Questions and Answers of Introduction To Probability Statistics

Given that $f(x)=\frac{k}{2^{x}}$ is a probability distribution for a random variable that can take on the values $x=$ $0,1,2,3$, and 4 , find $k$.
Check whether the following can define probability distributions and explain your answers.(a) $f(x)=\frac{1}{4}$for $x=10,11,12,13$(b) $f(x)=\frac{2 x}{5}$for $x=0,1,2,3,4,5$(c)
Determine whether the following can be probability distributions of a random variable which can take on only the values $1,2,3$, and 4 .(a) $f(1)=0.19, \quad f(2)=0.27, \quad f(3)=0.27$, and
An experiment consists of five draws from a pack of cards. Denoting the outcomes BRRBR, BRRRB,..., for black and red cards and assuming that all 32 outcomes are equally likely, find the probability
Amy commutes to work by two different routes A and B. If she comes home by route A, then she will be home no later than 6 P.M. with probability 0.8, but if she comes home by route B, then she will be
During the inspection of a rejected integrated circuit (IC), it was observed that the rejected IC could have an incorrect circuit, it could be bent, or it could have both defects. The probability of
Suppose that a probability of $\frac{1}{16}$ is assigned to each point of the sample space of part (a) of Exercise 3.1 on page 65 . Find the probability distribution of the total number of units of
Cumulative Poisson probabilities can be calculated using MINITAB.Output:Poisson with mean $=1.64$Find the cumulative Poisson probabilities for $x=2$ and $x=3$ when(a) $\lambda=2.73$;(b)
Suppose that the probabilities are, respectively, 0.40,0.40, and 0.20 that in city driving a certain kind of imported car will average less than 22 miles per gallon, anywhere from 22 to 25 miles per
As can easily be shown, the probabilities of getting 0 , 1 , or 2 heads with a pair of balanced coins are $\frac{1}{4}, \frac{1}{2}$, and $\frac{1}{4}$. What is the probability of getting 2 tails
Suppose the probabilities are 0.89,0.09, and 0.02 that the finish on a new car will be rated acceptable, easily repairable, or unacceptable. Find the probability that, among 20 cars painted one
Using the same sort of reasoning as in the derivation of the formula for the hypergeometric distribution, we can derive a formula which is analogous to the multi-nomial distribution but applies to
Simulate tossing a coin.(a) For a balanced coin, generate 100 flips.(b) For a coin with probability of heads 0.8, generate 100 flips.
The probabilities that a quality control team will visit $0,1,2,3$, or 4 production sites on a single day are 0.15,0.22,0.35,0.21, and 0.07.(a) Simulate the inspection team's visits on 30 days.(b)
Depending on the availability of parts, a company can manufacture 3, 4, 5, or 6 units of a certain item per week with corresponding probabilities of 0.10,0.40, 0.30, and 0.20. The probabilities
A manufacturer of smart phones has the following probability distribution for the number of defects per phone:(a) Determine the probability of 2 or more defects.(b) Is a randomly selected phone more
Upon reviewing recent use of conference rooms at an engineering consulting firm, an industrial engineer determined the following probability distribution for the number of requests for a conference
Refer to Exercise 4.80 and obtain the(a) mean;(b) variance;(c) standard deviation for the number of requests for conference rooms.Data From Exercise 4.80 4.80 Upon reviewing recent use of conference
Determine whether the following can be probability distributions of a random variable that can take on only the values of 0,1 , and 2 :(a) $f(0)=0.34 \quad f(1)=0.34$ and $f(2)=0.34$.(b)
Check whether the following can define probability distributions, and explain your answers.(a) $f(x)=\frac{x}{10}$, for $x=0,1,2,3,4$.(b) $f(x)=\frac{1}{3}$, for $x=-1,0,1$.(c)
An engineering student correctly answers $85 \%$ of all questions she attempts. What is the probability that the first incorrect answer was the fourth one?
