Questions and Answers of Statistics Principles And Methods

Define an appropriate sample space for each of the experiments described below:a. At the close of business each day, the Acme Department Store's accountant counts the number of customer transactions
For each of the sample spaces you have defined above, indicate whether the sample space is finite, countably infinite, or uncountably infinite. Justify your answers.
The sales team of a large car dealership in Seattle consists of the following individuals:A customer visiting the dealership randomly chooses one of the salespersons to discuss the purchase of a new
Assign probabilities to the events a to $f$ in the preceding question, but include the condition "given that the individual chosen is $\geq 30$ years old."
The manager of the cost accounting department of a large computer manufacturing firm always tells three jokes during her monthly report to the board of directors in an attempt to inject a bit of
Schneider's Plumbing and Heating, located in Fargo, North Dakota, has 300 accounts receivable distributed as follows:An auditor is coming to inspect Schneider's financial records. Included in the
A computer manufacturing firm produces three product lines: (1) desktop computer systems, (2) notebook computers, and (3) subnotebook computers. The sales department has convened its monthly meeting
A large electronics firm is attempting to hire six new electrical engineers. It has been the firm's experience that 35 percent of the college graduates who are offered positions with the firm have
A computer manufacturing firm accepts a shipment of CPU chips from its suppliers only if an inspection of 5 percent of the chips, randomly chosen from the shipment, does not contain any defective
The probability that a stereo shop sells at least one amplifier on a given day is 75 ; the probability of selling at least one CD player is .6; and the probability of selling at least one amplifier
Prove that the set function defined by P(ANB) P(A|B)= = for P(B) 0 P(B) is a valid probability set function in the probability space {B,YB, P(B)), where YB is the event space for the sample space B.
A large midwestern bank has devised a math aptitude test that it claims provides valuable input into the hiring decision for bank tellers. The bank's research indicates that 60 percent of all tellers
A large-scale firm specializing in providing temporary secretarial services to corporate clients has completed a study of the main reason why secretaries become dissatisfied with their work
A clerk is maintaining three different files containing job applications submitted for three different positions currently open in the firm at which the clerk is employed. One file contains two
A company manages three different mutual funds. Let $A_{\mathrm{i}}$ be the event that the ith mutual fund increases in value on a given day. Probabilities of various events relating to the mutual
Answer the following questions regarding the validity of probability assignments. If you answer false, explain why the statement is false.a. If $P(A)=.2, P(B)=.3$, and $A \cap B=\emptyset$, then
The ZAP Electric Co. manufactures electric circuit breakers. The circuit breakers are produced on two different assembly lines in the company's Spokane plant. Assembly line I is highly automated and
The ACME Computer Co. operates three plants that manufacture notebook computers. The plants are located in Seattle, Singapore, and New York. The plants produce 20, 30, and 50 percent of the company's
The diagram below indicates how probabilities have been assigned to various subsets of the sample space $S$ :a. Are the three events $A, B$, and $C$ pairwise independent events?b. Are the three
A large sack contains 1,000 flower seeds consisting of 300 carnations and 700 impatiens. Of the 300 carnation seeds, 200 will produce red flowers and 100 will produce white flowers. Of the 700
For each case below, determine whether or not the real-valued set function $P(A)$ is in fact a probability set function:a. Sample space $S=\{1,2,3,4,5,6,7,8\}$, Event space \(\Upsilon=\{A: A
The Smith Floor Wax Company manufactures and sells industrial-strength floor wax in the wholesale market for home care products. The factory produces $10,000 \mathrm{gal}$ of floor wax daily and
A box contains four different computer disks, labeled 1, 2, 3, and 4. Two disks are selected at random from the box "with replacement," meaning that after the first selection is made, the selected
The AJAX Microchip Company produces memory chips for personal computers. The company's entire production is generated from two assembly lines, labeled I and II. Assembly line I uses more rapid
