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mathematics
statistics

Questions and Answers of Statistics

The weather in Los Angeles is dry most of the time, but it can be quite rainy in the winter. The rainiest month of the year is February. The following table presents the annual rainfall in Los
The following table presents the number of students absent in a middle school in northwestern Montana for each school day in January 2008.a. Construct a box plot. b. There was a snowstorm on January
A sample of 100 men has average height 70 in. and standard deviation 2.5 in. A sample of 100 women has average height 64 in. and standard deviation 2.5 in. If both samples are combined, the standard
Following are summary statistics for two data sets, A and B.a. Compute the inter-quartile ranges for both A and B. b. Do the summary statistics for A provide enough information to construct a box
Refer to the asphalt data in Example 1.12.a. Construct a box plot for the asphalt data.b. Which values, if any, are outliers?c. Construct a dot plot for the asphalt data.d. For purposes of
For the following data:a. Make a scatter plot of y versus x. Is the relationship between x and y approximately linear, or is it nonlinear?b. Compute the natural logarithm of each y value. This is
A certain reaction was run several times using each of two catalysts, A and B. The catalysts were supposed to control the yield of an undesirable side product. Results, in units of percentage yield,
The following histogram presents the distribution of systolic blood pressure for a sample of women. Use it to answer the following questions.a. Is the percentage of women with blood pressures above
a. Using the class intervals in the table, construct a histogram in which the heights of the rectangles are equal to the frequencies.b. Using the class intervals in the table, construct a histogram
For each of the following histograms, determine whether the vertical axis has been labeled correctlya.b. c. d.
The article "Vehicle-Arrival Characteristics at Urban Uncontrolled Intersections" (V. Rengaraju and V. Rao, Journal of Transportation Engineering, 1995: 317-323) presents data on traffic
The article "Computing and Using Rural versus Urban Measures in Statistical Applications" (C. Goodall, K. Kafadar, and J. Tukey, The American Statistician, 1998:101-111) discusses methods to measure
Water scarcity has traditionally been a major concern in the Canary Islands. Water rights are divided into shares, which are privately owned. The article "The Social Construction of Scarcity the Case
The article "The Ball-on-Three-Ball Test for Tensile Strength: Refined Methodology and Results for Three Hohokam Ceramic Types" (M. Beck, American Antiquity, 2002:558-569) discusses the strength of
Integrated circuits consist of electric channels that are etched onto siliconwafers. Acertain proportion of circuits are defective because of "undercutting," which occurs when too much material is
The smallest number on a list is changed from 12.9 to 1.29. a. Is it possible to determine by how much the mean changes? If so, by how much does it change? b. Is it possible to determine by how much
There are 15 numbers on a list, and the mean is 25. The smallest number on the list is changed from 12.9 to 1.29. a. Is it possible to determine by how much the mean changes? If so, by how much does
Let S be the event that a randomly selected college student has taken a statistics course, and let C be the event that the same student has taken a chemistry course. Suppose P(S) = 0.4, P(C) = 0.3,
All the fourth-graders in a certain elementary school took a standardized test. A total of 85% of the students were found to be proficient in reading, 78%were found to be proficient in mathematics,
A system contains two components, A and B. The system will function only if both components function. The probability that A functions is 0.98, the probability that B functions is 0.95, and the
A section of an exam contains four True-False questions. A completed exam paper is selected at random, and the four answers are recorded. a. List all 16 outcomes in the sample space. b. Assuming the
A box contains four bolts. Two of them, labeled #1 and #2, are 5 mm in diameter, and two of them, labeled #3 and #4, are 7 mm in diameter. Bolts are randomly selected until a 5 mm bolt is obtained.
In a survey of households with television sets, the proportion of television sets in various types of rooms wasRoom
Among the cast aluminum parts manufactured on a certain day, 80% were flawless, 15% had only minor flaws, and 5% had major flaws. Find the probability that a randomly chosen part a. Has a flaw (major
One drawer in a dresser contains 8 blue socks and 6 white socks. A second drawer contains 4 blue socks and 2 white socks. One sock is chosen from each drawer. What is the probability that they match?
