) 0.64 C) 0.80 D) 0.896
16. Over the course of the season, he will attempt 100 free throws. Assuming free-throw attempts are independent, what is the probability that the number of free throws he makes exceeds 80?
A) 0.2000 B) 0.2266 C) 0.5000 D) 0.7734
17. Over the course of the season, he will attempt 100 free throws. Assuming free-throw attempts are independent, what is the probability that he makes at least 90 attempts? A) 0.0062
B) 0.2643 C) 0.72 D) 0.90
18. Let X be a binomial random variable with p = 0.35. What size sample would be required to make the standard deviation of the proportion pˆ X
n = equal to 0.04? A) 6 B) 33 C) 12 D) 143
19. It was reported that 18% of the residents of hospital-based continuing-care facilities in the Province of Ontario in 2004-2005 were under the age of 65. A study involving selecting a random sample of 300 residents of such facilities is to be conducted. What is the probability that between 15 and 20 percent of the individuals in the sample will be less than 65 years of age?
A) .762 B) .732 C) .711 D) .819
E) Not within ±0.005 of any of the above.
Use the following to answer questions 20-22:
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.
20. 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
B) 1 2
C) 2
D) 2
21. The quality control manager plans to take a simple random sample of size n from the production line. How big should n be so that the sampling distribution of X has standard deviation 0.01 oz?
A) 10 B) 100 C) 1000
D) Cannot be determined unless we know the population follows a Normal distribution.
22. 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 ten sampled chocolate bars will be less than 8.0 oz?
A) 0 B) 0.00078 C) 0.0316
Answer Key
1. A) X is B(7, 0.5)
B) X is approximately N(0.5, 0.0645) C) X is approximately N(54, 4.98)3 D) X is approximately N(0.5, 0.0598) 2. A) No, B) No, C) Yes, D) No
3. D 4. D 5. E 6. C 7. C 8. A 9. B 10. A 11. D 12. B 13. C 14. D 15. D 16. C 17. A 18. D 19. B 20. B 21. B 22. B