The management of the AJAX Microchip Company (mentioned in Problem 24) is interested in increasing quality control at the plant, and is considering the purchase of a testing device that can determine
Let $S=[0,5]$ be a sample space containing all possible values of the daily quantity demanded of electric power for a large midwestern city in the summer months. The units of measurement are
SUPERCOMP, a retail computer store, sells personal computers and printers. The number of computers and printers sold on any given day varies, with the probabilities of the various possible sales
Consider the experiment of tossing a fair coin (meaning heads and tails are equally likely on each toss) three times and observing the sequence of heads and tails that results. Let $H$ denote heads
BuyOnLine is a large internet-based online retailer that maintains four different teams of sales representatives. The ages of unpaid invoices from each of the four sales teams is summarized in the
The Port Authority of a large East Coast City is investigating the traffic flow in and out of a large train station in the middle of the city. There are six entry gates and six exit gates that
Let $A_{i}, i=1, \ldots, 4$, represent four events in a sample space, $S$. For each of the situations below, determine which assignment of probabilities are actually possible (i.e., do not
In each case below, determine whether the set function, $P$, is a probability set function.a. $P(A)=\frac{1}{91} \sum_{x \in A} x^{2}$ for $A \subset S, S=\{1,2,3,4,5,6\}$b. \(P(A)=\int_{x \in
Buy On Line is a large internet-based online retailer that maintains four different teams of sales representatives. The ages of unpaid invoices from each of the four sales teams is summarized in the
If $P(A)=.3, P(B)=.4, P(A \mid B)=.3$, what is the value ofa. $P(A \cap B)$b. $P(A \cup B)$c. $P(\bar{A} \mid B)$d. $P(A \mid \bar{B})$e. $P(\bar{A} \mid \bar{B})$f. $P(B \mid A)$g.
A regional airline implements a standard sales practice of "overbooking" their flights, whereby they sell more tickets for a flight then there are seats available for passengers. Their rationale for
An automobile manufacturer will accept a shipment of tires only if an inspection of 5 percent of the tires, randomly chosen from the shipment, does not contain any defective tires. The manufacturer
The table below indicates the probabilities of various outcomes with regard to the size of purchases and method of payment for customers that enter to a large New York electronics store:a. Is the
A new medical test has been developed by a major pharmaceutical manufacturer for detecting the incidence of a bacterial infection. Of the people who actually have the disease, the test will correctly
A large food processor operates three processing plants on the west coast. The plants, labeled 1,2, and 3, differ in size, and produce 20,35, and 45 percent of the food processor's total output of
This problem is the famous "Birthday Problem" in the statistics literature. The problem is the following: In a room of $n$ people, what is the probability that at least two people share the same
The Baseball World Series in the U.S. consists of seven games, and the first team to win four games is the winner of the series. Assume that the teams are evenly matched.a. What is the probability
The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first
Which of the following are valid PDFs? Justify your answer.a. $f(x)=(.2)^{x}(.6)^{1-x} I_{\{0,1\}}(x)$b. $f(x)=(.3)(.7)^{x} I_{\{0,1,2, \ldots\}}\{x\}$c. \(f(x)=.6 e^{-x / 4} I_{(0,
Graph each of the probability density functions in Problem 1.Problem 1.Which of the following are valid PDFs? Justify your answer.a. $f(x)=(.2)^{x}(.6)^{1-x} I_{\{0,1\}}(x)$b. \(f(x)=(.3)(.7)^{x}
Sparkle Cola, Inc., manufactures a cola drink. The cola is sold in 12 oz. bottles. The probability distribution associated with the random variable whose outcome represents the actual quantity of
A health maintenance organization (HMO) is currently treating 10 patients with a deadly bacterial infection. The best-known antibiotic treatment is being used in these cases, and this treatment is
Star Enterprises is a small firm that produces a product that is simple to manufacture, involving only one variable input. The relationship between input and output levels is given by $q=x^{.5}$,
The ACME Freight Co. has containerized a large quantity of 4-gigabyte memory chips that are to be shipped to a personal computer manufacturer in California. The shipment contains 1,000 boxes of
Intelligent Electronics, Inc., manufactures monochrome liquid crystal display (LCD) notebook computer screens. The number of hours an LCD screen functions until failure is represented by the outcome
