Questions and Answers of Statistics Engineers Scientists

The article “A Music Key Detection Method Based on Pitch Class Distribution Theory” (J. Sun, H. Li, and L. Ma, International Journal of Knowledge-based and Intelligent Engineering Systems,
The article €œReaction Modeling and Optimization Using Neural Networks and Genetic Algorithms: Case Study Involving TS-1-Catalyzed Hydroxylation of Benzene€ (S. Nandi, P.
The article “Influence of Penetration Rate on Penetrometer Resistance” (J. Oliveira, M. Almeida, et al., Journal of Geotechnical and Geoenvironmental Engineering, 2011:695–703) presents
Scores on an IQ test are normally distributed. A sample of 25 IQ scores had variance s2 = 64. The developer of the test claims that the population variance is σ2 = 225. Do these data provide
The article “Measurement of Complex Permittivity of Asphalt Paving Materials” (J. Shang, J. Umana, et al., Journal of Transportation Engineering, 1999:347–356) compared the dielectric constants
If P = 0.01, which is the best conclusion?i. H0 is definitely false.ii. H0 is definitely true.iii. There is a 1% probability that H0 is true.iv. H0 might be true, but it’s unlikely.v. H0 might be
The article “HIV-positive Smokers Considering Quitting: Differences by Race/Ethnicity” (E. Lloyd- Richardson, C. Stanton, et al., Am J Health Behav, 2008:3–15) surveyed 444 HIV-positive
The article €œEstimation of Mean Arterial Pressure from the Oscillometric Cuff Pressure: Comparison of Different Techniques€ (D. Zheng, J. Amoore, et al., Med Biol Eng Comput,
Refer to Exercise 6 in Section 5.10. Let Î¼ represent the population mean compressive strength, in MPa. Generate 1000 bootstrap samples.a. Using the bootstrap data you generated, which of these
A geologist is making repeated measurements (in grams) on the mass of a rock. It is not known whether the measurements are a random sample from an approximately normal population. Below are three
A simple random sample consists of 65 lengths of piano wire that were tested for the amount of extension under a load of 30 N. The average extension for the 65 lines was 1.102 mm and the standard
Resistors labeled as 100 Ω are purchased from two different vendors. The specification for this type of resistor is that its actual resistance be within 5% of its labeled resistance. In a sample of
The thicknesses of eight pads designed for use in aircraft engine mounts are measured. The results, in mm, are 41.83, 41.01, 42.68, 41.37, 41.83, 40.50, 41.70, and 41.42. Assume that the thicknesses
Refer to Exercise 1. The P-value for fertilizer C was 0.03. Use the Bonferroni correction to produce a reliable P-value for this fertilizer. Can you reject H0?Refer to Exercise 1An agricultural
A process for a certain type of ore is designed to reduce the concentration of impurities to less than 2%. It is known that the standard deviation of impurities for processed ore is 0.6%. Let μ
In a study of the relationship of the shape of a tablet to its dissolution time, 6 disk-shaped ibuprofen tablets and 8 oval-shaped ibuprofen tablets were dissolved in water. The dissolve times, in
A random sample of size 29 from a normal distribution has variance s2 = 24. Test H0 : σ2 ≥ 30 versus H1 : σ2 < 30.
A group of 15 swimmers is chosen to participate in an experiment to see if a new breathing style will improve their stamina. Each swimmer’s pulse recovery rate is measured after a 20 minute workout
At an assembly plant for light trucks, routine monitoring of the quality of welds yields the following data:Can you conclude that the quality varies among shifts?a. State the appropriate null
Integrated circuits consist of electric channels that are etched onto silicon wafers. A certain proportion of circuits are defective because of “undercutting,” which occurs when too much material
Refer to Exercise 6 in Section 5.10. Let µ represent the population mean compressive strength, in MPa. Consider the following null hypotheses:a. Using the bootstrap data presented in Exercise 6 in
The article €œImproved Bioequivalence Assessment of Topical Dermatological Drug Products Using Dermatopharmacokinetics€ (B. N€™Dri-Stempfer, W. Navidi, R. Guy, and A.
