In a manufacturing process that laminates several ceramic layers, 1% of the assemblies are defective. Assume that the assemblies are independent. a. What is the mean number of assemblies that need to be checked to obtain five defective assemblies? b. What is the standard deviation of the number of assemblies that need to be checked to obtain five defective assemblies? c. Determine the minimum number of assemblies that need to be checked so that the probability that at least one defective assembly is obtained exceeds 0.95.
> This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 2% of the components are id
> Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently. a. What i
> Samples of 20 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more t
> In 1898, L. J. Bortkiewicz published a book titled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution wi
> A computer system uses passwords that are exactly six characters and each character is one of the 26 letters (a–z) or 10 integers (0–9). Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replacement) 100,000 p
> Heart failure is due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 20 patients
> Samples of rejuvenated mitochondria are mutated (defective) in 1% of cases. Suppose that 15 samples are studied and can be considered to be independent for mutation. Determine the following probabilities. a. No samples are mutated. b. At most one sample
> Amultiple-choice test contains 25 questions, each with four answers. Assume that a student just guesses on each question. a. What is the probability that the student answers more than 20 questions correctly? b. What is the probability that the student an
> An electronic product contains 40 integrated circuits. The probability that any integrated circuit is defective is 0.01, and the integrated circuits are independent. The product operates only if there are no defective integrated circuits. What is the pro
> Determine the cumulative distribution function of a binomial random variable with n = 3 and p = 1/4.
> The random variable X has a binomial distribution with n = 10 and p = 0.5. Sketch the probability mass function of X. a. What value of X is most likely? b. What value(s) of X is(are) least likely? c. Repeat the previous parts with p = 0.01.
> The random variable X has a binomial distribution with n = 10 and p = 0.01. Determine the following probabilities. a. P(X = 5) b. P(X ≤ 2) c. P(X ≥ 9) d. P(3 ≤ X < 5)
> Let X be a binomial random variable with p = 0.1 and n = 10. Calculate the following probabilities. a. P(X ≤ 2) b. P(X > 8) c. P(X = 4) d. P(5 ≤ X ≤ 7)
> The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. a. What is the probability that there are no cracks that require repair
> For each scenario (a)–(j), state whether or not the binomial distribution is a reasonable model for the random variable and why. State any assumptions you make. a. A production process produces thousands of temperature transducers. Let X denote the numbe
> Trees are subjected to different levels of carbon dioxide atmosphere with 6% of them in a minimal growth condition at 350 parts per million (ppm), 10% at 450 ppm (slow growth), 47% at 550 ppm (moderate growth), and 37% at 650 ppm (rapid growth). What are
> The range of the random variable X is [0, 1, 2, 3, x] where x is unknown. If each value is equally likely and the mean of X is 6, determine x.
> Each multiple-choice question on an examhas four choices. Suppose that there are 10 questions and the choice is selected randomly and independently for each question. Let X denote the number of questions answered correctly. Does X have a discrete uniform
> Suppose that 1000 seven-digit telephone numbers within your area code are dialed randomly. What is the probability that your number is called?
> Thickness measurements of a coating process are made to the nearest hundredth of amillimeter. The thickness measurements are uniformly distributed with values 0.15, 0.16, 0.17, 0.18, and 0.19. Determine the mean and variance of the coating thickness for
> Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine the mean, variance, and standard deviation of the random variable Y = 5X and compare to the corresponding results for X.
> Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99. Determine the mean and variance of X.
> Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 675 to 700 nm. a. What are the mean and variance of the wavelength distribution for this radiation? b. If t
> An assembly consists of threemechanical components. Suppose that the probabilities that the first, second, and third components meet specifications are 0.95, 0.98, and 0.99, respectively. Assume that the components are independent. Determine the probabil
> Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in theMilkyWay Galaxy in the vicinity of our solar system is one star per 16 cubic light-years. a. What is the probability of two or more stars in
> Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful with probabilities 0.3, 0.6, and 0.1, respectively. The yearly revenue associated with a very successful, moderatel
> In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the probability that a wafer passes the test is 0.8 and that wafers are independent. Determine the probability mass functi
> An article in Knee Surgery, Sports Traumatology, Arthroscopy [“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique” (2005, Vol. 13, pp. 273–279)] cited a success rate of more than 90% for meniscal tears with a rim width
> a. P(X ≥ 2) b. P(X c. P(X = 1.5) d. P(X 2.1)
> Consider the hospital patients in Example 2.6. Two patients are selected randomly, with replacement, from the total patients at Hospital 1. What is the probability mass function of the number of patients in the sample who are admitted?