People Power, Inc., is a firm that specializes in providing temporary help to various businesses. Job applicants are administered an aptitude test that evaluates mathematics, writing, and manual
The weekly average price (in dollars/foot) and total quantity sold (measured in thousands of feet) of copper wire manufactured by the Colton Cable Co. can be viewed as the outcome of the bivariate
A personal computer manufacturer produces both desktop computers and notebook computers. The monthly proportions of customer orders received for desktop and notebook computers that are shipped within
A small nursery has seven employees, three of whom are salespersons, and four of whom are gardeners who tend to the growing and caring of the nursery stock.With such a small staff, employee
The joint density of the bivariate random variable $(X, Y)$ is given by$f(x, y)=x y I_{[0,1]}(x) I_{[0,2]}(y)$.a. Find the joint cumulative distribution function of $(X, Y)$. Use it to find the
The joint cumulative distribution function for (X,Y)(X,Y) is given
The cumulative distribution of the random variable XX is given by F(x)=(1−px+1)I{0,1,2,…}(x)F(x)=(1−px+1)I{0,1,2,…}(x), for some choice of p∈(0,1)p∈(0,1). a. Find the density function
The federal mint uses a stamping machine to make coins. Each stamping produces 10 coins. The number of the stamping at which the machine breaks down and begins to produce defective coins can be
The daily quantity demanded of unleaded gasoline in a regional market can be represented as $Q=100-10 p+E$, where $p \in[0,8]$, and $E$ is a random variable having a probability density given
For each of the cumulative distribution functions listed below, find the associated PDFs. For each CDF, calculate $P(x \leq 6)$.a. $F(b)=\left(1-\mathrm{e}^{-b / 6}ight) I_{(0, \infty)}(b)$b.
An economics class has a total of 20 students with the following age distribution:Two students are to be selected randomly, without replacement, from the class to give a team report on the state of
Let $X$ be a random variable representing the minimum of the two numbers of dots that are facing up after a pair of fair dice is rolled. Define the appropriate probability density for $X$. What
A package of a half-dozen light bulbs contains two defective bulbs. Two bulbs are randomly selected from the package and are to be used in the same light fixture. Let the random variable $X$
A committee of three students will be randomly selected from a senior-level political science class to present an assessment of the impacts of an antitax initiative to some visiting state
The Imperial Electric Co. makes high-quality portable compact disc players for sale in international and domestic markets. The company operates two plants in the United States, where one plant is
ACE Rentals, a car-rental company, rents three types of cars: compacts, mid-size sedans, and large luxury cars. Let $\left(x_{1}, x_{2}, x_{3}ight)$ represent the number of compacts, mid-size
If (X1,X2)(X1,X2) and (X3,X4)(X3,X4) are independent bivariate random variables, are X2X2 and X3X3 independent random variables? Why or why not?
The joint density function of the discrete trivariate random variable $\left(X_{1}, X_{2}, X_{3}ight)$ is given by\[\begin{aligned}f\left(x_{1}, x_{2}, x_{3}ight)= & .20 I_{\{0,1\}}\left(x_{1}ight)
SUPERCOMP, a retail computer store, sells personal computers and printers. The number of computers and printers sold on any given day varies, with the probabilities of the various possible sales
Given the function definitions below, determine which can be used as PDFs (PDFs) and which cannot. Justify your answers.a. \(f(x)= \begin{cases}\left(\frac{1}{4}ight)^{x} \text { for } & x=0,1,2,
Given the function definitions below, determine which can be used as cumulative distribution functions (CDFs) and which cannot. Justify your answers. a. F(c) = b. F(c) = ec 1 + ec { 0 for c (-,00) -x
For those functions in (28) that are actually cumulative distribution functions (CDFs), use the duality principle to derive the PDFs (PDFs) that are associated with the CDFs.
The daily quantity demanded of milk in a regional market, measured in 1,000 's of gallons, can be represented during the summer months as the outcome of the following random variable:\(Q=200-50
A small locally-owned hardware store in a western college town accepts both cash and checks for purchasing merchandise from the store. From experience, the store accountant has determined that 2
Let an outcome of the random variable $T$ represent the time, in minutes, that elapses between when an order is placed at a ticket counter by a customer and when the ticket purchase is completed.