A computer password consists of eight characters. a. How many different passwords is possible if each character may be any lowercase letter or digit? b. How many different passwords are possible if
In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within the specification are ready for installation. Those valves whose thicknesses
A particular automatic sprinkler system has two different types of activation devices for each sprinkler head. One type has a reliability of 0.9; that is, the probability that it will activate the
A population of 600 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification. The following table presents the
A geneticist is studying two genes. Each gene can be either dominant or recessive. A sample of 100 individuals is categorized as follows.a. What is the probability that a randomly sampled individual,
A driver encounters two traffic lights on the way to work each morning. Each light is red, yellow, or green. The probabilities of the various combinations of colors are given in the following
Nuclear power plants have redundant components in important systems to reduce the chance of catastrophic failure. Assume that a plant has two gauges to measure the level of coolant in the reactor
A lot of 10 components contain 3 that 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
Each day, a weather forecaster predicts whether or not it will rain. For 80% of rainy days, she correctly predicts that it will rain. For 90% of non-rainy days, she correctly predicts that it will
Refer to Exercise 28. Assume that both inspectors inspect every item and that if an item has no flaw, then neither inspector will detect a flaw. a. Assume that the probability that an item has a flaw
A box contains 10 fuses. Eight of them are rated at 10 amperes (A) and the other two are rated at 15 A. Two fuses are selected at random. a. What is the probability that the first fuse is rated at 15
Sickle-cell anemia is an inherited disease in which red blood cells are misshapen and sticky. Sickle cells tend to form clumps in blood vessels, inhibiting the flow of blood. Humans have two genes
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 signal is given. If
A system consists of four components, connected as shown in the diagram. Suppose that the components function independently, and that the probabilities of failure are 0.05 for A, 0.03 for B, 0.07 for
A system contains two components, C and D, connected in parallel as shown in the diagram.Assume C and D function independently. For the system to function, either C or D must function.a. If the
On graduation day at a large university, one graduate is selected at random. Let A represent the event that the student is an engineering major, and let B represent the event that the student took a
Suppose that start-up companies in the area of biotechnology have probability 0.2 of becoming profitable, and that those in the area of information technology have probability 0.15 of becoming
Of people in a certain city who bought a new vehicle in the past year, 12% of them bought a hybrid vehicle, and 5% of them bought a hybrid truck. Given that a person bought a hybrid vehicle, what is
Determine whether each of the following random variables is discrete or continuous. a. The number of heads in 100 tosses of a coin. b. The length of a rod randomly chosen from a day's production. c.
Resistors labeled 100Î© have true resistances that are between 80 Î© and 120 Î©. Let X be the mass of a randomly chosen resistor. The probability density function of X is given bya. What
The lifetime of a transistor in a certain application has a lifetime that is random with probability density functiona. Find the mean lifetime.b. Find the standard deviation of the lifetimes.c. Find
A competing process produces rings whose diameters (in centimeters) vary according to the probability density functionSpecifications call for the diameter to be 10.0±0.1 cm. Which process is better,
The level of impurity (in percent) in the product of a certain chemical process is a random variable with probability density functiona. What is the probability that the impurity level is greater
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
The thickness of a washer (in mm) is a random variable with probability density functiona. What is the probability that the thickness is less than 2.5 m?b. What is the probability that the thickness
The repair time (in hours) for a certain machine is a random variable with probability density functiona. What is the probability that the repair time is less than 2 hours?b. What is the probability
A chemical supply company ships a certain solvent in 10-gallon drums. Let X represent the number of drums ordered by a randomly chosen customer. Assume X has the following probability mass
A survey of cars on a certain stretch of highway during morning commute hours showed that 70% had only one occupant, 15% had 2, 10% had 3, 3% had 4, and 2% had 5. Let X represents the number of
A certain type of component is packaged in lots of four. Let X represents the number of properly functioning components in a randomly chosen lot. Assume that the probability that exactly x components
On 100 different days, a traffic engineer counts the number of cars that pass through a certain intersection between 5 P.M. and 5:05 P.M. The results are presented in the following table.a. Let X be
If X and Y are independent random variables with means μX = 9.5 and μY = 6.8 and standard deviations σX = 0.4 and σY = 0.1, find the means and standard deviations of the following: a. 3X b. Y −
The number of miles traveled per gallon of gasoline for a certain car has a mean of 25 and a standard deviation of 2. The tank holds 20 gallons.a. Find the mean number of miles traveled per tank.b.
In the article "An Investigation of the Ca-CO3-CaF2- K2SiO3-SiO2-Fe Flux System Using the Submerged Arc Welding Process on HSLA-100 and AISI-1018 Steels" (G. Fredrickson, M.S. thesis, Colorado School
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
The article "Abyssal Peridotites > 3800 Ma from Southern West Greenland: Field Relationships, Petrography, Geochronology, Whole-Rock and Mineral Chemistry of Dunite and Harzburgite Inclusions in the
The lifetime of a certain light bulb in a certain application has mean 700 hours and standard deviation 20 hours. As each bulb burns out, it is replaced with a new bulb. Find the mean and standard
A piece of plywood is composed of five layers. The layers are a simple random sample from a population whose thickness has mean 0.125 in. and standard deviation 0.005 in. a. Find the mean thickness
The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution (1 mole = 6.02 × 1023 molecules). If X is the molarity of a solution of magnesium chloride
The four sides of a picture frame consist of two pieces selected from a population whose mean length is 30 cm with standard deviation 0.1 cm, and two pieces selected from a population whose mean
In a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. For a particular summer season, let X be the number of days on which the
a. Find the conditional probability mass function pY |X (y | 4).b. Find the conditional probability mass function pX|Y (x | 3).c. Find the conditional expectation E(Y | X = 4).d. Find the conditional
Let Z represent the total number of vehicles that pass through the toll booth in a one-minute time interval.a. Find Î¼Z.b. Find ÏƒZ.c. Find P (Z = 2).In 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 = 3).d. Find the conditional
Refer to Exercise 18.a. Find Cov(X, Y).b. Find ÏX,Y.c. Find the conditional probability density function fY |X (y | 0.5).d. Find the conditional expectation E(Y | X = 0.5).In Exercise 18
The lifetime of a certain component, in years, has probability density functionTwo such components, whose lifetimes are independent, are available. As soon as the first component fails, it is
An investor has $100 to invest, and two investments between which to divide it. If she invests the entire amount in the first investment, her return will be X, while if she invests the entire amount
Let R denote the resistance of a resistor that is selected at random from a population of resistors that are labeled 100Ω. The true population means resistance is μR = 100 Ω, and the population
Let X and Y be random variables, and a and b be constants.a. Prove that Cov(aX, bY ) = ab Cov(X,Y ).b. Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation
Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that −1 ≤ ρX,Y ≤ 1. a. Express the quantity V(X − (σX/σY) Y) in terms of σX, σY,
a. Find the conditional probability mass function pY |X (y |0). b. Find the conditional probability mass function pX|Y (x |1). c. Find the conditional expectation E(Y | X = 0). d. Find the
Refer to Exercise 30. An equation to predict the ductility of a titanium weld is Y = 7.84C +11.44N + O − 1.58Fe, where Y is the oxygen equivalence used to predict ductility, and C, N, O, and Fe are
Let a, b, c, d be any numbers with aIn other words, f (x, y) is constant on the rectangle a a. Show that k = 1/((b - a)(d - c)).b. Show that the marginal density of X is fX (x) = 1/(b - a) for
The total number of assemblies that fail to meet specifications is X + Y. a. Find μX+Y. b. Find σX+Y . c. Find P(X + Y = 3).
