Chapter 4

If a typist averages one misspelling in every 3250 words, what are the chances that a 6000 -word report is free of all such errors? Answer the question two waysfirst, by using an exact binomial analysis, and second, by using a Poisson approximation. Does the similarity (or dissimilarity) of the two answers surprise you? Explain.

A medical study recently documented that 905 mistakes were made among the 289,411 prescriptions written during one year at a large metropolitan teaching hospital. Suppose a patient is admitted with a condition serious enough to warrant ten different prescriptions. Approximate the probability that at least one will contain an error.

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 babies born will have the mutation. (b) How many babies out of the next twenty thousand would have to be born with the mutation to convince you that the "one in ten thousand" estimate is too low?

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 require a price check? Calculate both a binomial answer and a Poisson answer. Is the binomial model "exact" in this case? Explain.

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 calendar year, and that each death requires the company to pay out \(\$ 50,000\) in benefits. Approximate the probability that the company will have to pay at least \(\$ 150,000\) in benefits next year.

According to an airline industry report (189), roughly one piece of luggage out of every two hundred that are checked is lost. Suppose that a frequent- flying businesswoman will be checking one hundred twenty bags over the course of the next year. Approximate the probability that she will lose two or more pieces of luggage.

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 any such solar system contains intelligent life. How small can \(p\) be and still give a fifty-fifty chance that we are not alone?

During the latter part of the nineteenth century, Prussian officials gathered information relating to the hazards that horses posed to cavalry soldiers. A total of ten cavalry corps were monitored over a period of twenty years. Recorded for each year and each corps was \(X\), the annual number of fatalities due to kicks. Summarized in the following table are the two hundred values recorded for \(X\) (14). Show that these data can be modeled by a Poisson pdf. Follow the procedure illustrated in Case Studies \(4.2 .2\) and \(4.2 .3 .\) \begin{array}{cc} \hline \text { No. of Deaths, } k & \begin{array}{c} \text { Observed Number of Corps-Years } \\ \text { in Which } k \text { Fatalities Occurred } \end{array} \\ \hline 0 & 109 \\ 1 & 65 \\ 2 & 22 \\ 3 & 3 \\ 4 & \frac{1}{200} \\ \hline \end{array}

A certain species of European mite is capable of damaging the bark on orange trees. The following are the results of inspections done on one hundred saplings chosen at random from a large orchard. The measurement recorded, \(X\), is the number of mite infestations found on the trunk of each tree. Is it reasonable to assume that \(X\) is a Poisson random variable? If not, which of the Poisson model assumptions is likely not to be true?

Assume that the number of hits, \(X\), that a baseball team makes in a nine- inning game has a Poisson distribution. If the probability that a team makes zero hits is \(\frac{1}{3}\), what are their chances of getting two or more hits?

Suppose a radioactive source is metered for two hours, during which time the total number of alpha particles counted is four hundred eighty-two. What is the probability that exactly three particles will be counted in the next two minutes? Answer the question two ways-first, by defining \(X\) to be the number of particles counted in two minutes, and second, by defining \(X\) to be the number of particles counted in one minute.

Suppose that on-the-job injuries in a textile mill occur at the rate of \(0.1\) per day. (a) What is the probability that two accidents will occur during the next (five-day) workweek? (b) Is the probability that four accidents will occur over the next two workweeks the square of your answer to part (a)? Explain.

Let \(X\) be a Poisson random variable with parameter \(\lambda\). Show that the probability that \(X\) is even is \(\frac{1}{2}(1+\) \(\left.e^{-2 \lambda}\right)\).

Let \(X\) and \(Y\) be independent Poisson random variables with parameters \(\lambda\) and \(\mu\), respectively. Example \(3.12 .10\) established that \(X+Y\) is also Poisson with parameter \(\lambda+\mu\). Prove that same result using Theorem \(3.8 .3\).

