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Questions and Answers of
Statistics
Fifty spotlights have just been installed in an outdoor security system. According to the manufacturer's specifications, these particular lights are expected to burn out at the rate of 1.1 per one
Five hundred people are attending the first annual "I was Hit by Lighting" Club. Approximate the probability that at most one of the five hundred was born on Poisson's birthday.
Suppose you want to invent a new superstition that "Bad things come in fours." Using the data given in Case Study 4.2.4 and the type of analysis described on p. 238, calculate the probability that
A chromosome mutation linked with colorblindness is known to occur, on the average, once in every ten thousand births.(a) Approximate the probability that exactly three of the next twenty thousand
Suppose that 1% of all items in a supermarket are not priced properly. A customer buys ten items. What is the probability that she will be delayed by the cashier because one or more of her items
A newly formed life insurance company has underwritten term policies on 120 women between the ages of forty and forty-four. Suppose that each woman has a 1/150 probability of dying during the next
According to an airline industry report (178), roughly 1 piece of luggage out of every 200 that are checked is lost. Suppose that a frequent-flying businesswoman will be checking 120 bags over the
Electromagnetic fields generated by power transmission lines are suspected by some researchers to be a cause of cancer. Especially at risk would be telephone linemen because of their frequent
Astronomers estimate that as many as one hundred billion stars in the Milky Way galaxy are encircled by planets. If so, we may have a plethora of cosmic neighbors. Let p denote the probability that
State Tech's basketball team, the Fighting Logarithms, have a 70% foul-shooting percentage.(a) Write a formula for the exact probability that out of their next one hundred free throws, they will make
A random sample of 747 obituaries published recently in Salt Lake City newspapers revealed that 344 (or 46%) of the decedents died in the three-month period following their birthdays (123). Assess
There is a theory embraced by certain parapsychologists that hypnosis can enhance a person's ESP ability. To test that hypothesis, an experiment was set up with fifteen hypnotized subjects (21). Each
If pX(k) = (10/k)(0.7)k(0.3)10−k, k = 0, 1, . . . , 10, is it appropriate to approximate P(4 ≤ X ≤ 8) by computing the following?Explain.
A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game against the Baltimore Orioles, the last before a long road trip. The ballpark's concession manager is trying
A fair coin is tossed two hundred times. Let Xi = 1 if the ith toss comes up heads and Xi = 0 otherwise, i = 1, 2, . . . , 200; X = Calculate the central limit theorem approximation for P(|X −
Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.
Let X be the amount won or lost in betting $5 on red in roulette. Then px(5) = 18/38 and px(−5) = 20/38. If a gambler bets on red one hundred times, use the central limit theorem to estimate the
If X1, X2, . . . , Xn are independent Poisson random variables with parameters λ1,λ2, . . . , λn, respectively, and if X = X1 + X2 + ・ ・ ・ + Xn, then X is a Poisson random variable with
An electronics firm receives, on the average, fifty orders per week for a particular silicon chip. If the company has sixty chips on hand, use the central limit theorem to approximate the probability
Let Z be a standard normal random variable. Use Appendix Table A.1 to find the numerical value for each of the following probabilities. Show each of your answers as an area under fZ(z). (a) P(0 ≤ Z
Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in 1957. Included as part of the test were some three thousand military and civilian
Econo-Tire is planning an advertising campaign for its newest product, an inexpensive radial. Preliminary road tests conducted by the firm's quality-control department have suggested that the
A large computer chip manufacturing plant under construction in West bank is expected to result in an additional fourteen hundred children in the county's public school system once the permanent
Records for the past several years show that the amount of money collected daily by a prominent televangelist is normally distributed with a mean (μ) of $20,000 and a standard deviation (σ) of
The following letter was written to a well-known dispenser of advice to the lovelorn (171): Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby
A criminologist has developed a questionnaire for predicting whether a teenager will become a delinquent. Scores on the questionnaire can range from 0 to 100, with higher values reflecting a
The cross-sectional area of plastic tubing for use in pulmonary resuscitators is normally distributed withμ = 12.5 mm2 and σ = 0.2 mm2.When the area is less than 12.0 mm2 or greater than 13.0 mm2,
At State University, the average score of the entering class on the verbal portion of the SAT is 565, with a standard deviation of 75. Marian scored a 660. How many of State's other 4250 freshmen did
A college professor teaches Chemistry 101 each fall to a large class of freshmen. For tests, she uses standardized exams that she knows from past experience produce bell-shaped grade distributions
Suppose the random variable Y can be described by a normal curve with μ=40. For what value of σ isP(20 ≤ Y ≤ 60) = 0.50
(a) Let 0 < a < b. Which number is larger?(b) Let a >0. Which number is larger?
