 4.1E: Let Y be a random variable with p(y) given in the table below. a Gi...
 4.2E: A box contains five keys, only one of which will open a lock. Keys ...
 4.3E: A Bernoulli random variable is one that assumes only two values, 0 ...
 4.4E: Let Y be a binomial random variable with n = 1 and success probabil...
 4.5E: Suppose that Y is a random variable that takes on only integer valu...
 4.6E: Consider a random variable with a geometric distribution (Section 3...
 4.7E: Let Y be a binomial random variable with n = 10 and p = .2.a Use Ta...
 4.8E: Suppose that Y has density function a Find the value of k that make...
 4.9E: A random variable Y has the following distribution function: a Is Y...
 4.10E: Refer to the density function given in Exercise 4.8.a Find the .95...
 4.11E: Suppose that Y possesses the density function a Find the value of c...
 4.12E: The length of time to failure (in hundreds of hours) for a transist...
 4.13E: A supplier of kerosene has a 150gallon tank that is filled at the ...
 4.14E: A gas station operates two pumps, each of which can pump up to 10,0...
 4.15E: As a measure of intelligence, mice are timed when going through a m...
 4.16E: Let Y possess a density function a Find c.b Find F(y).c Graph f (y)...
 4.17E: The length of time required by students to complete a onehour exam...
 4.18E: Let Y have the density function given by a Find c.b Find F(y).c Gra...
 4.19E: Let the distribution function of a random variable Y be a Find the ...
 4.20E: If, as in Exercise 4.16, Y has density function find the mean and v...
 4.21E: If, as in Exercise 4.17, Y has density function find the mean and v...
 4.22E: If, as in Exercise 4.18, Y has density function find the mean and v...
 4.23E: Prove Theorem 4.5.Reference
 4.24E: If Y is a continuous random variable with density function f (y), u...
 4.25E: If, as in Exercise 4.19, Y has distribution function find the mean ...
 4.26E: If Y is a continuous random variable with mean ? and variance ? 2 a...
 4.27E: For certain ore samples, the proportion Y of impurities per sample ...
 4.28E: The proportion of time per day that all checkout counters in a supe...
 4.29E: The temperature Y at which a thermostatically controlled switch tur...
 4.30E: The proportion of time Y that an industrial robot is in operation d...
 4.31E: Daily total solar radiation for a specified location in Florida in ...
 4.32E: Weekly CPU time used by an accounting firm has probability density ...
 4.33E: The pH of water samples from a specific lake is a random variable Y...
 4.34E: Suppose that Y is a continuous random variable with density f (y) t...
 4.35E: If Y is a continuous random variable such that E[(Y ?a)2] < ? for a...
 4.36E: Is the result obtained in Exercise 4.35 also valid for discrete ran...
 4.37E: If Y is a continuous random variable with density function f (y) th...
 4.38E: Suppose that Y has a uniform distribution over the interval (0, 1)....
 4.39E: If a parachutist lands at a random point on a line between markers ...
 4.40E: Suppose that three parachutists operate independently as described ...
 4.41E: A random variable Y has a uniform distribution over the interval (?...
 4.42E: The median of the distribution of a continuous random variable Y is...
 4.43E: A circle of radius r has area A = ? r 2. If a random circle has a r...
 4.44E: The change in depth of a river from one day to the next, measured (...
 4.45E: Upon studying low bids for shipping contracts, a microcomputer manu...
 4.46E: Refer to Exercise 4.45. Find the expected value of low bids on cont...
 4.47E: The failure of a circuit board interrupts work that utilizes a comp...
 4.48E: If a point is randomly located in an interval (a, b) and if Y denot...
 4.49E: A telephone call arrived at a switchboard at random within a onemi...
 4.50E: Beginning at 12:00 midnight, a computer center is up for one hour a...
 4.51E: The cycle time for trucks hauling concrete to a high way constructi...
 4.52E: Refer to Exercise 4.51. Find the mean and variance of the cycle tim...
 4.53E: The number of defective circuit boards coming off a soldering machi...
 4.54E: In using the triangulation method to determine the range of an acou...
 4.55E: Refer to Exercise 4.54. Suppose that measurement errors are uniform...
 4.56E: Refer to Example 4.7. Find the conditional probability that a custo...
 4.57E: According to Zimmels (1983), the sizes of particles used in sedimen...
 4.58E: Use Table 4, Appendix 3, to find the following probabilities for a ...
 4.59E: If Z is a standard normal random variable, find the value z0 such t...
 4.60E: A normally distributed random variable has density function Using t...
 4.61E: What is the median of a normally distributed random variable with m...
 4.62E: If Z is a standard normal random variable, what isa P ( Z 2 < 1)?b ...
 4.63E: A company that manufactures and bottles apple juice uses a machine ...
 4.64E: The weekly amount of money spent on maintenance and repairs by a co...
 4.65E: In Exercise 4.64, how much should be budgeted for weekly repairs an...
 4.66E: A machining operation produces bearings with diameters that are nor...
 4.67E: In Exercise 4.66, what should the mean diameter be in order that th...
 4.68E: The grade point averages (GPAs) of a large population of college st...