If the probability is 0.20 that a downtime of an automated production process will exceed 2 minutes, find the probability that 3 of 8 downtimes of the process will exceed 2 minutes using (a) the
If the probability is 0.90 that a new machine will produce 40 or more chairs, find the probabilities that among 16 such machines(a) 12 will produce 40 or more chairs;(b) at least 10 will produce 40
In 16 experiments studying the electrical behavior of single cells, 12 use micro-electrodes made of metal and the other 4 use micro-electrodes made from glass tubing. If 2 of the experiments are to
As can be easily verified by means of the formula for the binomial distribution, the probabilities of getting 0 , 1,2 , or 3 heads in 3 flips of a coin whose probability of heads is 0.4 are
With reference to Exercise 4.88, find the variance of the probability distribution using(a) the formula that defines $\sigma^{2}$;(b) the special formula for the variance of a binomial
Find the mean and the standard deviation of the distribution of each of the following random variables (having binomial distributions):(a) The number of heads in 440 flips of a balanced coin.(b) The
Use the Poisson distribution to approximate the binomial probability $b(1 ; 100,0.02)$.
With reference to Exercise 4.87, find the mean and the variance of the distribution of the number of microelectrodes made from glass tubing using(a) the probabilities obtained in that exercise;(b)
The daily number of orders filled by the parts department of a repair shop is a random variable with $\mu=142$ and $\sigma=12$. According to Chebyshev's theorem, with what probability can we
Records show that the probability is 0.00008 that a truck will have an accident on a certain highway. Use the formula for the Poisson distribution to approximate the probability that at least 5 of
The number of weekly breakdowns of a computer is a random variable having a Poisson distribution with $\lambda=0.2$. What is the probability that the computer will operate without a breakdown for 3
A manufacturer determines that a big screen HDTV set had probabilities of $0.8,0.15,0.05$, respectively, of being placed in the categories acceptable, minor defect, or major defect. If 3 HDTVs are
Suppose that the probabilities are $0.2466,0.3452$, $0.2417,0.1128,0.0395,0.0111,0.0026$, and 0.0005 that there will be $0,1,2,3,4,5,6$, or 7 polluting spills in the Great Lakes on any one day.
A candidate invited for a visit has probability 0.6 of being hired. Let $X$ be the number of candidates that visit before 2 are hired. Find(a) $P(X \leq 4)$;(b) $P(X \geq 5)$.
If among $n$ objects $k$ are alike and the others are all distinct, the number of permutations of these $n$ objects taken all together is $n ! / k !$.(a) How many permutations are there of
Determine the number of ways in which a software professional can choose 4 of 25 laptops to test a newly designed application.
How many ways can a company select 4 candidates to interview from a short list of 12 engineers?
A box of 15 spark plugs contains one that is defective. In how many ways can 4 spark plugs be selected so that(a) the defective one is selected;(b) the defective plug is not selected?
With reference to Exercise 3.25, suppose that three of the spark plugs are defective. In how many ways can 4 spark plugs be selected so that(a) one of the defective plugs is selected;(b) two of the
An engineering student has 6 different ball bearings and 9 different gears. In how many ways can 3 ball bearings and 3 gears be selected for an experiment on friction in machine parts?
(a) Among 880 smart phones sold by a retailer, 72 required repairs under the warranty. Estimate the probability that a new phone, which has just been sold, will require repairs under the warranty.
A car rental agency has 19 compact cars and 12 intermediate-size cars. If four of the cars are randomly selected for a safety check, what is the probability of getting two of each kind?
With reference to Exercise 3.34, suppose that the questionnaire filled in by one of the 150 persons is to be double-checked. If it is chosen in such a way that each questionnaire has a probability of
The probabilities that a TV station will receive $0,1,2,3, \ldots, 8$ or at least 9 complaints after showing a controversial program are, respectively,
A rotary plug valve needs to be replaced to repair a machine, and the probabilities that the replacement will be a flange style (low pressure), flange style (high pressure), wafer style, or lug style
If each point of the sample space of Figure 3.12 represents an outcome having the probability find, 32
The probability that a turbine will have a defective coil is 0.10, the probability that it will have defective blades is 0.15, and the probability that it will have both defects is 0.04.(a) What is
Given $P(A)=0.30, P(B)=0.62$, and $P(A \cap B)=$ 0.12 , find(a) $P(A \cup B)$;(b) $P(\bar{A} \cap B)$;(c) $P(A \cap \bar{B})$;(d) $P(\bar{A} \cup \bar{B})$.