An agricultural scientist tests six types of fertilizer, labeled A, B, C, D, E, and F, to determine whether any of them produces an increase in the yield of lima beans over that obtained with the
True or false:a. If we reject H0, then we conclude that H0 is false.b. If we do not reject H0, then we conclude that H0 is true.c. If we reject H0, then we conclude that H1 is true.d. If we do
A random sample of size 11 from a normal distribution has variance s2 = 96. Test H0 : σ2 ≤ 50 versus H1 : σ2 > 50.
For which P-value is the null hypothesis more plausible: P = 0.5 or P = 0.05?
Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number of defects X is presented in the following table.a. Find P(X
A process that manufactures piston rings produces rings whose diameters (in centimeters) vary according to the probability density function.a. Find the mean diameter of rings manufactured by this
The number of bytes downloaded per second on an information channel has mean 105 and standard deviation 104. Among the factors influencing the rate is congestion, which produces alternating periods
Human blood may contain either or both of two antigens, A and B. Blood that contains only the A antigen is called type A, blood that contains only the B antigen is called type B, blood that contains
A car dealer sold 750 automobiles last year. The following table categorizes the cars sold by size and color and presents the number of cars in each category. A car is to be chosen at random from the
A production facility contains two machines that are used to rework items that are initially defective. Let X be the number of hours that the first machine is in use, and let Y be the number of hours
The level of impurity (in percent) in the product of a certain chemical process is a random variable with probability density function.a. What is the probability that the impurity level is greater
True or false: If A and B are mutually exclusive,a. P(A ∪ B) = 0b. P(A ∩ B) = 0c. P(A ∪ B) = P(A ∩ B)d. P(A ∪ B) = P(A) + P(B)
The following table presents the 100 senators of the 113th U.S. Congress on January 3, 2013, classified by political party affiliation and gender.A senator is selected at random from this group.
The main bearing clearance (in mm) in a certain type of engine is a random variable with probability density functiona. What is the probability that the clearance is less than 0.02 mm?b. Find the
A flywheel is attached to a crankshaft by 12 bolts, numbered 1 through 12. Each bolt is checked to determine whether it is torqued correctly. Let A be the event that all the bolts are torqued
Refer to Exercise 19.a. Find Î¼X.b. Find Î¼Y.c. Find ÏƒX.d. Find ÏƒY.e. Find Cov(X,Y).f. Find ÏX,Y.Refer to Exercise 19. Tensile Strength
An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are
The error in the length of a part (absolute value of the difference between the actual length and the target length), in mm, is a random variable with probability density function.a. What is the
The concentration of a reactant is a random variable with probability density functiona. What is the probability that the concentration is greater than 0.5?b. Find the mean concentration.c. Find the
A certain plant runs three shifts per day. Of all the items produced by the plant, 50% of them are produced on the first shift, 30% on the second shift, and 20% on the third shift. Of all the items
Refer to Exercise 21. Is it possible for the probability that both gauges fail to be greater than 0.01? Explain.Refer to Exercise 21.Nuclear power plants have redundant components in important
Here are two random variables that are uncorrelated but not independent. Let X and Y have the following joint probability mass function:a. Use the definition of independence on page 141 to show that
Refer to Exercise 23. Assume that the probabilities that each of the three scenarios occurs are P(A) = 0.20, P(B) = 0.30, and P(C) = 0.50.a. Find the probability that scenario A occurs and that the
The height H and radius R (in cm) of a cylindrical can are random with joint probability density function.The volume of a can is V = Ï€ R2H. Find Î¼V . 19 < h < 21 and 3(h –
In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second
The diameter of a rivet (in mm) is a random variable with probability density function.a. What is the probability that the diameter is less than 12.5 mm?b. Find the mean diameter.c. Find the standard
A stock solution of hydrochloric acid (HCl) supplied by a certain vendor contains small amounts of several impurities, including copper and nickel. Let X denote the amount of copper and let Y denote
A fair coin is tossed five times. Which sequence ismore likely, HTTHH or HHHHH? Or are they equally likely? Explain.
