Questions and Answers of Statistics Principles And Methods

The total daily dollar sales in the ACME supermarket is represented by the outcome of the random variable $S$ having a mean of 20, where $s$ is measured in thousands of dollars.(a) The store
The first three moments about the origin for the random variable $Y$ are given as follows: $\mu_{1}^{\prime}=.5, \mu_{2}^{\prime}=.5, \mu_{3}^{\prime}$ $=.75$.(a) Define the first three moments
The random variable $Y$ has the $\operatorname{PDF} f(y)=y^{-2} I_{[1, \infty)}(y)$.(a) Find the mean of $Y$.(b) Can you find the first 100 moments about the origin (i.e., \(\mu_{1}^{\prime},
The moment-generating function of the random variable $Y$ is given by $M_{Y}(t)=(1-.25 t)^{-3}$ for \(t
A gas station sells regular and premium fuel. The two storage tanks holding the two types of gasoline are refilled every week. The proportions of the available supplies of regular and premium
Scott Willard, a famous weatherman on national TV, states that the temperature on a typical late fall day in the upper midwest, measured in terms of both the Celsius and Fahrenheit scales, can be
A fruit processing firm is introducing a new fruit drink, "Peach Passion," into the domestic market. The firm faces uncertain output prices in the initial marketing period and intends to make a
A Seattle newspaper intends to administer two different surveys relating to two different anti-tax initiatives on the ballot in November. The proportion of surveys mailed that will actually be
An automobile dealership sells two types of four-door sedans, the "Land Yacht" and the "Mini-Rover." The number of Land Yachts and Mini-Rovers sold on any given day varies, with the probabilities of
The season average price per pound, $p$, and total season quantity sold, $q$, of sweet cherries in a regional market can be represented as the outcome of a bivariate random variable $(P, Q)$
The yield/acre of wheat on a given parcel of land can be represented as the outcome of a random variable $Y$ defined by $Y=10 x^{1 / 3} e^{\varepsilon}$ for $x \in[8,100]$, where$Y=$ wheat
Let $X$ have the moment generating function $M_{X}(t)$. Show that(a) $M_{(X+a)}(t)=e^{a t} M_{X}(t)$.(b) $M_{b X}(t)=M_{X}(b t)$.(c) $M_{(X+a) / b}(t)=e^{(a /b) t} M_{X}(t / b)$.
The AJAX Disk Co. manufactures compact disks (CDs) for the music industry. As part of its quality-control program, the diameter of each disk is measured using an electronic measuring device. Letting
An investor wishes to invest $\$ 1,000$ and is examining two investment prospects. The net dollar return per dollar invested in the two projects can be represented as the outcome of a bivariate
The length of time in minutes for an individual to be served at a local restaurant is the outcome of a random variable, $T$, having a mean of 6 and a variance of 1.5 . How probable is the event
(a) Find the moment-generating function of a random variable $X$ having the density function$f(x)=\frac{1}{8}\left(\begin{array}{c}3 \\ x\end{array}ight) I_{(0,1,2,3)}(x)$.(Hint: Use of the
The Rockbed Insurance Company sells 1-year term life insurance policies for $\$ 10,000$ of annual coverage, where a 1 -year premium is charged to put the policy in force, and then if the insured
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
Define the mean, median, mode, and .10 and .90 quantiles of the random variable $X$ defined in the probability models $\{R(X), f(x)\}$ below: (a) f(x) = x + 4 50 (b) f(x) = .5e */1(R(x)} (x),
The regression curve of daily quantity demanded of tablet computers in a Midwestern market, measured in thousands of units and expressed as a function of price, all else held equal, is given
The yield per acre of a certain dwarf watermelon is highly dependent on the amount of rainfall that occurs during the growing season. Following standard cultivation practices, the relationship
For each probability density function below, determine the mean and variance, if they exist, and define the median and mode. (a) f(x) = [(1+x)](-00,00)(x) (b) f(x) = 4x [0,1](x) (c) f(x) = .3(.7)
The daily dollar sales of a large retail "Big Box" store, measured in 1,000 dollar units, is a random variable, $D$, that has an expectation of 20.(a) Provide an upper bound to the probability that
Given the following three moments about the origin, derive the first three moments about the mean, and determine whether the random variable has a probability density function that is skewed, if you
The bivariate random variable (P,Q) represents the weekly price, in dollars, and the quantity, in number of kegs, of an India Pale Ale beer, HopMeister, sold in the Pacific Northwest market. The
