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Questions and Answers of Statistics

In the case of a certain type of copper wire, it is known that, on the average, 1.5 flaws occur per millimeter. Assuming that the number of flaws is a Poisson random variable, what is the probability
Potholes on a highway can be a serious problem and are in constant need of repair. With a particular type of terrain and make of concrete, past experience suggests that there are, on the average, 2
Hospital administrators in large cities anguish about problems with traffic in emergency rooms in hospitals. For a particular hospital in a large city, the staff on hand cannot accommodate the
In airport luggage screening it is known that 3% of people screened have questionable objects in their luggage. What is the probability that a string of 15 people pass through successfully before an
Computer technology has produced an environment in which "robots" operate with the use of microprocessors. The probability that a robot fails during any 6 hour shift is 0.10. What is the probability
Refusal rate in telephone polls is known to be approximately 20%. A newspaper report indicates that 50 people were interviewed before the first refusal.(a) Comment, on the validity of the report. Use
During a manufacturing process 15 units are randomly selected each day from the production line to check the percent defective. From historical information it is known that the probability of a
An automatic welding machine is being considered for production. It will be considered for purchase if it is successful for 99% of its welds. Otherwise, it will not be considered efficient. A test is
A car rental agency at a local airport has available 5 Fords, 7 Chevrolets, 4 Dodges, 3 Hondas, and 4 Toyotas. If the agency randomly selects 9 of these cars to chauffeur delegates from the airport
Service calls come to a maintenance center according to a Poisson process and, on the average, 2.7 calls come per minute. Find the probability that(a) No more than 4 calls come in any minute;(b)
An electronics firm claims that the proportion of defective units of a certain process is 5%. A buyer has a standard procedure of inspecting 15 units selected randomly from a large lot. On a
An electronic switching device occasionally malfunctions and may need to be replaced. It is known that the device is satisfactory if it makes, on the average, no more than 0.20 error per hour. A
A company generally purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if two or more defective units are found in a random sample of 100 units.(a) What
A local drugstore owner knows that, on average, 100 people per hour stop by his store.(a) Find the probability that in a given 3-minutc period nobody enters the store.(b) Find the probability that in
(a) Suppose that you throw 4 dice, probability that you get at least one 1. Find(b) Suppose that you throw 2 dice 24 times. Find the probability that you get at least one (1, 1), that is, you roll
Suppose that 500 lottery tickets are sold. Among them, 200 tickets pay off at least the cost of the ticket. Now suppose that you buy 5 tickets. Find the probability that you will win back at least
Imperfections in computer circuit boards and computer chips lend themselves to statistical treatment. For a particular type of board the probability of a diode failure is 0.03. Suppose a circuit
The potential buyer of a particular engine requires (among other things) that the engine start successfully 10 consecutive times. Suppose the probability of a successful start is 0.990. Let us assume
The acceptance scheme for purchasing lots containing a large number of batteries is to test no more than 75 randomly selected batteries and to reject a lot if a single battery fails. Suppose the
An oil drilling company ventures into various locations, and its success or failure is independent from one location to another. Suppose the probability of a success at any specific location is
Consider the information of Review Exercise 5.94. The driller feels that he will "hit it big" if the second success occurs on or before the sixth attempt. What is the probability that the driller
A couple decides they will continue to have children until they have two males. Assuming that P(male) = 0.5, what is the probability that their second male is their fourth child?
It is known by researchers that 1 in 100 people carry a gene that leads to the inheritance of a certain chronic disease. From a random sample of 1000 individuals, what is the probability that fewer
A production process produces electronic component, parts. It has presumably been established that the probability of a defective part is 0.01. During a test of this presumption, 500 items are
A production process produces items in lots of 50. Sampling plans exist in which lots are pulled aside periodically and exposed to a certain type of inspection. It is usually assumed that the
Consider the situation of Review Exercise 5.99. It has been determined that the sampling plan should be extensive enough so that there is a high probability, say 0.9, that if as many as 2 defects
Homeland Security and missile defense technology make it paramount that we be able to detect incoming projectiles or missiles. To make the defense successful, multiple radar screens are: required.
Consider Review Exercise 5.101. Suppose if is important, that the overall system be as near perfect as possible. Assuming the: quality of the screens are: as indicated in Review Exercise 5.101,(a)
Go back to Review Exercise 5.99 (a) Re-compute the probability using the binomial distribution Comment.
There are two vacancies in a certain statistics department in the United States. Five individuals apply. Two have expertise in linear models and one has expertise in applied probability. The search
The manufacturer of a tricycle for children has received complaints about defective: brakes in the product. According to the design of the product and considerable: preliminary testing, it had been
Let X be a random variable with probability find the probability distribution of the random variable Y = 2 X ? 1.
