Questions and Answers of Statistics Principles And Methods

The daily closing price for a certain stock issue on the NYSE can be represented as the outcome of \(Y_{t}=Y_{t-1}+V_{t}\), where \(y_{t}\) is the value of the stock price on day \(t\), and \(V_{t}
A random sample of the gas mileage achieved by 20 domestic compact automobiles resulted in the following outcome:(25.52,24.90,22.24,22.36,26.62,23.46,25.46,24.98,25.82,
The time between work-related injuries at the Imperial Tool and Die Co. during a given span of time resulted in the following 20 observations, where time was measured in weeks:\((9.68,6.97,7.08,
A realtor randomly samples homeowners who have purchased homes in the last 2 years and records their income, \(y\), and home purchase price, \(p\) (the population is large enough that one can
The proportion of the work force of a large Detroit manufacturing firm that takes at least 1 day's sick leave in a given work week is assumed to be the outcome of a random variable whose PDF is
A large aircraft manufacturer produces a passenger jet having a navigation component consisting of three sequentially functioning redundant navigation systems that will allow the jet to be properly
A news agency wants to poll the population of registered voters in the United States (over 200,000,000) to find out how many would vote for the Republican candidate, Henry Washington, if the election
Your company manufactures LCD screens that are used in the production of a popular smartphone sold by a major wireless cell phone service provider. The engineering department suggests that the
The following outcomes were from a random sample of size 25 from the joint distribution of \((Y, X)\), where \(Y\) denotes yield, in bushels per acre, of a new variety of overlineley and \(X\)
The following 20 iid random variable outcomes arose from some unknown cumulative distribution \(F(t)\) :(a) Define the EDF for the random variable.(b) Define the range of the random variable together
The number of customers entering the lobby of a bank in any 5 minute interval during the lunch hour from noon until 1 p.m. follows a Poisson process, Poisson(10).(a) Using the uniform random outcomes
The operating time until failure, in 100,000 hour units, of a hard disk that your company manufactures follows an exponential distribution, Exponential(2.5).(a) Simulate time-until-failure outcomes
A ground-fault-protection circuit breaker is designed to "trip" and interrupt power whenever a ground fault is detected in a power line to which it attached. After it trips, it designed to be able to
The probability density function relating to the number of seconds that individuals remain on a website until they leave is given by the Pareto distribution,\(f(x)=\left(\begin{array}{c}\frac{\alpha
The monthly production of beer in a large Midwestern brewery can be viewed as the outcome of the following production process:\(\mathrm{Q}=100 \mathrm{I}^{25} \mathrm{k}^{.5} e^{V}\) and \(V \sim
The monthly wholesale price and quantity sold of fresh squeezed orange juice during the winter months by a small Florida growers' cooperative is represented by the following bivariate probability
The operating life of a certain type of math coprocessor installed in a personal computer can be represented as the outcome of a random variable having an exponential density function, as\(Z \sim f(z
Use the Lehman-Scheffé minimal sufficiency theorem, or some other argument, to find a set of minimal sufficient statistics for each case below.a. You are random sampling from a log-normal population
Identify which of the minimal sufficient statistics in (2) are complete sufficient statistics.
