Questions and Answers of Applied Statistics and Probability

A Web site contains 100 interconnected pages. The random variable is the number of unique pages viewed by a visitor to the Web site.
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.
Suppose that a table of part counts is generalized as follows:where a, b, and k are positive integers. Let A denote the event that a part is from supplier 1, and let B denote the event that a part
Suppose that a lot of washers is large enough that it can be assumed that the sampling is done with replacement. Assume that 60% of the washers exceed the target thickness.(a) What is the minimum
Suppose documents in a lending organization are selected randomly (without replacement) for review. In a set of 50 documents, suppose that 2 actually contain errors.(a) What is the minimum sample
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any
Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows:Let A denote the event that a sample has low
An article in the British Medical Journal [“Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy” (1986, Vol. 82,
Heart failures are due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances (73%) or foreign objects (27%). Natural occurrences are caused
In the 2012 presidential election, exit polls from the critical state of Ohio provided the following results:What is the probability a randomly selected respondent voted for Obama? Romney 45% Obama
A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9).
The article [“Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp.
The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,” [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two
Consider the bar code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five
Consider the well failure data given below.(a) What is the probability of a failure given there are more than 1,000 wells in a geological formation?(b) What is the probability of a failure given
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Determine
The number of content changes to a Web site follows a Poisson distribution with a mean of 0.25 per day.(a) What is the probability of two or more changes in a day?(b) What is the probability of no
In 1898, L. J. Bortkiewicz published a book entitled 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
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
Suppose that lesions are present at 5 sites among 50 in a patient. A biopsy selects 8 sites randomly (without replacement).(a) What is the probability that lesions are present in at least one
Suppose that a healthcare provider selects 20 patients randomly (without replacement) from among 500 to evaluate adherence to a medication schedule. Suppose that 10% of the 500 patients fail to
Consider the semiconductor wafer data in Table 2-1. Suppose that 10 wafers are selected randomly (without replacement) for an electrical test. Determine the following:(a) Probability that exactly 4
Consider the nonfailed wells in Exercises 3-35. Assume that four wells are selected randomly (without replacement) for inspection.(a) What is the probability that exactly two are selected from the
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that four visits that result in LWBS are to be randomly selected (without
The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by the type of transformation completed:A naturalist randomly selects three leaves from this
Customers visit a Web 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
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following:(a) Probability that the first patient
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
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that people independently arrive for service at hospital l.(a) What is the
A Web site randomly selects among 10 products to discount each day. The color printer of interest to you is discounted today.(a) What is the expected number of days until this product is again
In the process of meiosis, a single parent diploid cell goes through eight different phases. However, only 60% of the processes pass the first six phases and only 40% pass all eight. Assume that the
A computer system uses passwords constructed from 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
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement from the total for hospital 4. What is the smallest sample size needed so that the probability
Consider the time to recharge the flash in cell-phone cameras as in Example 3-2. Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. What is the
Consider the circuit in Example 2-34. Assume that devices fail independently. What is the probability mass function of the number of device failures? Explain why a binomial distribution does not
The probability that a visitor to a Web site provides contact data for additional information is 0.01. Assume that 1000 visitors to the site behave independently. Determine the following
Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.(a) What is the probability that exactly 12 reactions result in a final temperature less
Assume that a Web site changes its content according to the distribution in Exercise 3-34. Assume that 10 changes are made independently.(a) What is the probability that the change is made in less
Consider the patient data in Example 2-8. Suppose that five patients are randomly selected with replacement from the total for hospital 4. Determine the following probabilities:(a) Exactly one is
Consider the lengths of stay at a hospital’s emergency department in Exercise 3-33. Assume that five persons independently arrive for service.(a) What is the probability that the length of stay of
An article in Information Security Technical Report [€œMalicious Software€”Past, Present and Future€ (2004, Vol. 9, pp. 6€“18)] provided the following data on
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
Consider the hospital data in Example 2-8. Suppose a patient is selected randomly from the collection in the table. Let X denote the hospital number of the selected patient (either 1, 2, 3, or 4).
Each multiple-choice question on an exam has 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
The probability that data are entered incorrectly into a field in a database is 0.005. A data entry form has 28 fields, and errors occur independently for each field. The random variable X is the
Suppose that 1000 seven-digit telephone numbers within your area code are dialed randomly. What is the probability that your number is called?
Suppose that nine-digit Social Security numbers are assigned at random. If you randomly select a number, what is the probability that it belongs to one of the 300 million people in the United States?
The number of pages in a PDF document you create has a discrete uniform distribution from five to nine pages (including the end points). What are the mean and standard deviation of the number of
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
Let the random variable X have a discrete uniform distribution on the integers 1 ≤ x ≤ 3. Determine the mean and variance of X.
Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99. Determine the mean and variance of X.