Items are inspected for flaws by two quality inspectors. If a flaw is present, it will be detected by the first inspector with probability 0.9, and by the second inspector with probability 0.7.
Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) +Cov(Y, Z).
Two fair dice are rolled. Let X represent the number on the first die, and let Y represent the number on the second die. Find μXY .
Refer to Exercise 31. Assume the first card is not replaced before the second card is drawn.a. Find the joint probability mass function of X and Y.b. Find the marginal probability mass functions pX
A quality-control program at a plastic bottle production line involves inspecting finished bottles for flaws such as microscopic holes. The proportion of bottles that actually have such a flaw is
Let X and Y be jointly continuous with joint probability density function f (x, y) and marginal densities fX (x) and fY (y). Suppose that f (x, y) = g(x)h(y) where g(x) is a function of x alone, h(y)
Refer to Example 2.26. a. If a man tests negative, what is the probability that he actually has the disease?b. For many medical tests, it is standard procedure to repeat the test when a positive
A circle is drawn with radius R, where μR = 10 and σ2R = 1. The area of the circle is A = πR2. Find μA.
A system contains two components, A and B, connected in series, as shown in the diagram.Assume A and B function independently. For the system to function, both components must function.a. If the
If A and B are independent events, prove that the following pairs of events are independent: Ac and B, A and Bc, and Ac and Bc.
Refer to Exercise 15. Let E1 be the event that the wafer comes from Lot A, and let E2 be the event that the wafer is conforming. Are E1 and E2 independent? Explain.Refer to Exercise 15. Lot
A closet contains four pairs of shoes. If four shoes are chosen at random, what is the probability that the chosen shoes do not contain a pair?
A system contains two components, A and B. The system will function so long as either A or B functions. The probability that A functions is 0.95, the probability that B functions is 0.90, and the
The thickness X of a wooden shim (in mm) has probability density function.a. Find Î¼X.b. Find Ïƒ2X.c. Let Y denote the thickness of a shim in inches (1 mm = 0.0394 inches).
Measurements are made on the length and width (in cm) of a rectangular component. Because of measurement error, the measurements are random variables. Let X denote the length measurement and let Y
Refer to Exercise 12. a. Find the conditional probability mass function pY |X (y | 3). b. Find the conditional probability mass function pX|Y (x | 1). c. Find the conditional expectation E(Y | X =
Six hundred paving stones were examined for cracks, and 15 were found to be cracked. The same 600 stones were then examined for discoloration, and 27 were found to be discolored. A total of 562
The oxygen equivalence number of a weld is a number that can be used to predict properties such as hardness, strength, and ductility. The article “Advances in Oxygen Equivalence Equations for
Elongation (in percent) of steel plates treated with aluminum are random with probability density function.a. What proportion of steel plates have elongations greater than 25%?b. Find the mean
Refer to Exercise 12. Assume that the cost of an engine repair is $50, and the cost of a transmission repair is $100. Let T represent the total cost of repairs during a one-hour time interval. a.