Let X be a binomial random variable with probability distribution find the probability distribution of the random variable Y =X2.
Let X1 and X2 be discrete random variables with the joint multinomial distribution for x = 0, 1, 2; x2 = 0, 1, 2; x1 + x2
Let X1 and X2 be discrete random variables with joint probability distribution find the probability distribution of the random variable Y = X1X2.
Let X have the probability distribution show that the random variable Y = ?2 In A' has a chi-squared distribution with 2 degrees of freedom.
Given the random variable X with probability distribution find the probability distribution of Y = 8X3.
The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability distribution is given by where k is an appropriate constant and b depends on the absolute temperature
A dealer's profit, in units of $5000, on a new automobile is given by Y = X2, where X is a random variable having the density function(a.) Find the probability density function of the random variable
The hospital period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function(a) Find the probability density
The random variables X and Y, representing the weights of creams and toffees in 1-kilogram boxes of chocolates containing a mixture of creams, toffees, and cordials, have the joint density
The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Assume that the joint density function
Let X1 and X2 be independent random variables each having the probability distribution show that the random variables Y1 and Y2 are independent when Y1 = X1 + X2 and Y2 ? X1(X +X2).
A current of I amperes flowing through a resistance of R ohms varies according to the probability distribution if the resistance varies independently of the current according to the probability
Let X be a random variable with probability distribution find the probability distribution of the random variable Y = X2.
Let X have the probability distribution Find the probability distribution of the random variable Y =X2.
Show that the rth moment about the origin of the gamma distribution is
A random variable X has the discrete uniform distribution show that the moment-generating function of Xis
A random variable X has the geometric distribution g(x; p) = pqx?1 for x = 1, 2, 3 Show that the moment-generating function of X is and then use Mx(t) to find the mean and variance of the geometric
A random variable X has the Poisson distribution p(x; ?) = e?? ??/x for x = 0, 1, 2, Show that the moment-generating function of X is using Mx(t), find the mean and variance of the Poisson
The moment-generating function of a certain Poisson random variable X is given by find P(p – 2σ
Using the moment-generating function of Example 7.8, show that the mean and variance of the chi-squared distribution with v degrees of freedom are, respectively, v and 2v
Dy expanding et? in a Maclaurin series and integrating term by term, show that
If both A' and Y, distributed independently, follow exponential distributions with mean parameter 1. Find the distributions of(a) U = X + Y, and(b) V = X/(X + Y).
Show that the mean square error s2 = SSE/ k (n - l) for the analysis of variance in a one-way classification is an unbiased estimate of a2.
Prove Theorem.
Six different machines are being considered for use in manufacturing rubber seals. The machines are being compared with respect, to tensile strength of the product. A random sample of 4 seals from
The data in the following table represent the number of hours of relief provided by 5 different brands of headache tablets administered to 25 subjects experiencing fevers of 38?C or more. Perform
In an article Shelf-Space Strategy in Retailing, published in the Proceedings: Southern Marketing Association, the effect of shelf height on the supermarket sales of canned dog food is investigated.
Immobilization of free-ranging white-tailed deer by drugs allows researchers the opportunity to closely examine deer and gather valuable physiological information. In the study Influence of Physical
It has been shown that the fertilizer magnesium ammonium phosphate, MgNH4PO4, is an effective supplier of the nutrients necessary for plant growth. The compounds supplied by this fertilizer are
A study measures the sorption (either absorption or adsorption) rate of three different types of organic chemical solvents. These solvents are used to clean industrial fabricated-metal parts and are
The mitochondrial enzyme NAPH: NAD transhydrogenase of the common rat tapeworm (Hymenolepiasis diminuta) catalyzes hydrogen in transfer from NADYH to NAD, producing NADH. This enzyme is known to
For the data set in Exercise 13.7, use Bartlett's test to check whether the variances are equal.
Consider the data of Review Exercise 13.58. Make significance tests on the following contrasts:(a) B versus A, C, and D;(b) C versus A and D;(c) A versus D.
The study Loss of Nitrogen through Sweat by Preadolescent Boys Consuming Three Levels of Dietary Protein was conducted by the Department of Human Nutrition and Foods at the Virginia Polytechnic
The purpose of the study The Incorporation of a. Chelating Agent into a Flame Retardant Finish of a Cotton Flannelette and the Evaluation of Selected Fabric Properties conducted at the Virginia
Use Tukey’s test, with a 0.05 level of significance, to analyze the means of the 5 different brands of headache tablets in Exercise 13.4.