 4.69E: Refer to Exercise 4.68. If students possessing a GPA less than 1.9 ...
 4.70E: Refer to Exercise 4.68. Suppose that three students are randomly se...
 4.71E: Wires manufactured for use in a computer system are specified to ha...
 4.72E: One method of arriving at economic forecasts is to use a consensus ...
 4.73E: The width of bolts of fabric is normally distributed with mean 950 ...
 4.74E: Scores on an examination are assumed to be normally distributed wit...
 4.75E: A softdrink machine can be regulated so that it discharges an aver...
 4.76E: The machine described in Exercise 4.75 has standard deviation ? tha...
 4.77E: The SAT and ACT college entrance exams are taken by thousands of st...
 4.78E: Show that the maximum value of the normal density with parameters ?...
 4.79E: Show that the normal density with parameters ? and ? has inflection...
 4.80E: Assume that Y is normally distributed with mean ? and standard devi...
 4.81E:
 4.82E: Reference
 4.83E: Applet Exercise Use the applet Comparison of Gamma Density Function...
 4.84E: Applet Exercise Refer to Exercise 4.83. Use the applet Comparison o...
 4.85E: Applet Exercise Use the applet Comparison of Gamma Density Function...
 4.86E: Applet Exercise When we discussed the ? 2 distribution in this sect...
 4.87E: Applet Exercise Let Y and W have the distributions given in Exercis...
 4.88E: The magnitude of earthquakes recorded in a region of North America ...
 4.89E: If Y has an exponential distribution and P(Y > 2) = .0821, what isa...
 4.90E: Refer to Exercise 4.88. Of the next ten earthquakes to strike this ...
 4.91E: The operator of a pumping station has observed that demand for wate...
 4.92E: The length of time Y necessary to complete a key operation in the c...
 4.93E: Historical evidence indicates that times between fatal accidents on...
 4.94E: Onehour carbon monoxide concentrations in air samples from a large...
 4.95E: Let Y be an exponentially distributed random variable with mean ?. ...
 4.96E: Suppose that a random variable Y has a probability density function...
 4.97E: A manufacturing plant uses a specific bulk product. The amount of p...
 4.98E: Consider the plant of Exercise 4.97. How much of the bulk product s...
 4.99E: If ? > 0 and ? is a positive integer, the relationship between inco...
 4.100E: Let Y be a gammadistributed random variable where ? is a positive ...
 4.101E: Applet Exercise Refer to Exercise 4.88. Suppose that the magnitude ...
 4.102E: Applet Exercise Refer to Exercise 4.97. Suppose that the amount of ...
 4.103E: Explosive devices used in mining operations produce nearly circular...
 4.104E: The lifetime (in hours) Y of an electronic component is a random va...
 4.105E: Fourweek summer rainfall totals in a section of the Midwest United...
 4.106E: The response times on an online computer terminal have approximatel...
 4.107E: Refer to Exercise 4.106.a Use Tchebysheffâ€™s theorem to give an inte...
 4.108E: Annual incomes for heads of household in a section of a city have a...
 4.109E: The weekly amount of downtime Y (in hours) for an industrial machin...
 4.110E: If Y has a probability density function given by obtain E(Y ) and V...
 4.111E: Suppose that Y has a gamma distribution with parameters ? and ?.a I...
 4.112E: Suppose that Y has a ? 2 distribution with ? degrees of freedom. Us...
 4.113E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.114E: Applet Exercise Refer to Exercise 4.113. Use the applet Comparison ...
 4.115E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.116E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.117E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.118E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.119E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.120E: In Chapter 6 we will see that if Y is beta distributed with paramet...
 4.121E: Applet Exercise Use the applet Comparison of Beta Density Functions...
 4.122E: Applet Exercise Beta densities with ? < 1 and ? < 1 are difficult t...
 4.123E: The relative humidity Y, when measured at a location, has a probabi...
 4.124E: The percentage of impurities per batch in a chemical product is a r...
 4.125E: Refer to Exercise 4.124. Find the mean and variance of the percenta...
 4.126E: Suppose that a random variable Y has a probability density function...
 4.127E: Verify that if Y has a beta distribution with ? = ? = 1, then Y has...
 4.128E: The weekly repair cost Y for a machine has a probability density fu...
 4.129E: During an eighthour shift, the proportion of time Y that a sheetm...
 4.130E: Prove that the variance of a betadistributed random variable with ...
 4.131E: Errors in measuring the time of arrival of a wave front from an aco...
 4.132E: Proper blending of fine and coarse powders prior to copper sinterin...
 4.133E: The proportion of time per day that all checkout counters in a supe...
 4.134E: In the text of this section, we noted the relationship between the ...
 4.135E: Suppose that Y1 and Y2 are binomial random variables with parameter...
 4.136E: Suppose that the waiting time for the first customer to enter a ret...
 4.137E: Show that the result given in Exercise 3.158 also holds for continu...
 4.138E: Example 4.16 derives the momentgenerating function for Y ? ?, wher...