Derive the moment generating function of each of the random variables below, and use it to define the mean and variance of the random variable.(a) $f(x)=.2 e^{-.2 x} I_{(0, \infty)}(x)$(b) \(f(x)=2
A small manufacturing firm produces and sells a product in a market where output prices are uncertain. The owner of the firm wishes to make a short run production decision that will maximize her
The weekly number of MACs and PCs sold by a salesperson at the local computer store can be represented as the outcome of a bivariate random variable $(X, Y)$ with the nonzero values of its joint
An investor has $\$ 10,000$ to invest between two investment projects. The rate of return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable
The mean vector and covariance matrix of the trivariate random variable $\mathbf{X}$ is given by$\mathrm{E}(\mathbf{X})=\left[\begin{array}{c}-2 \\ 4 \\ 2\end{array}ight]$ and
The bivariate random variable (Y,X)(Y,X) has the following mean vector and covariance matrix: E[XY]=[105]E[XY]=[105] and Cov(X,Y)=[5222]Cov⁡(X,Y)=[5222] (a) Derive the values of aa and bb in
A shipment of 100 DVDs contains k defective disks. You randomly sample 20 DVDs, without replacement, from the shipment of 100 DVDs. Letting p = k/100, the probability that you will obtain less than
The daily price, $p$, and quantity, demanded, $q$, of gasoline on a European Wholesale spot market can be viewed (approximately) as the outcome of a bivariate normal random variable, where the
Show that the following probability density functions are members of the exponential class of densities:(a) Binomial family, for a fixed value of $n$.(b) Poisson family.(c) Negative binomial family
For each PDF family below, show whether or not the family belongs to the exponential class of densities.(a) $f(x ; \beta)=\beta x^{-(\beta+1)} I_{(1, \infty)}(x), \beta \in \Omega=(0, \infty)$.
Prove that if $X$ has the geometric density, then the "memoryless property" $P(x>s+t \quad \mid x>s)=P(x>t)$ holds for every choice of positive integers $s$ and $t$.
Prove that the CDFCDF of the geometric family of densities can be defined by F(b)=[1−(1−p)trunc(b)]F(b)=[1−(1−p)trunc⁡(b)] I[1,∞)(b)I[1,∞)(b).
The quantity of wheat demanded, per day, in a midwestern market during a certain marketing period is represented by$Q=100,000-12,500 P+V$ for $\mathrm{p} \in[2,6]$, where$Q$ is quantity
An investor has $\$ 10,000$ which she intends to invest in a portfolio of three stocks that she feels are good investment prospects. During the investor's planning horizon, the weekly closing
Let $Y$ have a chi-square distribution with 15 degrees of freedom, let $X$ have a chi-square distribution with 5 degrees of freedom, and let $Y=X+Z$, where $X$ and $Z$ are independent
Let $Y$ have the density $N(5,36), X$ have the density $N(4,25)$, let $Y$ and $X$ be independent random variables, and define $W=X-Y$.(a) Calculate $P(y>10)$.(b) Calculate
Let $\boldsymbol{X}$ be a bivariate random variable having the probability density $N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, with\(\boldsymbol{\mu}=\left[\begin{array}{l}5 \\
WAYSAFE, a large retail supermarket, has a standard inspection policy that determines whether a shipment of produce will be accepted or rejected. Specifically, they examine 5 percent of the objects
The FLAMES-ARE-US Co. manufactures butane cigarette lighters. Your top-of-the-line lighter, which has the brand name "SURE-FLAME," costs $\$ 29.95$. As a promotional strategy, the SURE-FLAME
An instructor in an introductory economics class has constructed a multiple-choice test for the mid-term examination. The test consists of 20 questions worth 5 points each. For each question, the
The liquid crystal display in the new Extime brand of digital watches is such that the probability it continues to function for at least $x$ hours before failure is constant (for any given choice
The Department of Transportation in a foreign county establishes gas-mileage standards that automobiles sold in must meet or else a "gas guzzler" tax is imposed on the sale of the offending types of
An appliance manufacturer is conducting a survey of consumer satisfaction with appliance purchases. All customers that have purchased one of the company's appliances within the last year will be
Customers arrive at the rate of four per minute at a large bank branch in downtown Seattle. In its advertising, the bank stresses that customers will receive service promptly with little or no
The accounts of the Excelsior company are being audited by an independent accounting firm. The company has 200 active accounts, of which 140 are current accounts, 45 are past due 60 or more days, and
