Exercise Questions. Q7. The random variable X is known to be uniformly distributed between 10 and

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1 Exercise Questions This exercise set only covers some topics discussed after the midterm. It does not mean that the problems in the final will be similar to these. Neither solutions nor answers will be posted. If you badly need the answer, visit me. Q1. Phone calls arrive at the rate of 48 per hour at the reservation desk for Rıhtım Restaurant. a) Compute the probability of receiving three calls in a five-minute interval of time. b) Compute the probability of receiving exactly 10 calls in 15 minutes. c) Suppose no calls are currently on hold. If the agent takes five minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the probability that none will be waiting? d) If no calls are currently being processed, what is the probability that the agent can take three minutes for personal time without being interrupted by a call? Q2. It is reported that air bag-related fatalities dropped to 18 in the year 2000 according to NSC. a) Compute the expected number of air bag-related fatalities per month. b) Compute the probability of no air bag-related fatalities in a month. c) Compute the probability of two or more air bag-related fatalities in a month. Q3. During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes. a) What is the expected number of calls in one hour? b) What is the probability of three calls in five minutes? c) What is the probability of no calls in a five-minute period? Q4. Cars arrive at a car wash randomly and independently; the probability of an arrival rate is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation? Q5. The number of typing errors made by a particular typist has a Poisson distribution with an average of four errors per page. If more than four errors show on a given page the typist must retype the whole page. What is the probability that a certain page does not have to be retyped? Q6. The random variable X is known to be uniformly distributed between 1.0 and 1.5. a) Compute P (X = 1.25). b) Compute P (1.0 X 1.25). c) Compute P (1.20 < X < 1.5). Q7. The random variable X is known to be uniformly distributed between 10 and 20. a) Compute P (X < 15). b) Compute P (12 X 18). c) Compute E(X). d) Compute Var(X). Q8. Izair quotes a flight time of 2 hours, 5 minutes for its flight from Ankara to Bakü. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes. a) What is the probability that the flight will be no more than 5 minutes late?

2 b) What is the probability that the flight will be more than 10 minutes late? c) What is the expected flight time? Q9. The label on a bottle of liquid detergent shows the contents to be 350 grams per bottle. The production operation fills the bottle uniformly according to the following probability density function. { 0.1 for 346 x 356 f(x) = 0 elsewhere a) What is the probability that a bottle will be filled between 350 and 354 grams? b) What is the probability that a bottle will be filled with 352 or more grams? Q10. Given that z is a standard normal random variable, compute the following a) P (z 1.0) b) P (z 1) c) P (z 1.5) d) P ( 2.5 z) e) P ( 3 < z 0) Q11. Given that z is a standard normal random variable, compute the following a) P (0 z.83) b) P ( 1.57 z 0) c) P (z >.44) d) P (z < 1.20) e) P (z.71) Q12. Given that z is a standard normal random variable, compute the following a) P ( 1.98 z.49) b) P (.52 z 1.22) c) P ( 1.75 z 1.04) Q13. Given that z is a standard normal random variable, find z for each situation. a) The area to the left of z is b) The area between 0 and z is c) The area to the left of z is d) The area to the right of z is e) The area to the left of z is f) The area to the right of z is Q14. Given that z is a standard normal random variable, find z for each situation. a) The area to the left of z is b) The area between z and z is c) The area between z and z is d) The area to the left of z is e) The area to the right of z is Q15. Given that z is a standard normal random variable, find z for each situation. a) The area to the right of z is.01. b) The area to the right of z is.025. c) The area to the right of z is.05. 2

3 d) The area to the right of z is.10. Q16. The average stock price for companies making up the S&P 500 is $30, and the standard deviation is $8.20. Assume that the stock prices are normally distributed. a) What is the probability a company will have a stock price of at least $40? b) What is the probability a company will have a stock price no higher than $20? c) How high does a stock price have to be put a company in the top 10%? Q17. The average amount of precipitation in Dallas during the month of April is 8.75 cm. Assume that a normal distribution applies and that the standard deviation is 2 cm. a) What percentage of the time does the amount of rainfall in April exceed 12.5 cm? b) What percentage of the time is the amount of rainfall in April less than 7.5 cm? c) A month is classified as extremely wet if the amount of rainfall is in the upper 10% for that month. How much precipitation must fall in April for it to be classified as extremely wet? Q18. A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-iq society. If IQ scores are normally distributed with a mean of 100 and standard deviation of 15, what score must a person have to qualify for Mensa? Q19. Consider the following exponential probability density function. f(x) = 1 8 e x/8 for x 0 a) Find P (X 6). b) Find P (X 4). c) Find P (X 6). d) Find P (4 X 6). Q20. Consider the following exponential probability density function. f(x) = 1 3 e x/3 for x 0 a) Find P (X 2). b) Find P (X 5). c) Find P (X 3). d) Find P (2 X 5). Q21. A magazine monitors ISPs and provides statistics on their performance. The average time to download a web page for free ISPs is approximately 20 seconds for European web pages. Assume the time to download a web page follows an exponential distribution. a) What is the probability it will take less than 10 seconds to download a web page? b) What is the probability it will take more than 30 seconds to download a web page? c) What is the probability it will take between 10 and 30 seconds to download a web page? 3