Assume that the cost of repairing an assembly whose clearance is too little is $2, and the cost of repairing an assembly whose clearance is too much is $3. a. Express the total cost of repairs in
Bolts manufactured for a certain purpose may be classified as acceptable (suitable for the intended purpose), downgraded (unsuitable for the intended purpose but acceptable for a different purpose),
A system consists of four components connected as shown.Assume A, B, C, and D function independently. If the probabilities that A, B, C, and D fail are 0.1, 0.2, 0.05, and 0.3, respectively, what is
There are two fuses in an electrical device. Let X denotes the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). Assume the joint probability density
A snowboard manufacturer has three plants, one in the eastern United States, one in the western United States, and one in Canada. Production records show that the U.S. plants each produced 10,000
Six new graduates are hired by an engineering firm. Each is assigned at random to one of six cubicles arranged in a row in the back of the room that houses the engineering staff. Two of the graduates
Let X and Y be independent random variables with μX = 2, σX = 1, μY = 2, and σY = 3. Find the means and variances of the following quantities. a. 3X b. X + Y c. X − Y d. 2X + 6Y
A steel manufacturer is testing a new additive for manufacturing an alloy of steel. The joint probability mass function of tensile strength (in thousands of pounds/in2) and additive concentration
Refer to Exercise 19.a. Compute the conditional mass function p Y|X (y | 0.06).b. Compute the conditional mass function p X|Y (x | 100).c. Compute the conditional expectation E(Y | X = 0.06).d.
The article "Uncertainty and Climate Change" (G. Heal and B. Kristrom, Environmental and Resource Economics, 2002:3-39) considers three scenarios, labeled A, B, and C, for the impact of global
A box contains four 75Wlightbulbs, three 60Wlightbulbs, and three burned-out light bulbs. Two bulbs are selected at random from the box. Let X represent the number of 75 W bulbs selected, and let Y
Refer to Exercise 26.a. Find μX.b. Find σ2X.c. Find Cov (X,Y).d. Find σX,σY.In Exercise 26
A penny and a nickel are tossed. The penny has probability 0.4 of coming up heads, and the nickel has probability 0.6 of coming up heads. Let X = 1 if the penny comes up heads, and let X = 0 if the
Silicon wafers are used in the manufacture of integrated circuits. Of the wafers manufactured by a certain process, 10% have resistances below specification and 5% have resistances above
A box contains three cards, labeled 1, 2, and 3. Two cards are chosen at random, with the first card being replaced before the second card is drawn. Let X represents the number on the first card, and
This exercise will lead you through a proof of Chebyshev’s inequality. Let X be a continuous random variable with probability density function f (x). Suppose that P(X < 0) = 0, so f (x) = 0 for
Blood is taken from each of n individuals to be tested for a certain disease. Rather than test each sample separately, a pooled method is used in an attempt to reduce the number of tests needed. Part
Four bricks are sampled from a large load and the crushing strength of each is measured to determine whether it meets a specification. If any of the four fail to meet the specification the load will
An electronic message consists of a string of bits (0s and 1s). The message must pass through two relays before being received. At each relay the probability is 0.1 that the bit will be reversed
Two dice are rolled. Given that two different numbers come up, what is the probability that one of the dice comes up 6?
The length of a rod was measured eight times. The measurements in centimeters, in the order they were taken, were 21.20, 21.22, 21.25, 21.26, 21.28, 21.30, 21.32, 21.35. a. Do these measurements
A person stands on a bathroom scale. The reading is 150 lb. After the person gets off the scale, the reading is 2 lb. a. Is it possible to estimate the uncertainty in this measurement? If so,

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