 4.139E: The momentgenerating function of a normally distributed random var...
 4.140E: Identify the distributions of the random variables with the followi...
 4.141E: If ?1 < ?2, derive the momentgenerating function of a random varia...
 4.142E: Refer to Exercises 4.141 and 4.137. Suppose that Y is uniformly dis...
 4.143E: The momentgenerating function for the gamma random variable is der...
 4.144E: Consider a random variable Y with density function given by a Find ...
 4.145E: A random variable Y has the density function a Find E(e3Y /2).b Fin...
 4.146E: A manufacturer of tires wants to advertise a mileage interval that ...
 4.147E: A machine used to fill cereal boxes dispenses, on the average, ? ou...
 4.148E: Find P(Y ? ? ? 2?) for Exercise 4.16. Compare with the correspond...
 4.149E: Find P(Y ? ? ? 2?) for the uniform random variable. Compare with ...
 4.150E: Find P(Y ? ? ? 2?) for the exponential random variable. Compare w...
 4.151E: Refer to Exercise 4.92. Would you expect C to exceed 2000 very ofte...
 4.152E: Refer to Exercise 4.109. Find an interval that will contain L for a...
 4.153E: Refer to Exercise 4.129. Find an interval for which the probability...
 4.154E: Suppose that Y is a ? 2 distributed random variable with ? = 7 degr...
 4.155E: A builder of houses needs to order some supplies that have a waitin...
 4.156E: The duration Y of longdistance telephone calls (in minutes) monito...
 4.157E: The life length Y of a component used in a complex electronic syste...
 4.158E: Consider the nailfiring device of Example 4.15. When the device wo...
 4.159E: A random variable Y has distribution function
 4.160SE: Let the density function of a random variable Y be given by a Find ...
 4.161SE: The length of time required to complete a college achievement test ...
 4.162SE: A manufacturing plant utilizes 3000 electric light bulbs whose leng...
 4.163SE: Refer to Exercise 4.66. Suppose that five bearings are randomly dra...
 4.164SE: The length of life of oildrilling bits depends upon the types of r...
 4.165SE: Let Y have density function a Find the value of c that makes f (y) ...
 4.166SE: Use the fact that to expand the momentgenerating function of Examp...
 4.167SE: Find an expression for = E(Y k ), where the random variable Y has a...
 4.168SE: The number of arrivals N at a supermarket checkout counter in the t...
 4.169SE: An argument similar to that of Exercise 4.168 can be used to show t...
 4.170SE: Refer to Exercise 4.168.a If U is the time until the second arrival...
 4.171SE: Suppose that customers arrive at a checkout counter at a rate of tw...
 4.172SE: Calls for dialin connections to a computer center arrive at an ave...
 4.173SE: Suppose that plants of a particular species are randomly dispersed ...
 4.174SE: The time (in hours) a manager takes to interview a job applicant ha...
 4.175SE: The median value y of a continuous random variable is that value su...
 4.176SE: If Y has an exponential distribution with mean ?, find (as a functi...
 4.177SE: Applet Exercise Use the applet Gamma Probabilities and Quantiles to...
 4.178SE: Graph the beta probability density function for ? = 3 and ? = 2.a I...
 4.179SE: A retail grocer has a daily demand Y for a certain food sold by the...
 4.180SE: Suppose that Y has a gamma distribution with ? = 3 and ? = 1.a Use ...
 4.181SE: Suppose that Y is a normally distributed random variable with mean ...
 4.182SE: A random variable Y is said to have a lognormal distribution if X ...
 4.183SE: If Y has a lognormal distribution with parameters ? and ? 2, it ca...
 4.184SE: Let Y denote a random variable with probability density function gi...
 4.185SE: Let f1(y) and f2(y) be density functions and let a be a constant su...
 4.186SE: The random variable Y, with a density function given by is said to ...
 4.187SE: Refer to Exercise 4.186. Resistors used in the construction of an a...
 4.188SE: Refer to Exercise 4.186.a What is the usual name of the distributio...
 4.189SE: If n > 2 is an integer, the distribution with density given by is c...
 4.190SE: A function sometimes associated with continuous nonnegative random ...
 4.191SE: Suppose that Y is a continuous random variable with distribution fu...
 4.192SE: The velocities of gas particles can be modeled by the Maxwell distr...
 4.193SE: Because has the properties of a distribution function, its derivati...
 4.194SE: We can show that the normal density function integrates to unity by...
 4.195SE: Let Z be a standard normal random variable and W = (Z 2 + 3Z )2.a U...
 4.196SE: Show that by writing by making the transformation y = (1/2)x2 and b...
 4.197SE: The function B(?, ?) is defined by
 4.198SE: The Markov Inequality Let g(Y ) be a function of the continuous ran...
 4.199SE: Let Z be a standard normal random variable.
 4.200SE: Suppose that Y has a beta distribution with parameters ? and ?.a If...
Solutions for Chapter 4: Mathematical Statistics with Applications 7th Edition
Full solutions for Mathematical Statistics with Applications  7th Edition
ISBN: 9780495110811
Solutions for Chapter 4
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811. Since 200 problems in chapter 4 have been answered, more than 259888 students have viewed full stepbystep solutions from this chapter. Chapter 4 includes 200 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