For the data used in Review Exercise 13.58, perform Tukey's test with a 0.05 level of significance to determine which laboratories differ, on average, in their analysis.
An investigation was conducted to determine the source of reduction in yield of a certain chemical product. It was known that the loss in yield occurred in the mother liquor, that is. The material
In the study An Evaluation of the Removal Method for Estimating Benthic Populations and Diversity conducted by the Virginia Polytechnic Institute and State University on the Jackson River, 5
The following data are values of pressure (psi) in a torsion spring for several settings of the angle between the legs of the spring in a free position. Compute a one-way analysis of variance for
In the following biological experiment 4 concentrations of a certain chemical are used to enhance the growth of a certain type of plant over time. Five plants are used at each concentration and the
The following table (A. Held Statistical theory with Engineering Applications John Wiley fc Sons New York. 1952) gives tensile1 st lengths of deviations from 340 for wires taken from nine cables to
The financial structure of a firm refers to the way the firm's assets are divided by equity and debt, and the financial leverage refers to the percentage of assets financed by debt. In the paper The
It is suspected that the environmental temperature in which batteries are activated affects their life. Thirty homogeneous batteries were tested six at each of five temperatures, and the data are
Do Duncan's test for paired comparisons for the data of Exercise 13.8. Discuss the results.
Show that the computing formula for SSB, in the analysis of variance of the randomized complete block design, is equivalent to the corresponding term in the identity ofTheorem.
For the randomized block design with k treatments and b blocks, showthat
Four kinds of fertilizer f1, f2, f3, and f4 are used to study the yield of beans. The soil is divided into 3 blocks each containing 4 homogeneous plots. The yields in kilograms per plot and the
Three varieties of potatoes are being compared for yield. The experiment is conducted by assigning each variety at random to 3 equal-size plots at each of 4 different locations. The following yields
The following data are the percents of foreign additives measured by 5 analysts for 3 similar brands of strawberry jam, A, B, and C. Perform the analysis of variance and test the hypothesis, at the
The following data represent the final grades obtained by 5 students in mathematics, English, French, and biology. Test the hypothesis that the courses are of equal difficulty. Use a P-value in your
In a study on The Periphyton of the South River, Virginia: Mercury Concentration, Productivity, and Autotropic Index Studies conducted, by the Department of Environmental Sciences and Engineering at
A nuclear power facility produces a vast amount of heat which is usually discharged into aquatic systems. This heat raises the temperature of the aquatic system, resulting in a greater concentration
In a study conducted by the Department of Health and Physical Education at the Virginia Polytechnic Institute and State University, 3 diets were assigned for a period of 3 days to each of 6 subjects
Organic arsenicals are used by forestry personnel as silvicides. The amount of arsenic that is taken into the body when exposed to these silvicides is a major health problem. It is important that the
Scientists in the Department of Plant Pathology at Virginia Tech devised an experiment in which 5 different treatments were applied to 6 different locations in an apple orchard to determine if there
In the paper "Self-Control and Therapist Control in the Behavioral Treatment of OverweightWomen" published in Behavioral Research and Therapy (Vol. 10, 1972), two reduction treatments and a control
Verify the sum-of-squares identity of Theorem.
For the r x r Latin square design, show that
The mathematics department of a large university wishes to evaluate the teaching capabilities of 4 professors. In order to eliminate any effects due to different mathematics courses and different
A manufacturing firm wants to investigate the effects of 5 color additives on the setting time of a new concrete mix. Variations in the setting times can be expected from day-to-day changes in
In the book Design of Experiments for the Quality Improvement published by the Japanese Standards Association (1989), a study on the amount of dye needed to get the best color for a certain type of a
An experiment was conducted to compare three types of coating materials for copper wire. The purpose of the coating is to eliminate "flaws" in the wire. Ten different specimens of length five
The following data show the effect of 4 operators, chosen randomly, on the output of a particular machine:(a) Perform a model II analysis of variance at the 0.05 level of significance.(b) Compute an
Assuming a random effects model, show that E (SSB) = (b – 1) σ2 + k (b – 1) σ23 for the randomized complete block design.
An experiment is conducted in which 4 treatments are to be compared in 5 blocks. The following data are generated:(a) Assuming a random effects model, test the hypothesis at the 0.05 level of
Assuming a random effects model, show that E (SST r) = (r – 1) (σ2 + rσ2r) for the Latin square design.
(a) Using a regression approach for the randomized complete block design, obtain the normal equations Ab = g in matrix form. (b) Show that R (β1, β2. β b ׀α1, α α

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