The operating life of a small electric motor manufactured by the AJAX Electric Co. can be represented as a random variable having a probability density given as\(Z \sim f(z ; \Theta)=\frac{1}{6
The number of customers that enter the corner grocery store during the noon hour has a Poisson distribution, i.e.,\(f(z ; \lambda)=\frac{e^{-\lambda} \lambda^{z}}{z !} I_{\{0,1,2,3,
Polly Pollster wants to estimate the proportion of voters in Washington State that are in favor of an antitax initiative. She will be using a random sample (with replacement) of 1,000 voters, and she
Two economics professors are arguing about the appropriate estimator to use in estimating the mean of the population distribution of incoming freshmen's I.Q.'s. The estimators will be based on a
The diameters of blank compact disks manufactured by the Dandy Disk Co. can be represented as outcomes of a random variable\(Z \sim f(z ; \Theta)=\frac{1}{\Theta} I_{(4,4+\Theta)}(z)\), for some
Your company sells trigger mechanisms for air bags that are used in many modern domestic and foreign-built passenger cars. The reliability of such trigger mechanisms is obviously critical in the
Suppose random sampling is from an exponential population distribution representing the waiting time between customer arrivals at a bank, so that the statistical model is given by\(f(x ;
One hundred one-acre test plots are being used to assess the yield potential of a new variety of wheat genetically engineered by Washington State University. The actual yield-per-acre observations on
In each case below, determine whether the estimator under consideration is unbiased, asymptotically unbiased, and/or consistent.a. The random sample \(\left(X_{1}, \ldots X_{n}ight)\) is generated
In Problem 12(b), above, consider the alternative estimator \(t^{\star}(\mathbf{X})=S^{2}\) for estimating \(\theta^{2}\). In a mean square error sense, which estimator would you prefer for
An incoming shipment of 1,000 toys from a toy manufacturer is received by a large department store for a pre-Christmas sale. The store randomly samples 50 toys from the shipment, without replacement,
The rates of return per dollar invested in two common stocks over a given investment period can be viewed as the outcome of a bivariate normal distribution \(N(\boldsymbol{\mu},
For estimating the population mean, \(\mu\), based on a size \(n\) iid random sample, \(\left(X_{1}, \ldots, X_{n}ight)\), from some population distribution, the following three estimators are being
Consider two estimators for the probability, \(p\), that a tossed coin will land on "heads", based on a iid random sample of size \(n\) from whatever Bernoulli distribution governs the probability of
In each case below, determine whether the estimator under consideration is unbiased, asymptotically unbiased, and/or consistent.a. The iid random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is
If you wanted to choose the estimator with the smaller expected squared distance from the value of \(\lambda\), which of the estimators in 18.c and 18.d above would you choose?
Let the size \(n\) random sample \(\mathbf{Y}=\left[Y_{1}, Y_{2}, \ldots Y_{n}ight]^{\prime}\) be such that \(\mathbf{Y}=\mathbf{x} \beta+\boldsymbol{\varepsilon}\), where \(\mathbf{x}\) is a \(n
An estimate is needed of the expected number of resets it takes to get a certain brand of 15amp15amp ground fault protecting circuit breaker to fail to reset. A random sample of nn circuit breakers
The daily production of electricity generated by a coal-fired power plant operating in the midwest can be represented as\(Y_{i}=\beta_{1} l_{i}^{\beta_{2}} m_{i}^{\beta_{3}}
Smith's Dairy is contemplating the profitability of utilizing a new bovine growth hormone for increasing the milk production of its cows. Smith's randomly selectscows and administers given dosages of
Suppose \(\mathbf{Y}_{(n \times 1)}=\mathbf{x}_{(n \times k)} \boldsymbol{\beta}_{(k \times 1)}+\boldsymbol{\varepsilon}_{(n \times 1)}\), where \(\boldsymbol{\varepsilon} \sim\) \(N\left(\mathbf{0},
An economist is analyzing the relationship between disposable income and food expenditure for the citizens of a developing country. A survey of the citizens has been taken, and data have been
The research department of the Personal Computer Monthly magazine is analyzing the operating life of computer chips that are produced by a major manufacturer located in the Silicon Valley. The
A large commercial bank intends to analyze the accuracy with which their bank tellers process cash transactions. In particular, it desires an estimate of the expected proportion of daily cash
The Personnel Department of the ACME Textile Co. administers an aptitude test to all prospective assembly line employees. The average number of garments per hour that an employee can produce is
A business consultant to the ACME Textile Co. suggests that the estimator\(\hat{\beta}^{*}=\left(\mathbf{x}^{\prime} \mathbf{x}+kight)^{-1} \mathbf{x}^{\prime} \mathbf{Y}\)might be useful to consider
Henri Theil, a famous economist/econometrician, analyzed the demand for textiles in the Netherlands during the period 1923-1939, using a general linear model framework. In particular, he specified
In each case below, determine whether, and under what assumptions, the stated relationship between \(Y\) and \(x\) can be represented in general linear model form such that the least squares
The following statistical model is postulated for representing the relationship between real aggregate disposable income and real aggregate expenditure on nondurable goods:\(Y_{i}=\exp