> f (x) = 2x + 1 25 , x = 0, 1, 2, 3, 4 a. P(X = 4) b. P(X ≤ 1) c. P(2 ≤ X < 4) d. P(X > −10)
> f (x) = (8∕7)(1∕2)x , x = 1, 2, 3 a. P(X ≤ 1) b. P(X > 1) c. P(2 < X < 6) d. P(X ≤ 1 or X > 1)
> The sample space of a random experiment is {a, b, c, d, e, f }, and each outcome is equally likely. A random variable is defined as follows: Determine the probability mass function of X. Use the probability mass function to determine the following probab
> In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states. Assume the following proportions of the states: a. Determine the cumulative distribution function of the nickel charge. b.
> Determine the mean and variance of the random variable in Exercise 3.1.13.
> Data from www.centralhudsonlabs.com determined the mean number of insect fragments in 225-gram chocolate bars was 14.4, but three brands had insect contamination more than twice the average. See the U.S. Food and Drug Administration–Center for Food Safet
> Determine the mean and variance of the random variable in Exercise 3.1.12.
> Determine the mean and variance of the random variable in Exercise 3.1.11.
> Determine the mean and variance of the random variable in Exercise 3.1.10.
> It is suspected that some of the totes containing chemicals purchased from a supplier exceed the moisture content target. Assume that the totes are independent with respect to moisture content. Determine the proportion of totes from the supplier that mus
> Saguaro cacti are large cacti indigenous to the southwesternUnited States and Mexico.Assume that the number of saguaro cacti in a region follows a Poisson distribution with a mean of 280 per square kilometer. Determine the following: a. Mean number of ca
> An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose that orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,
> From 500 customers, a major appliance manufacturer randomly selects a sample without replacement. The company estimates that 25% of the customers will reply to the survey. If this estimate is correct, what is the probability mass function of the number o
> Each main bearing cap in an engine contains 4 bolts. The bolts are selected at random without replacement froma parts bin that contains 30 bolts from one supplier and 70 bolts from another. a. What is the probability that a main bearing cap contains all
> Assume that the number of errors along a magnetic recording surface is a Poisson random variable with amean of one error every 105 bits. A sector of data consists of 4096 eight-bit bytes. a. What is the probability of more than one error in a sector? b.
> The random variable X has the following probability distribution: Determine the following: a. P(X ≤ 3) b. P(X > 2.5) c. P(2.7 d. E(X) e. V(X)
> Suppose that the number of customers who enter a store in an hour is a Poisson random variable, and suppose that P(X = 0) = 0.05. Determine the mean and variance of X.
> Determine the probability mass function for the random variable with the following cumulative distribution function:
> A manufacturer of a consumer electronics product expects 2% of units to fail during the warranty period. A sample of 500 independent units is tracked for warranty performance. a. What is the probability that none fails during thewarranty period? b. What
> Patient response to a generic drug to control pain is scored on a 5-point scale where a 5 indicates complete relief. Historically, the distribution of scores is Two patients, assumed to be independent, are each scored. a. What is the probability mass fun
> The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1. Suppose that 20 independent individuals suffering from this illness are treated with a drug and 4 recover in a one-week time period. If the dr
> The number of errors in a textbook follows a Poisson distribution with a mean of 0.01 error per page. What is the probability that there are three or fewer errors in 100 pages?
> The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent. a. If you call 10 times, what is the probability that exactly nine of your calls are answered within 30 seconds? b. If y
> The number of messages that arrive at aWeb site is a Poisson random variable with a mean of five messages per hour. a. What is the probability that five messages are received in 1.0 hour? b. What is the probability that 10 messages are received in 1.5 ho
> Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of six cars per minute. To set the timing of a traffic signal, the following probabilities are used.
> A shipment of chemicals arrives in 15 totes. Three of the totes are selected at random without replacement for an inspection of purity. If two of the totes do not conform to purity requirements, what is the probability that at least one of the nonconform
> The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. Determine the following probabilities: a. exactly 5 calls in one hour b. 3 or fewer calls
> An electronic scale in an automated filling operation stops themanufacturing line after three underweight packages are detected. Suppose that the probability of an underweight package is 0.001 and each fill is independent. a. What is the mean number of f
> The probability that an eagle kills a rabbit in a day of hunting is 10%. Assume that results are independent for each day. a. What is the distribution of the number of days until a successful hunt? b. What is the probability that the first successful hu
> A particularly long traffic light on your morning commute is green on 20% of the mornings. Assume that each morning represents an independent trial. a. What is the probability that the first morning that the light is green is the fourth morning? b. What
> A congested computer network has a 1%chance of losing a data packet that must be resent, and packet losses are independent events. An e-mail message requires 100 packets. a. What is the distribution of the number of packets in an e-mail message that must
> An automated egg carton loader has a 1% probability of cracking an egg, and a customer will complain if more than one egg per dozen is cracked. Assume that each egg load is an independent event. a. What is the distribution of cracked eggs per dozen? Incl
> Let X denote the number of bits received in error in a digital communication channel, and assume that X is a binomial random variable with p = 0.001. If 1000 bits are transmitted, determine the following: a. P(X = 1) b. P(X ≥ 1) c. P(X ≤ 2) d. mean and
> Let the random variable X be equally likely to assume any of the values 1/8, 1/4, or 3/8. Determine the mean and variance of X.