4 Q22. The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds. a) What is the probability that the arrival time between vehicles is 12 seconds or less? b) What is the probability that the arrival time between vehicles is 6 seconds or less? c) What is the probability of 30 or more seconds between vehicle arrivals? Q23. The lifetime (hours) of an electronic device is a random variable with the following density function. 4 f(x) = 1 50 e x/50 for x 0 a) What is the mean lifetime of the device? b) What is the probability that the device will fail in the first 25 hours of operation? c) What is the probability that the device will operate 100 or more hours before failure? Q24. Assume that the test scores from a college admissions test are normally distributed wth a mean of 450 and a standard deviation of 100. a) What percentage of the people taking the test score between 400 and 500? b) Suppose someone receives a score of 630. What percentage of the people taking the test score better? What percentage score worse? c) If a particular university will not admit anyone scoring below 480, what percentage of the people taking the test would be acceptable to the university? Q25. A machine fills containers with a particular product. The standard deviation of filling weights is known from past data to be 17 grams. If only 2% of the containers hold less than 500 grams, what is the mean filling weight for the machine? (Assume the filling weights have a normal distribution.) Q26. In a population with a mean of 200 and a standard deviation of 50, suppose a simple random sample of size 100 is selected and X is used to estimate µ. a) What is the probability that the sample mean will be within ±5 of the population mean? b) What is the probability that the sample mean will be within ±10 of the population mean? Q27. Last year a company began a program to compensate its employees for unused sick days, paying each employee a bonus of one-half the usual wage earned for each unused sick day. The question that naturally arises is Did this policy motivate employees to use fewer sick days? Before last year, the number of sick days used by employees had a distribution with a mean of 7 days and a standard deviation of 2 days. a) Assuming that these parameters did not change last year, find the approximate probability that the sample mean number of sick days used by 100 employees chosen at random was less than or equal to 6.4 last year. b) How would you interpret the result if the sample mean for the 100 employees was 6.4?

5 Q28. A random sample of n = 68 observations is selected from a population with µ = 19.6 and σ = 3.2. Approximate each of the following a) P ( X 19.6) b) P ( X 19) c) P ( X 20.1) d) P (19.2 X 20.6) Q29. It is known that the average salary of a travel management professional is $ Assume that the standard deviation of such salaries is $ Consider a random sample of 50 travel management professionals and let X represent the mean salary for the sample. a) What is µ X? b) What is σ X? c) Find the z-score for the value X = $ d) Find P ( X > ). Q30. A new method was used in Statistics course last year. A standardized test, administered at the end of the year, was used to measure the effectiveness of the new method. The distribution of past scores on the standardized test produced a mean of 75 and a standard deviation of 10. If the new method is no different from the old method, what is the approximate probability that the mean score X of a random sample of 36 students will be greater than 79? Q31. The amount of time a bank teller spends with each customer has a population mean 3.10 minutes and standard deviation 0.40 minutes. If you select a random sample of 32 customers, a) what is the probability that the mean time spent per customer is at least 3 minutes? b) there is an 85% chance that the sample mean is below how many minutes? c) If you select a random sample of 128 customers, there is an 85% chance that the sample mean is below how many minutes? Q32. For an audience of 600 people attending a concert, the average time on the journey to the concert was 32 minutes, and the standard deviation was 10 minutes. A random sample of 150 audience members was taken. a) What is the probability that the sample mean journey time was more than 31 minutes? b) What is the probability that the sample mean journey time was less than 33 minutes? c) What is the probability that the sample mean journey time was not between 31 and 33 minutes? Q33. Scores on an examination taken by a very large group of students are normally distributed with mean 700 and standard deviation 120. a) An AA is awarded for a score higher than 820. What proportion of all students obtain an AA? b) A BB is awarded for scores between between 730 and 770. An instructor has a section of 100 students who can be viewed as a random sample of all students in the large group. Find the expected number of students in this section who will obtain a BB. c) It is decided to give a failing grade to 5% of students with the lowest scores. What is the minimum score needed to avoid a failing grade? 5