If \(X_{1}\) is a Poisson random variable for which \(E\left(X_{1}\right)=\lambda\) and if the conditional pdf of \(X_{2}\) given that \(X_{1}=x_{1}\) is binomial with parameters \(x_{1}\) and \(p\), show that the marginal pdf of \(X_{2}\) is Poisson with \(E\left(X_{2}\right)=\lambda p\).

Suppose that commercial airplane crashes in a certain country occur at the rate of \(2.5\) per year. (a) Is it reasonable to assume that such crashes are Poisson events? Explain. (b) What is the probability that four or more crashes will occur next year? (c) What is the probability that the next two crashes will occur within three months of one another?

Records show that deaths occur at the rate of \(0.1\) per day among patients residing in a large nursing home. If someone dies today, what are the chances that a week or more will elapse before another death occurs?

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 hundred hours. What is the expected number of bulbs that will fail to last for at least seventy-five hours?

(a) Let \(00\). Which number is larger? \(\int_{a}^{a+1} \frac{1}{\sqrt{2 \pi}} e^{-z^{2} / 2} d z\) or \(\int_{a-1 / 2}^{a+1 / 2} \frac{1}{\sqrt{2 \pi}} e^{-z^{2} / 2} d z\)

(a) Evaluate \(\int_{0}^{1.24} e^{-z^{2} / 2} d z\). (b) Evaluate \(\int_{-\infty}^{\infty} 6 e^{-z^{2} / 2} d z\).

Assume that the random variable \(Z\) is described by a standard normal curve \(f_{Z}(z)\). For what values of \(z\) are the following statements true? (a) \(P(Z \leq z)=0.33\) (b) \(P(Z \geq z)=0.2236\) (c) \(P(-1.00 \leq Z \leq z)=0.5004\) (d) \(P(-z

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 race ends in a tie. (b) the challenger scores an upset victory

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 between seventy-five and eighty, inclusive. (b) Approximate the probability asked for in part (a)

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 (131). Assess the statistical significance of that finding by approximating the probability that \(46 \%\) or more would die in that particular interval if deaths occurred randomly throughout the year. What would you conclude on the basis of your answer?

If \(p_{X}(k)=\left(\begin{array}{c}10 \\\ k\end{array}\right)(0.7)^{k}(0.3)^{10-k}, k=0,1, \ldots, 10\), is it appropriate to approximate \(P(4 \leq X \leq 8)\) by computing the following? $$ P\left[\frac{3.5-10(0.7)}{\sqrt{10(0.7)(0.3)}} \leq Z \leq \frac{8.5-10(0.7)}{\sqrt{10(0.7)(0.3)}}\right] $$ Explain.

A fair coin is tossed two hundred times. Let \(X_{i}=1\) if the \(i\) th toss comes up heads and \(X_{i}=0\) otherwise, \(i=1,2, \ldots, 200 ; X=\sum_{i=1}^{200} X_{i}\). Calculate the Central Limit Theorem approximation for \(P(|X-E(X)| \leq 5)\). How does this differ from the DeMoivre-Laplace approximation?

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 \(p_{x}(5)=\frac{18}{38}\) and \(p_{x}(-5)=\frac{20}{38}\). If a gambler bets on red one hundred times, use the Central Limit Theorem to estimate the probability that those wagers result in less than \(\$ 50\) in losses.

Suppose \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) are independent Poisson random variables, each with parameter \(\lambda=3\). Let \(S=X_{1}+X_{2}+X_{3}+X_{4}\) (a) Use the Central Limit Theorem to approximate the probability that \(13 \leq S \leq 14\). (b) Calculate the exact probability that \(13 \leq S \leq 14\).

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 that they will be unable to fill all their orders for the upcoming week. Assume that weekly demands follow a Poisson distribution.

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 "observers." Now, more than fifty years later, eight cases of leukemia have been diagnosed among those three thousand. The expected number of cases, based on the demographic characteristics of the observers, was three. Assess the sta- tistical significance of those findings. Calculate both an exact answer using the Poisson distribution as well as an approximation based on the Central Limit Theorem.