> Consider the time to recharge the flash in cell-phone cameras as in Example 3.1. Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. What is the smallest sample size needed so that the probability of at
> The article “An Association Between Fine Particles and Asthma Emergency Department Visits for Children in Seattle” [Environmental Health Perspectives June 1999, Vol. 107(6)] used Poisson models for the number of asthma emergency department (ED) visits pe
> The probability that a visitor to aWeb site provides contact data for additional information is 0.01. Assume that 1000 visitors to the site behave independently. Determine the following probabilities: a. No visitor provides contact data. b. Exactly 10 vi
> An array of 30 LED bulbs is used in an automotive light. The probability that a bulb is defective is 0.001 and defective bulbs occur independently. Determine the following: a. Probability that an automotive light has two or more defective bulbs. b. Expec
> Customers visit aWeb site, and the probability of an order if a customer views five or fewer pages is 0.01. However, if a customer views more than five pages, the probability of an order is 0.1. The probability a customer views five or more pages is 0.25
> Consider the time to recharge the flash in cell-phone cameras as in Example 3.1. Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. Determine the following: a. Probability that the second failure occur
> If the range of X is the set {0, 1, 2, 3, 4} and P(X = x) = 0.2, determine the mean and variance of the random variable.
> a. P(X ≤ 50) b. P(X ≤ 40) c. P(40 ≤ X ≤ 60) d. P(X e. P(0 ≤ X f. P(−10
> a. P(X ≤ 3) b. P(X ≤ 2) c. P(1 ≤ X ≤ 2) d. P(X > 2)
> Determine the cumulative distribution function for the random variable in Exercise 3.1.16.
> Determine the cumulative distribution function for the random variable in Exercise 3.1.13.
> Determine the cumulative distribution function for the random variable in Exercise 3.1.12; also determine the following probabilities: a. P(X < 1.5) b. P(X ≤ 3) c. P(X > 2) d. P(1 < X ≤ 2)
> Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour. Suppose that you exit the workplace at 6:00 P.M. Determine the following: a. Probability that you wait more than 10 minutes for a cab. b. Probability that you wait
> Determine the cumulative distribution function for the random variable in Exercise 3.1.11.
> Determine the cumulative distribution function of the random variable in Exercise 3.1.10.
> Determine the cumulative distribution function for the random variable in Exercise 3.1.9.
> A healthcare provider schedules 30 minutes for each patient’s visit, but some visits require extra time. The random variable is the number of patients treated in an eight-hour day.
> The number of mutations in a nucleotide sequence of length 40,000 in a DNA strand after exposure to radiation is measured. Each nucleotide may be mutated.
> A group of 10,000 people are tested for a gene called Ifi202 that has been found to increase the risk for lupus. The random variable is the number of people who carry the gene.
> The random variable is the number of surface flaws in a large coil of galvanized steel.
> The random variable is the number of computer clock cycles required to complete a selected arithmetic calculation.
> A batch of 500 machined parts contains 10 that do not conform to customer requirements. Parts are selected successively, without replacement, until a nonconforming part is obtained. The random variable is the number of parts selected.
> A batch of 500 machined parts contains 10 that do not conform to customer requirements. The random variable is the number of parts in a sample of five parts that do not conform to customer requirements.
> Inclusions are defects in poured metal caused by contaminants. The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Determine the following: a. Probability of at least one inclusion in a cu
> Suppose that X has a Poisson distribution. Determine the following probabilities when the mean of X is 4 and repeat for a mean of 0.4: a. P(X = 0) b. P(X ≤ 2) c. P(X = 4) d. P(X = 8)
> What is the difference between a confidence interval estimate of the mean response, μY|X = Xi, and a prediction interval of YX = Xi?
> When and how do you use the Durbin-Watson statistic?
> How do you evaluate the assumptions of regression analysis?
> What are the assumptions of regression analysis?
> Why should you always carry out a residual analysis as part of a regression model?
> Use the following information from a multiple regression analysis: n = 20 b1 = 4 b2 = 3 Sb1 = 1.2 Sb2 = 0.8 a. Which variable has the largest slope, in units of a t statistic? b. Construct a 95% confidence interval estimate of the population slope, b1.
> When is the explained variation (i.e., regression sum of squares) equal to 0?
> When is the unexplained variation (i.e., error sum of squares) equal to 0?
> What is the interpretation of the coefficient of determination?
> What is the interpretation of the Y intercept and the slope in the simple linear regression equation?
> Assume that you are working with the results from Problems 11.15 and 11.16. a. What is the value of the FSTAT test statistic for the interaction effect? b. What is the value of the FSTAT test statistic for the factor A effect? c. What is the value of the