`error (or `risk)
In hypothesis testing, an error incurred by rejecting a null hypothesis when it is actually true (also called a type I error).

Average run length, or ARL
The average number of samples taken in a process monitoring or inspection scheme until the scheme signals that the process is operating at a level different from the level in which it began.

Axioms of probability
A set of rules that probabilities deined on a sample space must follow. See Probability

Block
In experimental design, a group of experimental units or material that is relatively homogeneous. The purpose of dividing experimental units into blocks is to produce an experimental design wherein variability within blocks is smaller than variability between blocks. This allows the factors of interest to be compared in an environment that has less variability than in an unblocked experiment.

Chance cause
The portion of the variability in a set of observations that is due to only random forces and which cannot be traced to speciic sources, such as operators, materials, or equipment. Also called a common cause.

Conditional mean
The mean of the conditional probability distribution of a random variable.

Conditional probability
The probability of an event given that the random experiment produces an outcome in another event.

Conidence interval
If it is possible to write a probability statement of the form PL U ( ) ? ? ? ? = ?1 where L and U are functions of only the sample data and ? is a parameter, then the interval between L and U is called a conidence interval (or a 100 1( )% ? ? conidence interval). The interpretation is that a statement that the parameter ? lies in this interval will be true 100 1( )% ? ? of the times that such a statement is made

Conidence level
Another term for the conidence coeficient.

Critical region
In hypothesis testing, this is the portion of the sample space of a test statistic that will lead to rejection of the null hypothesis.

Crossed factors
Another name for factors that are arranged in a factorial experiment.

Cumulative distribution function
For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.

Design matrix
A matrix that provides the tests that are to be conducted in an experiment.

Discrete uniform random variable
A discrete random variable with a inite range and constant probability mass function.

Distribution free method(s)
Any method of inference (hypothesis testing or conidence interval construction) that does not depend on the form of the underlying distribution of the observations. Sometimes called nonparametric method(s).

Distribution function
Another name for a cumulative distribution function.

Expected value
The expected value of a random variable X is its longterm average or mean value. In the continuous case, the expected value of X is E X xf x dx ( ) = ?? ( ) ? ? where f ( ) x is the density function of the random variable X.

Fisherâ€™s least signiicant difference (LSD) method
A series of pairwise hypothesis tests of treatment means in an experiment to determine which means differ.

Frequency distribution
An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on

Geometric mean.
The geometric mean of a set of n positive data values is the nth root of the product of the data values; that is, g x i n i n = ( ) = / w 1 1 .