The Stonebridge Tire Co. manufactures passenger car tires. The manufacturing process results in tires that are either first-quality tires, blemished tires, or defective tires. The proportions of the
KoShop, a large retail department store, has a standard inspection policy that determines whether a shipment of products will be accepted or rejected. Specifically, they examine a randomly chosen
Customers arrive, on average, at the rate of two per minute at a bank in downtown Portland.(a) Define an appropriate probability model for the experiment of observing the number of customers entering
The accounts of Pullman Plumbers, Inc. are being audited by an independent accounting firm. The company currently has 100 active accounts, of which 50 are current accounts, 35 are past due 60 or more
An instructor in an introductory economics class has a true-false section on the mid-term examination that consists of 10 questions worth 3 points each. Jeff Nostudy, a student in the class, has not
In the board game of "Aggravation", the game begins with each player having their four game pieces in a "base" holding area and a player is unable to place one of her game pieces on the board for
The Central Processing Unit (CPU) in a laptop computer that your company manufactures is known to have the following service life characteristics:$P(x>s+t \mid x>s)=P(x>t) \forall s$ and
The LCD screen on a popular smartphone has the following operating life characteristic where outcomes, x, of the random variable X measure the operating life of the screen, measured in 100,000
Let (X1; . . . ;X20) be 20 iid N(2; 4) random variables. Also, define the random variables Zi Xi –2/2 for i 1; . . . ; 20. Answer the following questions relating to these
The grades assigned to students taking a midterm in a large principles of economics class is assumed to be normally distributed with a mean of 75 and a standard deviation of 7. The Professor teaching
The production function for the number of toy robots that your company manufactures in a week, expressed in 100 's of robots, is represented as follows:\(Q=\mathbf{a}^{\prime}
The weekly average price, in dollars per gallon, and quantity sold of organic milk, measured in thousands of gallons, in a large west coast market in the fall is represented by the outcomes of a
Given current technology, the production of active matrix color screens for notebook computers is a difficult process that results in a significant proportion of defective screens being produced. At
Central limit theorems have important applications in the area of quality control. One such application concerns so-called control charts, and in particular, $\bar{X}$ charts, which are used to
The lifetime of a certain computer chip that your company manufactures is characterized by the population distribution f(z;θ)=1θe−z/θI(0,∞)(z)f(z;θ)=1θe−z/θI(0,∞)(z), where zz is
In each case below, the outcome of some function, T(X(n))T(X(n)), of nn iid random variables X(n)=(X1,…,Xn)X(n)=(X1,…,Xn) is being considered for providing an estimate of some function of
The daily number of customers entering a large grocery store who purchase one or more dairy products is given by the outcome of a binomial random variable Xt with parameters p and nt for day t. The
Let $\left(X_{1}, \ldots, X_{n}ight)$ be iid random variables with \(\sigma^{2}
Let the random variables in the sequence $\left\{X_{n}ight\}$ be iid with a gamma density having parameters $\alpha=2$ and $\beta=3$.(a) What is the probability density for $\bar{X}_{n}$ ?(b)
A pharmaceutical company claims that it has a drug that is 75 percent effective in generating hair growth on the scalps of balding men. In order to generate evidence regarding the claim, a consumer
A political candidate has hired a polling firm to assess her chances of winning a senatorial election in a large eastern state. She wants an estimate of the proportion of registered voters that would
Let observations on the quantity supplied of a certain commodity be generated by $Y_{i}=x_{i} \beta+V_{i}$, where $\left|x_{i}ight| \in[a$, b] $\forall i$ are scalar observations on fixed
Let X1,…,XnX1,…,Xn be iid random variables having continuous uniform distributions of the form f(z)=I(0,1)(z)f(z)=I(0,1)(z). (a) Define an asymptotic distribution for
A company produces a popular beverage product that is distributed nationwide. The aggregate demand for the product during a given time period can be represented by n n Q = Q = ( i B p + V) i=1 i=1
The daily tonnage of garbage handled by the EnviroSafe Landfill Co. is represented as the outcome of a random variable having some triangular distribution, asThis distribution is represented
A statistician wants to use iid outcomes from some exponential distribution f(x;θ)=(1/θ)e−x/θI(0,∞)(x)f(x;θ)=(1/θ)e−x/θI(0,∞)(x) to generate an estimate of the variance of the
