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# Probability Calculations in Statistics

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The time required to complete a certain construction project follows a normal distribution with a mean of 43 weeks and a standard deviation of 3.7 weeks. Use this information to answer questions 1 through 3.

1. What is the probability of the completing the project in no more than 48 weeks?

2. What is the probability of the completing the project in more than 37.9 weeks?

3. What is the probability of completing the project between 36 weeks and 49 weeks?

The students in a certain college course recently completed an exam. 14 students earned an A, 11 students earned a B, 10 students earned a C, 11 students earned a D and 11 students earned an F on the exam. The students were queried regarding the number of hours they had devoted to studying for the exam. 12 of the students who earned an A, 9 of the students who earned a B, 7 of the students who earned a C, 3 of the students who earned a D, and 1 of the students who earned an F reported that they had devoted more than 8 hours to studying for the exam. The remaining students reported that they had devoted no more than 8 hours to studying for the exam. Use this information to answer questions 4 through 6.

4. What is the probability of a randomly selected student having devoted no more than 8 hours to studying for the exam?

5. What is the probability of a randomly selected student having earned a C, D or F on the exam?

6. What is the probability of a randomly selected student having earned a C, D or F on the exam given they devoted more than 8 hours to studying for the exam?

Your professor tells you that if you earn a score of 90% or better on your midterm exam, then there is an 80% chance you will earn an A for the course. You estimate that you have only a 55% chance of earning a score of 90% or better on the midterm exam. Use this information to answer question 7.

7. What is the probability that your score on the midterm exam will be 90% or better and you will receive an A for the course?

A sample consisting of freshman college students was surveyed regarding the average number of hours of television they expected to watch daily during the current semester. The student responses are summarized in the table below. Assume that the sample was representative of all freshman students. Use this information to answer question 8.

Number of Hours. Number of Students
5 or more. 53
4 41
3. 37
2. 25
1. 10
0 1

8. What is the probability that a randomly selected freshman student will watch two or fewer hours of television daily during the current semester?

A student has an important exam coming up and is contemplating not studying for the exam in order to attend a concert with his friends. The student knows the exam will consist of twenty-five multiple choice questions with four possible answers for each question. The student also knows that he must earn a minimum score of 76% on the exam in order to successfully achieve his desired grade for the course. Use this information to answer questions 9 and 10 based upon modeling this situation as a binomial experiment.

9. What is the probability that the student will successfully earn exactly the required minimum score of 76% on the exam based solely upon randomly guessing the answer for each question?

10. What is the probability that the student will earn less than the required minimum score of 76% on the exam based solely upon randomly guessing the answer for each question?

A standard deck of playing cards consists of fifty-two cards. The cards in each deck consist of four suits, namely spades (?), clubs (?), diamonds (?) and hearts (?). Each suit consists of thirteen cards, namely ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2. Use this information to answer question 11.

11. What is the probability of selecting four cards of the same suit in four consecutive draws from a randomly shuffled deck of playing cards, assuming the specific cards that must be selected are not important, the order in which the cards must be selected is not important, and the suit is not important?

Customers arrive at a supermarket check-out counter following a Poisson distribution with an average arrival rate of 5 customers per hour. Use this information to answer questions 12 through 14.

12. What is the probability of less than 5 customers arriving at the supermarket check-out counter in a given one hour period?

13. What is the probability of exactly 5 customers arriving at the supermarket check-out counter in a given one hour period?

14. What is the probability of more than 5 customers arriving at the supermarket check-out counter in a given one hour period?

A local commuter bus service advertises that its buses run every 15 minutes along a certain route. Assume that the bus service follows an exponential distribution. Use this information to answer questions 15 through 17.

15. What is the probability of a bus picking up the passengers at a given bus stop in no more than 15 minutes?

16. What is the probability of a bus picking up the passengers at a given bus stop in more than 15 minutes?

17. What is the probability of a bus picking up the passengers at a given bus stop in between 15 and 25 minutes?
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