The probability that a construction company will get the tender for constructing a flyover is 0.33, the probability that it will get the tender for constructing an underpass is 0.28, and the
It can be shown that for any three events $A, B$, and $C$, the probability that at least one of them will occur is given by\[\begin{aligned}P(A \cup B \cup C)= & P(A)+P(B)+P(C) \\& -P(A \cap
Suppose that in the maintenance of a large medicalrecords file for insurance purposes the probability of an error in processing is 0.0010, the probability of an error in filing is 0.0009, the
If the probability of event A is p, then the odds that it will occur are given by the ratio of p to 1−p. Odds are usually given as a ratio of two positive integers having no common factor, and if
When we roll a pair of balanced dice, what are the probabilities of getting(a) 7 ;(b) 11 ;(c) 7 or 11 ;(d) 3 ;(e) 2 or 12 ;(f) 2,3 , or 12 ?
The registration numbers for the candidates of an entrance test are numbered from 000001 to 200000. What is the probability that a candidate will get a registration number divisible by 40 ?
In a group of 160 graduate engineering students, 92 are enrolled in an advanced course in statistics, 63 are enrolled in a course in operations research, and 40 are enrolled in both. How many of
With reference to Exercise 3.1, suppose that the points $(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)$, $(2,0),(2,1),(2,2),(2,3),(3,0),(3,1),(3,2)$, and $(3,3)$ have the probabilities
An experiment has the four possible mutually exclusive outcomes $A, B, C$, and $D$. Check whether the following assignments of probability are permissible:(a) \(P(A)=0.38, P(B)=0.16, P(C)=0.11,
With reference to Exercise 3.7, suppose that each point $(i, j)$ of the sample space is assigned the probability $\frac{420 / 401}{2(i+j)}$.(a) Verify that this assignment of probabilities is
Explain why there must be a mistake in each of the following statements:(a) The probability that a mineral sample will contain silver is 0.38 and the probability that it will not contain silver is
If $A$ and $B$ are mutually exclusive events, $P(A)=0.45$, and $P(B)=0.30$, find(a) $P(\bar{A})$;(b) $P(A \cup B)$;(c) $P(A \cap \bar{B})$;(d) $P(\bar{A} \cap \bar{B})$.
An article in a civil engineering magazine asks "How Strong Are the Pillars of Our Overhead Bridges?' and goes on to say that samples were collected of materials being used in the construction of 294
A television channel announced a vote for their viewers' favorite television show. Viewers were asked to visit the channel's website and vote online for their favorite show. Identify the population
Consider the population of all cars owned by women in your neighborhood. You want to know the model of the car.(a) Specify the population unit.(b) Specify the variable of interest.(c) Specify the
Identify the statistical population, sample, and variable of interest in each of the following situations:(a) Tensile strength is measured on 20 specimens of super strength thread made of the same
For ceiling fans to rotate effectively, the bending angle of the individual paddles of the fan must remain between tight limits. From each hour's production, 25 fans are selected and the angle is
Ten seniors have applied to be on the team that will build a high-mileage car to compete against teams from other universities. Use Table 7 of random digits to select 5 of the 10 seniors to form the
Refer to the slot depth data in Table 1.1. After the machine was repaired, a sample of three new ceramic parts had slot depths 215, 216, and 213 (thousandths of an inch).(a) Redraw the $X$-bar chart
A Canadian manufacturer identified a critical diameter on a crank bore that needed to be maintained within a close tolerance for the product to be successful. Samples of size 4 were taken every hour.
Losses at an oil refinery (in millions of dollars) due to excess heat can be divided according to the reason behind the generation of excessive heat.oversupplying fuel202excess air124carelessness of
Tests were conducted to measure the running temperature for engines (in ${ }^{\circ} \mathrm{F}$ ). A sample of 15 tests yielded the temperature
To determine the strengths of various detergents, the following are 20 measurements of the total dissolved salts (parts per million) in
Civil engineers help municipal wastewater treatment plants operate more efficiently by collecting data on the quality of the effluent. On seven occasions, the amounts of suspended solids (parts per
A dam on a river holds water in its reservoir to generate electricity. Because the dam is in a rainforest area, the flow of water is highly uncertain. In December last year, the overflow from the
Physicists first observed neutrinos from a supernova that occurred outside of our solar system when the detector near Kamiokande, Japan, recorded twelve arrivals. The times(seconds) between the
The power generated (MW) by liquid hydrogen turbo pumps, given to the nearest tenth, is grouped into a table having the classes [40.0, 45.0), [45.0, 50.0), [50.0,55.0),[55.0,60.0) and [60.0,65.0),
With reference to the preceding exercise, is it possible to determine from the grouped data how many turbo pumps have a power generation of(a) more than 50.0?(b) less than 50.0?(c) at most 60.0?(d)
To continually increase the speed of computers, electrical engineers are working on ever-decreasing scales.The size of devices currently undergoing development is measured in nanometers
Convert the distribution obtained in the preceding exercise into a cumulative "less than or equal to" distribution and graph its ogive.
