Sample Problems for the Final Exam 1. Hydraulic landing assemblies coming from an aircraft rework facility are each inspected for defects. Historical records indicate that 8% have defects in shafts only, 6% have defects in bushings only and 2% have defects in both shafts and bushings. a) Draw a Venn Diagram displaying this information b) One of the hydraulic assemblies is selected randomly. What is the probability that the assembly has i) defect in bushing defect ii) a defect in bushing or shaft iii) a defect neither in a bushing nor a shaft iv) a defect in bushing or shaft, but not both. 2. A business off ice orders paper supplies from one of three vendors: 1, 2 or 3. Orders are to be placed on two successive days, one order per day. Thus (2, 3) might denote vendor 2 gets the order on the first day and vendor 3 gets the order on the second day. a) Find the sample space (i.e. list all possibilities) in this experiment of ordering paper supplies on two successive days. Use set notation. b) Assume the vendors are selected at random each day. Let A denote the event that the same vendor gets both orders, and B denote the event that vendor 2 gets at least one order. i) List the elements of A Use set notation ii) List the elements of B Use set notation iii) Find the following probabilities: c c P (A) P (B) P( A B) P( A B )
3. Suppose that a grocery store rejects a sack of apples from a local farm if at least 3 bad apples are found in the sack. In general 10% of the apples produced at this farm are bad. A store manager randomly selects a sack of apples for inspection. a) What is the probability that the 10 th apple inspected will be the first bad apple. b) What is the probability that the 10 th apple selected will be the second bad apple. c) If in a sack, there are 10 apples, what is the probability that this sack of 10 apples will be accepted by the store? 4. Let the random variable X be the number of repair calls that an appliance repair shop may receive during an hour. The distribution of X is given below. Value of X 0 1 2 3 4 Probability 0.2 0.3 0.12 0.18 0.2 a) Write the cumulative distribution function (cdf) of X. b. Sketch the graph of the cumulative distribution function for X. c) What is the expected number of repair calls during an hour? 5. Let Y be the number of flaws on the surface of a randomly selected boiler of a certain type. Suppose X has a Poisson distribution with parameter µ = 5. Find, a) P(X = 6) b) P(5 X 7). 6. A tennis coach has a basket containing 25 balls; 15 of these are Penn balls and the other ten are Wilsons. Twelve balls are selected for a match. a) What probability model is used to solve this problem? b) What is the probability that exactly eight of the balls selected are Penn balls? 7. The shopping times of 64 randomly selected customers at a local super market were recorded. The average and variance of the 64 shopping times were 33 minutes and 256 respectively. a) Estimate the true average shopping time per customer with a confidence interval using a 92% confidence level. b) What is the margin of error?
c) What sample size is required to reduce the margin of error (from b) by a quarter? Assume all other values are the same. 8. Suppose we draw a random sample of size n = 50 from bank accounts in a large city. We are interested in the average amount of saving per 50 accounts. The distribution of individual saving is very skewed to the right. Suppose we know the population average saving is µ = $3000 and σ = $2000. a) Describe the distribution of sample means, where each observation is the average of 50 accounts drawn from this population. You need to give information on the shape, center and spread of the distribution of the means. b) Based on what theorem can you answer a)? 9. A random sample of size 25 is chosen from a normal population with known mean, µ =8, and standard deviation σ = 4. a) Determine the probability of having sample mean between 7 and 9. b) Determine the probability of having sample mean greater than 7. c) What is the 75 th percentile of the sample mean? 10. A nationally distributed college newspaper conducts a survey among students nationwide every year. This year, responses from a simple random sample of 204 college students to the question About how many CDs do you own? resulted in a sample mean x = 72.8. Based on data from previous years, the editors of the newspaper will assume that = 7.2. A 95% confidence interval for the mean number of CDs owned by all college students is found to be (71.8, 73.8). Answer each of the following questions with YES, NO, or CAN T TELL. A) Does the sample mean lie in the 95% confidence interval? B) Does the population mean lie in the 95% confidence interval? C) If we were to use a 92% confidence level, would the confidence interval from the same data produce an interval wider than the 95% confidence interval? D) With a smaller sample size, all other things being the same, would the 95% confidence interval be wider? 11. New laptops manufactured at a computer manufacturing plant may experience one of three defects, type I, type II or type III. Let A be the event that a randomly selected laptop experiences defect type I. Let B be the event that a randomly selected laptop experiences defect type II, and C be the event that a randomly selected laptop experiences defect type III. The following probabilities are given:
P, P ( A) 0.12, P ( B) 0. 07, P ( C) 0. 05 ( A B) 0. 13 P ( A C) 0.14, P ( B C) 0. 10 P ( A B C) 0. 01 a) What is the probability that a randomly selected laptop does not have a type I defect? b) What is the probability that a randomly selected laptop has a type I and a type II, but not a type III defect? c) What is the probability that a randomly selected laptop has at most two of these defects? 12. Let A and B be two events with ( A B) 7 / 8 P ( A' ) 5 / 8. a) Find P (B) P( A B' b) Find ) P, ( A B) 1 / 4 P and 13. Three machines A, B, C produce respectively 40%, 25% and 35% of the total number of items in a factory. The percentage of defective products from each machine is 2%, 3% and 4% respectively. a) If a product is selected at random, find the probability that the product is defective. b) If a defective product is selected at random, what s the probability that it was produced by machine B. 14. Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 20 failed keyboards, 8 of which have electrical defects and 12 of which have mechanical defects. a) How many ways are there to select 5 keyboards for thorough inspection (order is not important). b) In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect? c) If a random sample of 5 keyboards is selected, what is the probability that at least 4 of these will have a mechanical failure? 15. Let A and B be two independent events with ( A ) 0. 5 following probabilities. P b) P( A' B' ) c) P( A B' ) a) ( A B) P, P ( B ) 0. 2. Find the
16..In order to verify the accuracy of their financial accounts, companies use auditors on a regular basis to verify accounting entries. The company s employees make erroneous entries 5% of the time. Suppose a company auditor randomly checks three accounts. Let X be the number of errors detected by the auditor. a) Find the probability mass function (pmf) of X. You can assume independence here. b) Find the probability that the auditor detects more than one error. 17. The telephone lines serving an airline reservation office are all busy about 60% of the time. a) If you are calling this office, what is the probability that you will complete your call in i) the first try ii) the second try iii) the third try b) If you and a friend must complete calls to this office, what is the probability that a total of four tries will be necessary assuming that your friend will complete the call before you. 18. A particular sale involves four items randomly selected from a large lot that is known to contain 10% defectives. Let Y denote the number of defectives among the four sold. The purchaser of the item will return the defectives for repair and the repair cost is given by C Find the expected repair cost. 3Y 2 Y 2. 19. Suppose that a radio contains six transistors, two of which are defectives. Three transistors are selected at random, removed from the radio, and inspected. Let Y be the number of defective transistors observed. Find the probability mass function of Y. Y P(Y) f ( x) k x( x 1 if 0 x 2. 19. Suppose X has the density function ) a) Find the value of k 1 3 b) Find P X 2 2 c) Find the cumulative distribution function of X d) Find E(X), the expected value of X. 20. Let X ~ N(8,2), i.e. X has a normal density with mean 8 and standard deviation 2. a) Find P 7 X 9.5 b) Find P X 9.56 c) Find P X 6.8
d) Find the 95 th percentile of X e) What two values of X capture the middle 80% of the data under this normal distribution? 21. One hour carbon monoxide concentration in air samples from a large city have an approximately exponential distribution with mean 3.6 parts per million. a) Find the probability that the carbon monoxide concentration exceeds 9 parts per million during a randomly selected 1-hour period b) Find the probability that the concentration is between 4 and 12 parts per million. 22. Let X and Y be discrete random variables, each taking values 0, 1 or 2. Their joint mass distribution table is given below. X Y 0 1 2 0 1/9 2/9 1/9 1 2/9 2/9 0 2 1/9 0 0 a) Find E (X ) b) Find E (Y ) c) Find E( X Y) d) Find E( X Y ) 23. Let X and Y be continuous random variables with joint density function f ( x, y) k if 0 x 2, 0 y 1, 2y x. a) Find k. b) Compute P( X Y) c) Find the marginal density function f X (x) d) Find the marginal density function f Y (x) e) Find the expected value of E( X Y) 24. A sprinkler system is being installed in a newly renovated building on campus. The average activation time is supposed to be at most 20 seconds. A series of 12 fire alarm/sprinkler system tests result in an average activation time of 21.5 seconds. Do these data indicate that the design specifications have not been met? Assume that activation times for this system are normally
distributed with = 3 seconds. a) What is the parameter for this problem? b) Formulate the appropriate null and alternative hypotheses c) Calculate an appropriate test statistic d) What is the P-value for this test statistic? e) What do you conclude at 5% significance level? f) If the true average activation time of the sprinkler system is, in fact, equal to 20 seconds, was your conclusion from e) an error? If so what type? If not, why? 25. A highway safety researcher is studying the average maximum distance at which drivers are able to read a highway traffic sign. The designer of the sign claims it will be over 450 feet. Sixteen randomly selected drivers are asked at what distance they can still read the sign. The average maximum distance (in feet) of these 16 drivers is 498 feet with a standard deviation of 95 feet. The highway safety researcher wishes to use these data to test if the mean maximum distance at which drivers are able to read the sign is greater than 450 feet at a significance level of = 0.05. a) What is the parameter for this problem? b) Formulate the appropriate null and alternative hypotheses c) Calculate an appropriate test statistic d) The P-value for this test statistic is between 0.03 and 0.04. Based on this, what do you conclude at 5% significance level? 26. The heights of young American women, in inches, are normally distributed with mean and standard deviation = 2.4. a) A simple random sample of four young American women is selected and their heights measured. The four heights, in inches, are 63 69 62 66 Based on these data, what is a 99% confidence interval for? b) The margin of error for a 99% confidence interval is desired to be ±1 inch. What should be the sample size?