It is estimated that 80% of all eighteen-year old women have weights ranging from 103.5 to 144.5 lb. Assuming the weight distribution can be adequately modeled by a normal curve and that 103.5 and
Recall the breath analyzer problem described in Example 4.3.5. Suppose the driver's blood alcohol concentration is actually 0.09% rather than 0.075%. What is the probability that the breath analyzer
If a random variable Y is normally distributed with mean μ and standard deviation σ, the Z ratio Y – μ/σ is often referred to as a normed score: It indicates the magnitude of y relative to the
The IQs of nine randomly selected people are recorded. Let Y denote their average. Assuming the distribution from which the Yi's were drawn is normal with a mean of 100 and a standard deviation of
Let Y1, Y2, . . . , Yn be a random sample from a normal distribution where the mean is 2 and the variance is 4. How large must n be in order thatP(1.9 ≤ Y̅≤ 2.1) ≥ 0.99
A circuit contains three resistors wired in series. Each is rated at 6 ohms. Suppose, however, that the true resistance of each one is a normally distributed random variable with a mean of 6 ohms and
The cylinders and pistons for a certain internal combustion engine are manufactured by a process that gives a normal distribution of cylinder diameters with a mean of 41.5 cm and a standard deviation
Use moment-generating functions to prove the two corollaries to Theorem 4.3.3.
Let Y1, Y2, . . . , Y9 be a random sample of size 9 from a normal distribution where μ = 2 and σ = 2. Let Y∗1, Y∗2, . . . , Y∗9 be an independent random sample from a normal distribution
(a) Evaluate(b) Evaluate
Assume that the random variable Z is described by a standard normal curve fZ(z). For what values of z are the following statements true?(a) P(Z ≤ z) = 0.33(b) P(Z ≥ z) = 0.2236(c) P(−1.00 ≤ Z
Let zα denote the value of Z for which P(Z ≥ zα) = α. By definition, the interquartile range, Q, for the standard normal curve is the difference Q = z.25 − z.75 Find Q.
Oak Hill has 74,806 registered automobiles. A city ordinance requires each to display a bumper decal showing that the owner paid an annual wheel tax of $50. By law, new decals need to be purchased
Hertz Brothers, a small, family-owned radio manufacturer, produces electronic components domestically but subcontracts the cabinets to a foreign supplier. Although inexpensive, the foreign supplier
Fifty-five percent of the registered voters in Sheridanville favor their incumbent mayor in her bid for re-election. If four hundred voters go to the polls, approximate the probability that (a) The
Because of her past convictions for mail fraud and forgery, Jody has a 30% chance each year of having her tax returns audited. What is the probability that she will escape detection for at least
The factorial moment-generating function for any random variable W is the expected value of tw. Moreover dr/dtr E(tW)|t=1 = E[W(W ˆ’ 1)・・・(W ˆ’ r + 1)]. Find the factorial
A teenager is trying to get a driver's license. Write out the formula for the pdf px(k), where the random variable X is the number of tries that he needs to pass the road test. Assume that his
Is the following set of data likely to have come from the geometric pdf pX(k) = (3/4)k−1・(1/4), k = 1, 2, . . .? Explain.
Recently married, a young couple plans to continue having children until they have their first girl. Suppose the probability that a child is a girl is 1 2 , the outcome of each birth is an
Show that the cdf for a geometric random variable is given by FX(t) = P(X ≤ t) = 1 − (1 − p)[t], where [t] denotes the greatest integer in t, t ≥ 0.