The following are the sizes of particles of cement dust (given to the nearest hundredth of a micron) in a cement
Convert the distribution obtained in Exercise 2.12 into a cumulative "less than" distribution and plot its ogive.Data From Exercise 2.12 2.12 The following are the sizes of particles of cement dust
An engineer uses a thermocouple to monitor the temperature of a stable reaction. The ordered values of 50 observations (Courtesy of Scott Sanders), in tenths of ${ }^{\circ} \mathrm{C}$, areGroup
Convert the distribution obtained in Exercise 2.14 into a cumulative "less than" distribution and plot its ogive.Data From Exercise 2.14 2.14 An engineer uses a thermocouple to monitor the tem-
The following are the number of transistors failing a quality check per hour during 72 observed hours of production:246812185461018234125118219142568171491824118553091971887772712735885990Group these
Given a set of observations $x_{1}, x_{2}, \ldots, x_{n}$, we define their empirical cumulative distribution as the function whose values $F(x)$ equals the proportion of the observations less than or
Referring to Exercise 2.17, graph the empirical cumulative distribution for the data in Exercise 2.16.Data From Exercise 2.17Data From Exercise 2.16Data From Exercise 2.3 2.17 Given a set of
The pictogram of Figure 2.11 is intended to illustrate the fact that per capita income in the United States doubled from $\$ 21,385$ in 1993 to $\$ 42,643$ in 2012. Does this pictogram convey a fair
Categorical distributions are often presented graphically by means of pie charts, in which a circle is divided into sectors proportional in size to the frequencies (or percentages) with which the
Convert the distribution of nanopillar heights on page 26 into a distribution having the classes (205, 245],(245,325],(325,365],(365,405], where the right-hand endpoint is included. Draw two
List the data that correspond to the following stems of stem-and-leaf displays: (b) 62 | 35589 (a) 4 011257 Leaf unit = 1.0 Leaf unit = 1.0 (c) 8 | 01 23 62 91 First leaf digit unit = 10.0 (d) 2.28 |
To construct a stem-and-leaf display with more stems than there would be otherwise, we might repeat each stem. The leaves $0,1,2,3$, and 4 would be attached to the first stem and leaves $5,6,7,8$,
If the double-stem display has too few stems, we create 5 stems where the first holds leaves 0 and 1 , the second holds 2 and 3, and so on. The resulting stem-and-leaf display is called a five-stem
In each of the following situations, should your value be near the average or an outlier? If an outlier, should it be too large or too small?(a) Income on your starting job(b) Your score on the final
In each of the following situations, should your value be near the average or an outlier? If outlier, should it be too large or too small?(a) The time you take to complete a lab assignment next
Is the influence of a single outlier greater on the mean or the median? Explain.
Is the influence of a single outlier greater on the sample range or the interquartile range? Explain.
Referring to Exercise 1.8 in Chapter 1, we see that the sample of 4 deviations (observation - specification) during the second hour for a critical crank-bore diameter is\[\begin{array}{llll}-6 &
At the end of 2012, nine skyscrapers in the world were over 300 meters tall. The ordered values of height are$\begin{array}{lllllllll}366 & 381 & 442 & 452 & 484 & 492 & 508 & 601 &
Engineers are developing a miniaturized robotic capsule for exploration of a human gastrointestinal tract. One novel solution uses motor-driven legs. The engineers' best design worked for a few
A contract for the maintenance of a leading manufacturer's computers was given to a team of specialists. After six months, the supervisor of the team felt that computer performance could be improved

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