In each case, indicate whether the statement regarding the relationship \(\mathbf{Y}=\mathbf{x} \boldsymbol{\beta}+\boldsymbol{\varepsilon}\) is true or false, and justify your answer.(a) Let the
Your company markets a disposable butane lighter called "surelight." In your product advertising, you use the slogan "lights on the first try-everytime!" As a quality check, you intend to examine a
A regional telephone company is analyzing the number of telephone calls that are connected to wrong numbers at its telephone exchange. It collects the number of wrong telephone connections on each of
The number of minutes past the scheduled departure time that jets with no mechanical problems leave the terminal in an overcrowded airport in the northeast are iid outcomes from a uniform population
Define MLEs for the following problems: (a) Estimating the p′ispi′s based on a random sample from a multinomial population distribution. (b) Estimating the unknown parameters in the mean vector
Define MOM estimators in each of the following cases. Are estimators consistent? Are they asymptotically normal? (a) Estimating \(\mu\) and \(\sigma^{2}\) based on a random sample from a normal
(Generalized Least Squares Estimator) Consider the linear model \(\mathbf{Y}=\mathbf{x} \boldsymbol{\beta}+\boldsymbol{\varepsilon}\) in which all of the classical assumptions apply, except that
(Linear and Nonlinear Models) Determine which of the following models can be transformed into linear models and which cannot, and identify the transformation in the cases where it is possible.(a)
(Nonlinear Least Squares Estimator) Consider the nonlinear regression model\(\mathbf{Y}=\mathbf{g}(\mathbf{x}, \boldsymbol{\beta})+\boldsymbol{\varepsilon}=\beta_{1} \mathbf{1}_{n}+\beta_{2}
A shipment of 20 projection screen television sets is at the receiving dock of a large department store. The department store has a policy of not accepting shipments that contain more than 10 percent
A pharmaceutical company is analyzing the effectiveness of a new drug that it claims can stimulate hair growth in balding men. For the purposes of an advertising campaign, the marketing department
In each case below, identify whether the null and the alternative hypotheses are simple or composite hypotheses.(a) You are random sampling from a gamma population distribution and you are testing
A large metropolitan branch of a savings and loan is examining staffing issues and wants to test the hypothesis that the expected number of customers requiring the services of bank personnel during
The annual proportion of new restaurants that survive in business for at least 1 year in a U.S. city with population \(\geq 500,000\) people is assumed to be the outcome of some Beta population
A complaint has been lodged against a major domestic manufacturer of potato chips stating that their \(16 \mathrm{oz}\) bags of chips are being underfilled. The manufacturer claims that their filling
In a random sample of size 10 from a Bernoulli population distribution, how many (nonrandomized) critical regions can you define that have size \(\leq .10\) and that are also unbiased for testing the
Randomized Test It was demonstrated in Example 9.10 that the choices of size for most powerful tests of the hypothesis \(H_{0}: p=.2\) versus \(H_{a}: p=.8\) was quite limited. Suppose that a .05
The number of work-related injuries per week that occur at the manufacturing plant of the Excelsior Corporation is a Poisson-distributed random variable with mean \(\lambda \geq 3\), according to
Being both quality and cost conscious, a major foreign manufacturer of compact disk players is contemplating their warranty policy. The standard warranty for compact disk players sold by competing
Your company supplies an electronic component that is critical to the navigational systems of large jet aircraft. The operating life of the component has an exponential distribution with some mean
The Pareto distribution\(f(\mathbf{x} ; \theta, c)=c^{\theta} \theta x^{-(1+\theta)} I_{(c, \infty)}(x)\)for \(\theta>1\) and \(c>0\) has been used to model the distribution of incomes in a given
Suppose that a random sample of size \(n\) is drawn from a normal population distribution for which \(\sigma^{2}\) is assumed to be known and equal to the given value \(\sigma_{*}^{2}\). Define UMPU
Suppose that a random sample of size \(n\) is drawn from a normal population distribution for which \(\mu\) is assumed to be known and equal to the given value \(\mu_{*}\). Define UMPU level
Control Charting: A large mail-order house has initiated a quality control program. They randomly sample 100 of each day's orders and monitor whether or not the mail order-taking process is "under
The Gibralter Insurance Co. is reevaluating the premiums it charges on car insurance and is analyzing classifications of cars into high risk, average risk, and low risk on the basis of frequency of
A certain business uses national telephone solicitation to sell its product. Its sales staff have individual weekly sales quotas of 10 sales that they must meet or else their job performance is
A manufacturer of breakfast cereals has been accused of systematically underfilling their cereal packages. The manufacturer claims that the complaint must stem from settling of the cereal that occurs
Testing for Differences in Two Populations: Variances. After the firm in question (1) was inspected by the Dept. of Weights and Measures, a new random sample of forty observations on fill weights was
Testing for Differences in Two Populations: Means} (Equal Variances).Referring to the data obtained in both preceding questions, consider testing whether the two populations means are the same.