Suppose three fair dice are tossed repeatedly. Let the random variable X denote the roll on which a sum of 4 appears for the first time. Use the expression for Fx(t) given in Question 4.4.5 to
Let Y be an exponential random variable, where fY(y) = λe−λy, 0 ≤ y. For any positive integer n, show that P(n ≤ Y ≤ n + 1) = e−λn(1 − e−λ). If p = 1 − e−λ, the "discrete"
Sometimes the geometric random variable is defined to be the number of trials, X, preceding the first success. Write down the corresponding pdf and derive the moment-generating function for X two
Differentiate the moment-generating function for a geometric random variable and verify the expressions given for E(X) and Var(X) in Theorem 4.4.1.
A door-to-door encyclopedia salesperson is required to document five in-home visits each day. Suppose that she has a 30% chance of being invited into any given home, with each address representing an
Suppose that X1, X2, . . . , Xk are independent negative binomial random variables with parameters r1 and p, r2 and p, . . ., and rk and p, respectively. Let X = X1 + X2+・ ・ + Xk. Find MX(t),
An underground military installation is fortified to the extent that it can withstand up to three direct hits from air-to-surface missiles and still function. Suppose an enemy aircraft is armed with
Darryl's statistics homework last night was to flip a fair coin and record the toss, X, when heads appeared for the second time. The experiment was to be repeated a total of one hundred times. The
When a machine is improperly adjusted, it has probability 0.15 of producing a defective item. Each day, the machine is run until three defective items are produced. When this occurs, it is stopped
For a negative binomial random variable whose pdf is given by Equation 4.5.1, find E(X) directly by evaluating pr(1 − p)k−r.
Let the random variable X denote the number of trials in excess of r that are required to achieve the rth success in a series of independent trials, where p is the probability of success at any given
Calculate the mean, variance, and moment generating function for a negative binomial random variable X whose pdf is given by the expression
Let X1, X2, and X3 be three independent negative binomial random variables with pdfsfor i = 1, 2, 3. Define X = X1 + X2 + X3. Find P(10 ≤ X ≤ 12). (Use the moment-generating functions of X1, X2,
Differentiate the moment-generating function MX(t) = [pet/1−(1−p)et]r to verify the formula given in Theorem 4.5.1 for E(X).
An Arctic weather station has three electronic wind gauges. Only one is used at any given time. The lifetime of each gauge is exponentially distributed with a mean of one thousand hours. What is the
Differentiate the gamma moment-generating function to show that the formula for E(Ym) given in Question 4.6.8 holds for arbitrary r > 0.
A service contact on a new university computer system provides twenty-four free repair calls from a technician. Suppose the technician is required, on the average, three times a month. What is the
Suppose a set of measurements Y1, Y2, . . . , Y100 is taken from a gamma pdf for which E(Y) = 1.5 and Var(Y) = 0.75. How many Yi's would you expect to find in the interval [1.0, 2.5]?
Demonstrate that λ plays the role of a scale parameter by showing that if Y is gamma with parameters r and λ, then λY is gamma with parameters r and 1.
Show that a gamma pdf has the unique mode r − 1/λ; that is, show that the function fY(y) = λr/ Г(r)yr−1e−λy takes its maximum value at ymode = r−1/λ and at no other point.
Prove that Г(1/2) = √π. [Consider E(Z2), where Z is a standard normal random variable.]
Show that Г(7/2) = 15/8 √π.
If the random variable Y has the gamma pdf with integer parameter r and arbitrary λ > 0, show that E(Ym) = (m + r −1)!/(r −1)!λm
Differentiate the gamma moment-generating function to verify the formulas for E(Y) and Var(Y) given in Theorem 4.6.3.
A random sample of size 8 – X1 = 1, X2 = 0, X3 = 1, X4 = 1, X5 = 0, X6 = 1, X7 = 1, and X8 = 0 – is taken from the probability function pX(k; θ) = θk(1 − θ)1−k, k = 0, 1; 0 < θ < 1.
(a) Based on the random sample Y1 = 6.3, Y2 = 1.8, Y3 = 14.2, and Y4 = 7.6, use the method f maximum likelihood to estimate the parameter θ in the uniform pdf fY(y; θ) = 1/θ, 0 ≤ y ≤
Find the maximum likelihood estimate for θ in the pdf fY(y; θ) = 2y/1 − θ2, θ ≤ y ≤ 1if a random sample of size 6 yielded the measurements 0.70, 0.63, 0.92, 0.86, 0.43, and 0.21.