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 lifetimes of these tires will be normally distributed with an average of thirty thousand miles and a standard deviation of five thousand miles. The marketing division would like to run a commercial that makes the claim that at least nine out of ten drivers will get at least twenty-five thousand miles on a set of EconoTires. Based on the road test data, is the company justified in making that assertion?

A large computer chip manufacturing plant under construction in Westbank is expected to result in an additional fourteen hundred children in the county's public school system once the permanent workforce arrives. Any child with an IQ under 80 or over 135 will require individualized instruction that will cost the city an additional \(\$ 1750\) per year. How much money should Westbank anticipate spending next year to meet the needs of its new special ed students? Assume that IQ scores are normally distributed with a mean \((\mu)\) of 100 and a standard deviation \((\sigma)\) of \(16 .\)

Records for the past several years show that the amount of money collected daily by a prominent televangelist is normally distributed with a mean \((\mu)\) of \(\$ 20,000\) and a standard deviation \((\sigma)\) of \(\$ 5000\). What are the chances that tomorrow's donations will exceed \(\$ 30,000 ?\)

Among the many letters sent to a popular adviceto-the-lovelorn columnist, was one involving a paternity issue that raised an interesting statistical question. The distraught writer-call her "San Diego Reader" - said her husband is in the military and that she got pregnant the last day before he left for an extended tour of duty. Ten months and four days later the baby was born - usually a happy occasion - but her husband, accustomed to pregnancies being nine months long, became obsessed with the possibility that he might not be the child's father. DNA testing was not yet available. The only relevant information known at the time was that pregnancy durations are normally distributed with a mean ( \(\mu\) ) of 266 days and a standard deviation \((\sigma)\) of 16 days. For the benefit of San Diego Reader's husband, how would you associate a probability with a pregnancy lasting 10 months and 4 days? Do you think San Diego Reader is telling the truth?

The cross-sectional area of plastic tubing for use in pulmonary resuscitators is normally distributed with \(\mu=12.5 \mathrm{~mm}^{2}\) and \(\sigma=0.2 \mathrm{~mm}^{2}\). When the area is less than \(12.0 \mathrm{~mm}^{2}\) or greater than \(13.0 \mathrm{~mm}^{2}\), the tube does not fit properly. If the tubes are shipped in boxes of one thousand, how many wrong-sized tubes per box can doctors expect to find?

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 better? Assume that the scores are normally distributed.

It is estimated that \(80 \%\) of all eighteen-year-old women have weights ranging from \(103.5\) to \(144.5 \mathrm{lb}\). Assuming the weight distribution can be adequately modeled by a normal curve and that \(103.5\) and \(144.5\) are equidistant from the average weight \(\mu\), calculate \(\sigma\).

If a random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), the \(Z\) ratio \(\frac{Y-\mu}{\sigma}\) is often referred to as a normed score: It indicates the magnitude of y relative to the distribution from which it came. "Norming" is sometimes used as an affirmative-action mechanism in hiring decisions. Suppose a cosmetics company is seeking a new sales manager. The aptitude test they have traditionally given for that position shows a distinct gender bias: Scores for men are normally distributed with \(\mu=62.0\) and \(\sigma=7.6\), while scores for women are normally distributed with ? = 76.3 and ? = 10.8. Laura and Michael are the two candidates vying for the position: Laura has scored 92 on the test and Michael 75. If the company agrees to norm the scores for gender bias, whom should they hire?

The IQs of nine randomly selected people are recorded. Let \(\bar{Y}\) denote their average. Assuming the distribution from which the \(Y_{i}\) 's were drawn is normal with a mean of 100 and a standard deviation of 16 , what is the probability that \(\bar{Y}\) will exceed 103 ? What is the probability that any arbitrary \(Y_{i}\) will exceed 103 ? What is the probability that exactly three of the \(Y_{i}\) 's will exceed 103 ?