We have shown that if {Yn} is a sequence of w2 random variables, where Yn ~, then (Yn - n)/2nN(0, 1). Since Y we know that P(y2534.3816) Ply 50 63.1671) = P(y100118.498) = .90. Assign (approx-
Let $\left\{X_{n}ight\}$ be a sequence of random variables having binomial densities, where $X_{n}$ has a binomial density with parameters $n$ and $p$, i.e.,\(X_{n}
The Nevada Gaming Commission has been directed to check the fairness of a roulette wheel used by the WINBIG Casino. In particular, a complaint was lodged stating that a "red" slot occurs more
The Elephant Memory Chip Co. (EMC for short) instructs its resident statistician to investigate the operating-life characteristics of their new 4 gigabyte memory chip in order to provide product
Let (X1,…,Xn)(X1,…,Xn) be a random sample from a Poisson population distribution. Derive the limiting distribution of T=(¯Xn−μ)(S2n/n)1/2T=(X¯n−μ)(Sn2/n)1/2 where
Liquid crystal displays (LCDs) that your wholesaling company is marketing for a large Japanese electronics firm are known to have a distribution of lifetimes of the following Gamma-distribution
In each case below, determine whether the random variable sequence $\left\{Y_{n}ight\}$ converges in probability and/or in mean square, and if so, define what is being converged to.(a) \(\quad
In each case below, derive the probability limit and an asymptotic distribution for $n^{-1} \sum_{i=1}^{n} X_{i}$ and a limiting distribution for the random variable $Y_{n}$, if they can be
Let $X$ be a random sample of size $n$ from a $N\left(\mu, \sigma^{2}ight)$ population distribution representing the weights, in ounces, of cereal placed in cereal boxes for a certain brand and
Let $X$ and $Y$ be two independent random samples of sizes $n_{X}$ and $n_{y}$, respectively, from two normal population distributions that do not necessarily have the same means or
Let $X$ be a random sample of size $\mathrm{n}$ from a $N\left(\mu, \sigma^{2}ight)$ population distribution representing the yield per acre, in pounds, of a new strain of hops used in the
The shipping and receiving department of a large toy manufacturer is contemplating two strategies for sampling incoming parts deliveries and estimating the proportion, $p$, of defective parts in a
GenAG, Inc., a genetics engineering laboratory specializing in the production of better seed varieties for commercial agriculture, is analyzing the yield response to fertilizer application for a new
The Always Ready Battery Co. has developed a new "Failsafe" battery which incorporates a small secondary battery that becomes immediately functional upon failure of the main battery. The operating
The seasonal catch of a commercial fishing vessel in a certain fishery in the southern hemisphere can be represented by $Q=c(\mathbf{z}) V$, where $\mathbf{z}$ is a vector of characteristics of
A company markets its line of products directly to consumers through telephone solicitation. Salespersons are given a base pay that depends on the number of documented phone calls made plus incentive
The daily quantity of a commodity that can be produced using a certain type of production technology is given by the outcome of the random variable $Q$, defined as $Q=10 x_{1}^{35} x_{2}^{5} V$,
The daily price, $p$, and quantity sold, $q$, of ground beef produced by the Red Meat Co. can be represented by outcomes of the bivariate random variable $(P, Q)$ having bivariate density
Let $X=\left(X_{1}, \ldots, X_{26}ight)$ and $Y=\left(Y_{1}, \ldots, Y_{31}ight)$ represent two independent random samples from two normal population distributions. Let $S_{X}^{2}$ and
The daily price, $p$, and daily quantity sold, $q$, of pink salmon produced by the AJAX Fish Packing Co. can be represented by outcomes of the bivariate random variable $(P, Q)$ with density
The probability that a customer entering an electronics store will make a purchase is equal to $p=.15$, and customers' decisions whether to purchase electronics equipment are jointly independent
The number of times that a copy machine malfunctions in a day is the outcome of a Poisson process with $\lambda=.1$.(a) Simulate the operating behavior of the copy machine regarding the daily
The length of time that a 4 gigabyte $\mathrm{PC}$ computer memory module operates until failure is the outcome of an exponential random variable with mean $\mathrm{E} X=3.25$, where $x$ is
The monthly proportion of purchases paid by check to a large grocery store that are returned because of insufficient funds can be viewed as the outcome of a Beta $(1,20)$ distribution.(a) Simulate

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