Outcomes of the random variable $Z$ represent the number of customers that are waiting in a queue to be serviced at Fast Lube, a quick stop automobile lubrication business, when the business opens
The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable $(P, Q)$ having the
The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first
The following function is proposed as a cumulative distribution function for the bivariate random variable $(X, Y)$ :\(F(x, y)=\left(1+e^{-(x / 10+y / 20)}-e^{-x / 10}-e^{-y / 20}ight) I_{(0,
For each of the joint PDFs listed below, determined which random variables are independent and which are not.a. $f(x, y)=e^{-(x+y)} I_{[0, \infty)}(x) I_{[0, \infty)}(y)$b. \(f(x,
The daily wholesale price and quantity sold of ethanol in a Midwestern regional market during the summer months is represented by the outcome of a bivariate random variable $(P, Q)$ having the
Let the random variable $X$ represent the product of the number of dots facing up on each die after a pair of fair dice is rolled. Let $Y$ represent the sum of the number of dots facing up on the
The production of a certain volatile commodity is the outcome of a stochastic production function given by $Y$ $=L^{5} K^{25} e^{V}$, where $V$ is a random variable having the cumulative
A small domestic manufacturer of television sets places a three-year warranty on its picture tubes. During the warranty period, the manufacturer will replace the television set with a new one if the
A small rural bank has two branches located in neighboring towns in eastern Washington. The numbers of certificates of deposit that are sold at the branch in Tekoa and the branch in Oakesdale in any
The weekly number of luxury and compact cars sold by "Honest" Abe Smith at the Auto Mart, a local car dealership, can be represented as the outcome of a bivariate random variable $(X, Y)$ with the
The yield, in bushels per acre, of a certain type of feed grain in the midwest can be represented as the outcome of the random variable $Y$ defined by $Y=3 x_{l}^{30} x_{k}^{45} e^{U}$where
The daily price/gallon and quantity sold (measured in millions of gallons) of a lubricant sold on the wholesale spot market of a major commodity exchange is the outcome of a bivariate random variable
The short-run production function for a particular agricultural crop is critically dependent on the level of rainfall during the growing season, the relationship being $Y=30+3 X-.075 X^{2}$, where
For each of the densities below, indicate whether the mean and variance of the associated random variable exist. In addition, find the median and mode, and indicate whether or not each density is
The daily price of a certain penny stock is a random variable with an expected value of $\$ 2$. Then the probability is $\leq .20$ that the stock price will be greater than or equation to \(\$
The miles per gallon attained by purchasers of a line of pickup trucks manufactured in Detroit are outcomes of a random variable with a mean of 17 miles per gallon and a standard deviation of .25
The daily quantity of water demanded by the population of a large northeastern city in the summer months is the outcome of a random variable, $X$, measured in millions of gallons and having a MGF
The annual return per dollar for two different investment instruments is the outcome of a bivariate random variable $\left(X_{1}, X_{2}ight)$ with joint moment-generating function
Stanley Statistics, an infamous statistician, wants you to enter a friendly wager with him. For $\$ 1,000$, he will let you play the following game. He will continue to toss a fair coin until the
The city of Megalopolis operates three sewage treatment plants in three different locations throughout the city. The daily proportion of operating capacity exhibited by the three plants can be
The average price and total quantity sold of an economy brand of ballpoint pen in a large western retail market during a given sales period is represented by the outcome of a bivariate random
A game of chance is considered to be "equitable" or "fair" if a player's expected payoff is equal to zero. Examine the following games:(a) The player rolls a pair of fair dice. Let $Z$ represent
The manager of a bakery is considering how many chocolate cakes to bake on any given day. The manager knows that the number of chocolate cakes that will be demanded by customers on any given day is a
The daily price and quantity sold of wheat in a Northwestern market during the first month of the marketing year is the outcome of a bivariate random variable $(P, Q)$ having the probability
In each case below, calculate the expected value of the random variable $Y$ :(a) $\mathrm{E}(Y \mid X)=2 x^{2}+3, f_{X}(x)=e^{-x} I_{(0, \infty)}(x)$.(b) \(\mathrm{E}(Y \mid \mathrm{X})=3 X_{1}

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