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) 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 that $$ P(1.9 \leq \bar{Y} \leq 2.1) \geq 0.99 $$

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 a standard deviation of \(0.3\) ohm. What is the probability that the combined resistance will exceed 19 ohms? How "precise" would the manufacturing process have to be to make the probability less than \(0.005\) that the combined resistance of the circuit would exceed 19 ohms?

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 \(41.5 \mathrm{~cm}\) and a standard deviation of \(0.4 \mathrm{~cm}\). Similarly, the distribution of piston diameters is normal with a mean of \(40.5 \mathrm{~cm}\) and a standard deviation of \(0.3 \mathrm{~cm}\). If the piston diameter is greater than the cylinder diameter, the former can be reworked until the two "fit." What proportion of cylinder-piston pairs will need to be reworked?

The town of Willoughby has just been selected as the site for a new automobile assembly complex. City officials estimate that the anticipated influx of jobseeking families will add some four hundred students to Willoughby's public high schools. (a) What family of random variables is the number 400 likely to represent, and what is its standard deviation? Explain. (b) Every new high school student in Willoughby is given a diagnostic test whose scores tend to be normally distributed with a mean \((\mu)\) of 200 and a standard deviation \((\sigma)\) of 40 . Any student scoring below 120 is provided with remedial instruction, and those scoring above 290 are offered additional honors courses. What is the probability that a new student qualifies for special instruction? (c) Every student requiring special instruction costs the city an additional \(1500.00. Suppose four hundred new high school students do, in fact, enroll next Fall. The Willoughby School Board has budgeted \)20,400.00 to cover the cost of providing whatever additional instruction they might need. Is that amount likely to be sufficient? Explain.

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 three years? Assume that she exaggerates, distorts, misrepresents, lies, and cheats every year.

A teenager is trying to get a driver's license. Write out the formula for the pdf \(p_{x}(k)\), where the random variable \(X\) is the number of tries that he needs to pass the road test. Assume that his probability of passing the exam on any given attempt is \(0.10\). On the average, how many attempts is he likely to require before he gets his license?

Is the following set of data likely to have come from the geometric pdf \(p x(k)=\left(\frac{3}{4}\right)^{k-1} \cdot\left(\frac{1}{4}\right), k=1,2, \ldots ?\) Explain. \(\begin{array}{llllllllll}2 & 8 & 1 & 2 & 2 & 5 & 1 & 2 & 8 & 3 \\ 5 & 4 & 2 & 4 & 7 & 2 & 2 & 8 & 4 & 7 \\ 2 & 6 & 2 & 3 & 5 & 1 & 3 & 3 & 2 & 5 \\ 4 & 2 & 2 & 3 & 6 & 3 & 6 & 4 & 9 & 3 \\ 3 & 7 & 5 & 1 & 3 & 4 & 3 & 4 & 6 & 2\end{array}\)

Recently married, a young couple plans to continue having children until they have their first girl. Suppose the probability is \(\frac{1}{2}\) that a child is a girl, the outcome of each birth is an independent event, and the birth at which the first girl appears has a geometric distribution. What is the couple's expected family size? Is the geometric pdf a reasonable model here? Discuss.

Show that the cdf for a geometric random variable is given by \(F_{X}(t)=P(X \leq t)=1-(1-p)^{[r]}\), where \([t]\) denotes the greatest integer in \(t, t \geq 0\).

Let \(Y\) be an exponential random variable, where \(f_{Y}(y)=\lambda e^{-\lambda y}, 0 \leq y\). For any positive integer \(n\), show that \(P(n \leq Y \leq n+1)=e^{-\alpha n}\left(1-e^{-\lambda}\right)\). Note that if \(p=1-e^{-\lambda}\), the "discrete" version of the exponential pdf is the geometric pdf.