Testing for Differences in Two Populations: Means (Unequal Variances) (The Behrens-Fisher Problem).Consider the hypotheses in Problem (3) in the case where it is not assumed that the variances of the
Regarding the asymptotically valid Wald and \(Z\)-tests for the value of \(\sigma^{2}\) in the case of nonnormally distributed populations and the GLM as discussed at the end of Section 10.6, justify
Recall the analysis of the demand for textiles in the Netherlands performed by renown econometrician Henri Theil, as discussed in Problem 8.9. With reference to the least squares-based estimate of
Testing The Equality of Two Exponential Population Distributions. The Reliable Computer Co. is considering purchasing CPU chips from one of two different suppliers to use in the production of
The Huntington Chemical Co. has been accused of dumping improperly treated waste in a local trout lake, and litigation may be imminent. The central allegation is that the trout population has been
Use a Wald-Wolfowitz runs test to assess whether the data provided by the cereal manufacturer in Problem 1 is the outcome of a random sample from some population distribution. The observations
Test whether the data of the cereal manufacturer provided in Problem 1 can be interpreted as an outcome of a random sample from the specific normal population distribution claimed, i.e. test \(H_{0}:
Paired Comparisons of Means. A family counseling service offers a 1-day program of training in meal planning which they claim is effective in reducing cholesterol levels of individuals completing the
The production function for a commodity can be approximated over a restricted range of relevant input levels by a linear relationship of the form \(\mathbf{Y}=\mathbf{x}
The production department of a paper products manufacturer is analyzing the frequency of breakdowns in a certain type of envelope machine as it contemplates future machinery repair and replacement
The relationship between sales and level of advertising expenditure is hypothesized to be quadratic over the relevant range of expenditure levels being examined, i.e., \(Y_{t}=\beta_{1}+\beta_{2}
An analyst is investigating the effect of certain policy events on common stock prices in a given industry. In an attempt to isolate abnormal from normal returns of firms in the industry, the
Testing for Independence in a Bivariate Normal Population Distribution. Let \(\left(X_{i}, Y_{i}ight), i=1, \ldots, n\) be a random sample from some bivariate normal population distribution
Test of the Equality of Proportions. Let \(X_{1}\) and \(X_{2}\) be independent random variables with Binomial distributions \(\operatorname{Binomial}\left(n_{i}, p_{i}ight), i=1,2\), where the
Testing Whether Random Sample is From a Discrete Uniform Population Distribution. Let \(X_{i}, i=1, \ldots, n\) be random sample from some discrete integer-valued population distribution. Let the
Describe how you would test one-sided and two-sided hypotheses, using both finite sample and asymptotically valid testing procedures, in each of the following cases. Use whatever procedures you feel
The weekly proportion of storage tank capacity that is utilized at a regional hazardous liquid waste receiving site is an outcome of a \(\operatorname{Beta}(\theta, 1)\) random variable. At the end
Follow the proof of Theorem 10.14 to demonstrate that when random sampling from the continuous PDF \(f(z ; \Theta)\) with scalar parameter \(\Theta\),\(-2 \sum_{i=1}^{n} \ln \left[1-F\left(X_{i} ;
Regional daily demand for gasoline in the summer driving months is assumed to be the outcome of a \(N\left(\mu, \sigma^{2}ight)\) random variable. Assume you have 40 iid daily observations on daily
The population distribution of income in a populous developing country is assumed to be given (approximately) by the continuous PDF \(f(x ; \Theta)=\Theta(1+x)^{-(\Theta+1)} I_{(0, \infty)}(x)\)
Examine if the following are legitimate probability distributions.a.b.c.d. x -1 234 f(x) 1 243
For a standard normal random variable Z, find(a) P[Z < 1.56 )(b) P [ Z > 1.245)(c) P[ .61 < Z < 1.92)(d) P [ -1.47 < Z < 1.055)
In a recent year, 35 sites around the world experienced earthquakes of magnitude greater than 6.5.Construct a histogram using equal-length intervals starting with (6.5, 6.8 ], where the right-hand

Showing 200 - 300 of 509