A random sample of size n is taken from the pdf fY(y; θ) = 2y/θ2, 0 ≤ y ≤ θ Find an expression for ˆθ, the maximum likelihood estimator for θ.
If the random variable Y denotes an individual's income, Pareto's law claims that P(Y ≥ y) = (k/y)θ, where k is the entire population's minimum income. It follows that FY(y) = 1− (k/y)θ, and,
The exponential pdf is a measure of lifetimes of devices that do not age. However, the exponential pdf is a special case of the Weibull distribution, which measures time to failure of devices where
Suppose a random sample of size n is drawn from a normal pdf where the mean μ is known but the variance σ2 is unknown. Use the method of maximum likelihood to find a formula for ˆσ2. Compare your
Let y1, y2, . . . , yn be a random sample of size n from the pdf fY(y; θ) = 2y/θ2, 0 ≤ y ≤ θ. Find a formula for the method of moments estimate for θ. Compare the values of the method of
Use the method of moments to estimate θ in the pdffY(y; θ) = (θ2 + θ)yθ−1(1 − y), 0 ≤ y ≤ 1Assume that a random sample of size n has been collected.
Find the method of moments estimate for λ if a random sample of size n is taken from the exponential pdf,fY(y; λ) = λe−λy, y ≥ 0.
The number of red chips and white chips in an urn is unknown, but the proportion, p, of reds is either 1/3 or 1/2. A sample of size 5, drawn with replacement, yields the sequence red, white, white,
Suppose that Y1 = 8.3, Y2 = 4.9, Y3 = 2.6, and Y4 = 6.5 is a random sample of size 4 from the two-parameter uniform pdf,fY(y; θ1, θ2) = 1/2θ2, θ1 − θ2 ≤ y ≤ θ1 + θ2Use the method of
Find a formula for the method of moments estimate for the parameter θ in the Pareto pdf,fY(y; θ) = θkθ (1/y)θ+1, y ≥ k; θ ≥ 1Assume that k is known and that the data consist of a
Calculate the method of moments estimate for the parameter θ in the probability functionpX(k; θ) = θk(1 − θ)1−k, k = 0, 1if a sample of size 5 is the set of numbers 0, 0, 1, 0, 1.
Find the method of moments estimates for μ and σ2, based on a random sample of size n drawn from a normal pdf, where μ= E(Y) and σ2 = Var(Y). Compare your answers with the maximum likelihood
Use the method of moments to derive formulas for estimating the parameters r and p in the negative binomial pdf,
Bird songs can be characterized by the number of clusters of "syllables" that are strung together in rapid succession. If the last cluster is defined as a "success," it may be reasonable to treat the
Let y1, y2, . . . , yn be a random sample from the continuous pdf fY(y; θ1, θ2). Let ˆσ2 = 1/n Show that the solutions of the equations E(Y) = y and Var(Y)= ˆσ2 for θ1 and θ2 give the
Use the sample Y1 = 8.2, Y2 = 9.1, Y3 = 10.6, and Y4 = 4.9 to calculate the maximum likelihood estimate for λ in the exponential pdffY(y; λ) = λe−λy, y ≥ 0
Suppose a random sample of size n is drawn from the probability modelpX(k; θ) = θ2ke−θ2/k! , k = 0, 1, 2, . . .Find a formula for the maximum likelihood estimator, ˆθ.
Given that Y1 = 2.3, Y2 = 1.9, and Y3 = 4.6 is a random sample fromfY(y; θ) = y3e−y/θ/6θ4, y ≥ 0calculate the maximum likelihood estimate for θ.
Use the method of maximum likelihood to estimate θ in the pdffY(y; θ) = θ/2√y e−θ√y, y ≥ 0Evaluate θe for the following random sample of size 4: Y1 = 6.2, Y2 = 7.0, Y3 = 2.5, and Y4 =
An engineer is creating a project scheduling program and recognizes that the tasks making up the project are not always completed on time. However, the completion proportion tends to be fairly high.
The following data show the number of occupants in passenger cars observed during one hour at a busy intersection in Los Angeles (69). Suppose it can be assumed that these data follow a geometric
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