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Statistics: Various Probability Questions

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Chapter 5

8. A sample of 2,000 licensed drivers revealed the following number of speeding violations.
Number of Violations Number of Drivers
0 1910
1 46
2 18
3 12
4 9
5 or more 5
Total 2000
a. What is the experiment?

b. List one possible event.

c. What is the probability that a particular driver had exactly two speeding violations?

d. What concept of probability does this illustrate?

66. A survey of undergraduate students in the School of Business at Northern University revealed
the following regarding the gender and majors of the students:

a. What is the probability of selecting a female student?

b. What is the probability of selecting a finance or accounting major?

c. What is the probability of selecting a female or an accounting major? Which rule of addition
did you apply?

d. Are gender and major independent? Why?

e. What is the probability of selecting an accounting major, given that the person selected
is a male?

f. Suppose two students are selected randomly to attend a lunch with the president of the
university. What is the probability that both of those selected are accounting majors?

Chapter 7

38. The accounting department at Weston Materials, Inc., a national manufacturer of unattached
garages, reports that it takes two construction workers a mean of 32 hours and a
standard deviation of 2 hours to erect the Red Barn model. Assume the assembly times follow
the normal distribution.

a. Determine the z values for 29 and 34 hours. What percent of the garages take between
32 hours and 34 hours to erect?

b. What percent of the garages take between 29 hours and 34 hours to erect?
c. What percent of the garages take 28.7 hours or less to erect?

d. Of the garages, 5 percent take how many hours or more to erect?

44. The number of passengers on the Carnival Sensation during one-week cruises in the
Caribbean follows the normal distribution. The mean number of passengers per cruise is
1,820 and the standard deviation is 120.

a. What percent of the cruises will have between 1,820 and 1,970 passengers?
b. What percent of the cruises will have 1,970 passengers or more

c. What percent of the cruises will have 1,600 or fewer passengers?
d. How many passengers are on the cruises with the fewest 25 percent of passengers?
60. In establishing warranties on HDTV sets, the manufacturer wants to set the limits so that
few will need repair at manufacturer expense. On the other hand, the warranty period must
be long enough to make the purchase attractive to the buyer. For a new HDTV the mean
number of months until repairs are needed is 36.84 with a standard deviation of 3.34
months. Where should the warranty limits be set so that only 10 percent of the HDTVs need
repairs at the manufacturer's expense?

The manufacturer of salad dressings uses machines to dispense the dressing into bottles that move along an assembly line. The machine is working well when 8 ounces is dispensed. The standard deviation of the process is 0.15 ounce. A sample of 50 bottles is selected periodically and the assembly line is stopped when there is evidence that the average amount dispensed is less than 8 ounces. Suppose that a sample of 50 bottles reveals an average of 7.983 ounces.

1. State the null and alternative hypotheses.
2. At the .05 level of significance, is there evidence that the average amount dispensed is less than 8 ounces?

A machine being used for packaging raisins has been set so that, on average, 15 ounces of raisins will be packaged per box. The operations manager wishes to test the machine setting and selects a sample of 30 consecutive raisin packages filled during the production process. The sample mean is 15.18 and the sample standard deviation is 0.4909. Use the t-distribution. Is there evidence that the mean weight per box is different from 15 ounces? (use level of significance equal to .05)

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Solving various questions on fundemantals of statistics

12. Which of the following numbers could be the probability of an event?

1.5 , 1/2 , 3/4 , 2/3 , 0 , -1/4

34. For some diseases, such as sickle-cell anemia, an individual will get the disease only if he or she receives both recessive alleles. This is not always the case. For example, huntington's disease only requires one dominant gene for an individual to contract the disease. Suppose that a husband and wife, who both have a dominant Huntington's disease allele (S) and a normal recessive allele (s), decide to have a child.

a) List the possible genotypes of their offspring.
b) What is the probability that the offspring will not have Huntington's disease? In other words, what is the probability that the offspring will have genotype ss? interpret this probability.
c) What is the probability that the offspring will have Huntington's disease?

40. Which of the assignments of probabilities should be used if the coin is known to be fair?

Sample Spaces
Assignments HH HT TH TT
A 1/4 1/4 1/4 1/4
B 0 0 0 1
C 3/16 5/16 5/16 3/16
D 1/2 1/2 -1/2 1/2
E 1/4 1/4 1/4 1/8
F 1/9 2/9 2/9 4/9

48. Determine whether the probabilities on the following page are computed using classical methods, empirical methods, or subjective methods.

a) The probability of having eight girls in an eight-child family is 0.390625%
b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is 0.54%
c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about 30%
d) On the basis of clinical trials, the probability of efficacy of a new drug is 75%

26. The following probability model shows the distribution of doctoral degrees from U.S. universities in 2009 by area of study.

Area of Study Probability
Engineering 0.154
Physical Sciences 0.087
Life Sciences 0.203
Mathematics 0.031
Computer sciences 0.033
Social sciences 0.168
Humanities 0.094
Education 0.132
Professional and other fields 0.056
Health 0.042
Source: US National Science Foundation

a) Verify that this is a probability model.
b) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science or life science? Interpret this probability.
c) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science, life science, mathematics, or computer science? Interpret this probability.
d) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 did not study mathematics? Interpret this probability.
e) Are doctoral degrees in mathematics unusual? Does this result surprise you?

32. A standard deck of cards contains 52 cards. One card is randomly selected from the deck.
a) Compute the probability of randomly selecting a two or three from a deck of cards.
b) Compute the probability of randomly selecting a two or three or four from a deck of cards.
c) Compute the probability of randomly selecting a two or club from a deck of cards.

8. Determine whether the events E and F are independent or dependent. Justify your answer.
a) E: The battery in your cell phone is dead.
F: The batteries in your calculator are dead.
b) E: Your favorite color is blue.
F: Your friend's favorite hobby is fishing.
c) E: You are late for school.
F: Your car runs out of gas.

18. The probability that a randomly selected 40-year-old female will live to be 41 years old is 0.99855 according to the National Vital Statistics Report, Vol. 56, No. 9.
a) What is the probability that two randomly selected 40-year-old females will live to be 41 years old?
b) What is the probability that five randomly selected 40-year old females will live to be 41 years old?
c) What is the probability that at least one of five randomly selected 40-year-old females will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old females did not live to be 41 years old?

34. In how many ways can 15 students be lined up?

46. In how many ways can the top 2 horses finish in a 10-horse race?

52. How many different random samples of size 7 can be obtained from a population whose size is 100?

56. How many distinguishable DNA sequences can be formed using one A, four Cs, three Gs, and four Ts?

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