Laura and Philip each fire one shot at a target. Laura has probability 0.5 of hitting the target, and Philip has probability 0.3. The shots are independent.a. Find the probability that the target is
The article “Traps in Mineral Valuations—Proceed With Care” (W. Lonegan, Journal of the Australasian Institute of Mining and Metallurgy, 2001:18–22) models the value (in millions of dollars)
In the article €œAn Investigation of the Ca€“CO3€“CaF2€“K2SiO3€“SiO2€“Fe Flux System Using the Submerged Arc Welding Process on HSLA-100
Refer to Exercise 12. Let Z = X + Y represent the total number of repairs needed. a. Find μZ. b. Find σZ. c. Find P(Z = 2). Refer to Exercise 12. y 1 2 3 0.13 0.10 0.07 0.03 1 0.12 0.16 0.08 0.04
Let V be the event that a computer contains a virus, and let W be the event that a computer contains a worm. Suppose P(V) = 0.15, P(W) = 0.05, and P(V ∪ W) = 0.17.a. Find the probability that the
The Needleman-Wunsch method for aligning DNA sequences assigns 1 point whenever a mismatch occurs, and 3 points whenever a gap (insertion or deletion) appears in a sequence. Assume that under certain
A drawer contains 6 red socks, 4 green socks, and 2 black socks. Two socks are chosen at random. What is the probability that they match?
Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F)
Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the
Sarah and Thomas are going bowling. The probability that Sarah scores more than 175 is 0.4, and the probability that Thomas scores more than 175 is 0.2. Their scores are independent.a. Find the
Let A and B be events with P(A) = 0.3 and P(A ∪ B) = 0.7.a. For what value of P(B) will A and B be mutually exclusive?b. For what value of P(B) will A and B be independent?
A certain commercial jet plane uses a mean of 0.15 gallons of fuel per passenger-mile, with a standard deviation of 0.01 gallons.a. Find the mean number of gallons the plane uses to fly 8000 miles if
A quality-control engineer samples 100 items manufactured by a certain process, and finds that 15 of them are defective. True or false:a. The probability that an item produced by this process is
Refer to Exercise 10. Let Y be the number of chips tested up to and including the second acceptable chip.a. What is the smallest possible value for Y ?b. What is the probability that Y takes on that
The article “High Cumulative Risk of Lung Cancer Death among Smokers and Nonsmokers” (P. Brennan, et al. American Journal of Epidemiology, 2006:1233–1241) states that the probability is 0.24
A gas station earns $2.60 in revenue for each gallon of regular gas it sells, $2.75 for each gallon of midgrade gas, and $2.90 for each gallon of premium gas. Let X1, X2, and X3 denote the numbers of
A company has hired 15 new employees, and must assign 6 to the day shift, 5 to the graveyard shift, and 4 to the night shift. In how many ways can the assignment be made?
Microprocessing chips are randomly sampled one by one from a large population, and tested to determine if they are acceptable for a certain application. Ninety percent of the chips in the population
Of all failures of a certain type of computer hard drive, it is determined that in 20% of them only the sector containing the file allocation table is damaged, in 70% of them only nonessential
In a lot of 10 components, 2 are sampled at random for inspection. Assume that in fact exactly 2 of the 10 components in the lot are defective. Let X be the number of sampled components that are
An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are
A machine that fills bottles with a beverage has a fill volume whose mean is 20.01 ounces, with a standard deviation of 0.02 ounces. A case consists of 24 bottles randomly sampled from the output of
In a certain state, license plates consist of three letters followed by three numbers. a. How many different license plates can be made?b. How many different license plates can be made in which
After manufacture, computer disks are tested for errors. Let X be the number of errors detected on a randomly chosen disk. The following table presents values of the cumulative distribution function
The reading given by a thermometer calibrated in ice water (actual temperature 0°C) is a random variable with probability density functionwhere k is a constant.a. Find the value of k.b. What is
A test consists of 15 questions. Ten are true-false questions, and five are multiple-choice questions that have four choices each. A student must select an answer for each question. In how many ways
Two independent measurements are made of the lifetime of a charmed strange meson. Each measurement has a standard deviation of 7 × 10−15 seconds. The lifetime of the meson is estimated by
A college math department consisting of 10 faculty members must choose a department head, an assistant department head, and a faculty senate representative. In how many ways can this be done?
The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the
Refer to Exercise 5. Two candidates are randomly selected.a. List the equally likely outcomes.b. What is the probability that both are qualified?c. What is the probability that exactly one is

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