An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the Transaction
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:Nickel Charge
Consider the circuit in Example 2-32. Assume that devices fail independently. What is the probability mass function of the number of failed devices?
Consider the wafers with contamination particles in Example 2-17. Assume that wafers are independent with respect to contamination particles. Wafers are selected until one with five or more
The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp.
The distribution of the time until a Web site changes is important to Web crawlers that search engines use to maintain current information about Web sites. The distribution of the time until change
Actual lengths of stay at a hospital€™s emergency department in 2009 are shown in the following table (rounded to the nearest hour). Length of stay is the total of wait and service times.
The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:Final Temperature Conditions
The space shuttle flight control system called Primary Avionics Software Set (PASS) uses four independent computers working in parallel. At each critical step, the computers “vote” to determine
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)] cites a
Two suppliers each supplied 2000 parts that were evaluated for conformance to specifications. One part type was more complex than the other. The proportion of nonconforming parts of each type are
Natural red hair consists of two genes. People with red hair have two dominant genes, two regressive genes, or one dominant and one regressive gene. A group of 1000 people was categorized as
A computer system uses passwords that are exactly seven characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used.(a) How many passwords
A batch contains 36 bacteria cells. Assume that 12 of the cells are not capable of cellular replication. Of the cells, 6 are selected at random, without replacement, to be checked for replication.(a)
An article in Genome Research [€œAn Assessment of Gene Prediction Accuracy in Large DNA Sequences€ (2000, Vol. 10, pp. 1631€“1642)], considered the accuracy
Energy released from cells breaks the molecular bond and converts ATP (adenosine triphosphate) into ADP (adenosine diphosphate). Storage of ATP in muscle cells (even for an athlete) can sustain
A congested computer network has a 0.002 probability of losing a data packet, and packet losses are independent events. A lost packet must be resent.(a) What is the probability that an e-mail message
A researcher receives 100 containers of oxygen. Of those containers, 20 have oxygen that is not ionized, and the rest are ionized. Two samples are randomly selected, without replacement, from the
Shafts are classified in terms of the machine tool that was used for manufacturing the shaft and conformance to surface finish and roundness.(a) If a shaft is selected at random, what is the
The probabilities of poor print quality given no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0, 0.3, 0.4, and 0.6, respectively. The probabilities of no printer
An e-mail filter is planned to separate valid e-mails from spam. The word free occurs in 60% of the spam messages and only 4% of the valid messages. Also, 20% of the messages are spam.
The article [“Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp.
Two Web colors are used for a site advertisement. If a site visitor arrives from an affiliate, the probabilities of the blue or green colors being used in the advertisement are 0.8 and 0.2,
Consider the well failure data given below. Use Bayes€™ theorem to calculate the probability that a randomly selected well is in the gneiss group given that the well has failed. Wells
Consider the hospital emergency room data given below. Use Bayes€™ theorem to calculate the probability that a person visits hospital 4 given they are LWBS. Hospital Total 5640 Total 6991
Consider the endothermic reactions given below. Use Bayes€™ theorem to calculate the probability that a reaction's final temperature is 271 K or less given that the heat absorbed is above
The following table provides data on wafers categorized by location and contamination levels. More generally, let the number of wafers with low contamination from
The following table provides data on wafers categorized by location and contamination levels. Let A denote the event that contamination is low, and let B denote the event
An integrated circuit contains 10 million logic gates (each can be a logical AND or OR circuit). Assume the probability of a gate failure is p and that the failures are independent. The integrated
Consider the code 39 is a common bar code system that consists of narrow and wide bars(black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars
A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit
A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit
Consider the endothermic reactions given below. Let A denote the event that a reaction's final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Use
A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. Let A denote the event that a
Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from 100 strands are as follows:(a) If a strand is randomly selected, what is the probability that
In the article “ACL Reconstruction Using Bone-Patellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical Results” in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 248–255),
A computer system uses passwords that contain exactly eight characters, and each character is one of 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote
Suppose that a patient is selected randomly from the those described ,The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,” [Gastroenterology
A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose that an operating room needs to schedule three knee, four hip, and five shoulder surgeries.
Consider the bar code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars
Consider the well failure data is given below. Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed. Determine the
Consider the hospital emergency room data is given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital).Determine the
A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit
Consider the endothermic reaction’s table given below. Let A denote the event that a reaction's final temperature is 271 K or less. Let B denote the event that the heat absorbed is above
An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP
Magnesium alkyls are used as homogenous catalysts in the production of linear low-density polyethylene (LLDPE), which requires a finer magnesium powder to sustain a reaction. Redox reaction
In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states and that is usually found in the following states:Nickel Charge
In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt
A computer system uses passwords that contain exactly eight characters, and each character is 1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω

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