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introduction to probability statistics

Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions

5.80 Problems with Your New Smartphone? A new study by Square Trade indicates that smartphones are 50% more likely to malfunction than simple phones over a three-year period.14 Of smartphone failures, 30% are related to internal components not working, and overall, there is a 31% chance of having
5.79 Diabetes in Children Insulin-dependent diabetes (IDD) is a common chronic disorder of chil- dren. This disease occurs most frequently in persons of northern European descent, but the incidence ranges from a low of 1-2 cases per 100,000 per year to a high of more than 40 per 100,000 in parts of
5.78 Football Coin Tosses During the 1992 football season, the Los Angeles Rams (now the St. Louis Rams) had a bizarre streak of coin-toss losses. In fact, they lost the call 11 weeks in a row. 12a. The Rams' computer system manager said that the odds against losing 11 straight tosses are 2047 to
5.77 Dominant Traits The alleles for black (B) and white (b) feather colour in chickens show incomplete dominance; individuals with the gene pair Bb have "blue" feathers. When one individual that is homozygous dominant (BB) for this trait is mated with an individual that is homozygous reces- sive
5.76 Plant Genetics A peony plant with red petals was crossed with another plant having streaky petals. The probability that an offspring from this cross has red flowers is 0.75. Let X be the number of plants with red petals resulting from ten seeds from this cross that were collected and
5.75 Plant Density One model for plant competi- tion assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the
5.74 What's for Breakfast? A packaging experiment is conducted by placing two different package designs for a breakfast food side by side on a supermarket shelf. The objective of the experiment is to see whether buyers indicate a preference for one of the two package designs. On a given day, 25
5.73 Probability of Rain Most weather forecasters protect themselves very well by attaching probabili- ties to their forecasts, such as "The probability of rain today is 40%." Then, if a particular forecast is incor- rect, you are expected to attribute the error to the random behaviour of the
5.72 Grey Hair on Campus University campuses are greying! According to a recent article, one in four college students is aged 30 or older. Many of these students are women updating their job skills. Assume that the 25% figure is accurate, that your university is representative of universities at
5.71 Student Fees A student union states that 80% of all students favour an increase in student fees to subsidize a new recreational area. A random sample of n = 25 students produced 15 in favour of increased fees. What is the probability that 15 or fewer in the sample would favour the issue if the
5.70 Psychosomatic Problems A psychiatrist believes that 80% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 25 patients at random to test her theory.a. Assuming that the psychiatrist's theory is true, what is the expected value of X, the number of the
5.69 Reality TV Reality TV (Survivor, Fear Factor, etc.) is a relatively new phenomenon in television programming, with contestants escaping to remote locations, taking dares, breaking world records, or racing across the country. Of those who watch reality TV, 50% say that their favourite reality
5.68 Vacation Homes Approximately 60% of Canadians rank "owning a vacation home nestled on a beach or near a mountain resort" as their number one choice for a status symbol. A sample of n = 400 Canadians is randomly selected.a. What is the average number in the sample who would rank owning a
5.67 Income Splitting According to an Ipsos Reid survey (February 27, 2007), conducted on behalf of CanWest/Global News, most Canadians (77%) are in favour of "income splitting" for couples. 10 Suppose that we randomly select n = 15 Canadians and let x be the number who are in favour of income
5.66 Integers II Refer to Exercise 5.65. Twenty people are asked to select a number from 0 to 9. Eight of them choose a 4, 5, or 6.a. If the choice of any one number is as likely as any other, what is the probability of observing eight or more choices of the numbers 4, 5, or 6?b. What conclusions
5.65 Integers If a person is given the choice of an integer from 0 to 9, is it more likely that he or she will choose an integer near the middle of the sequence than one at either end?a. If the integers are equally likely to be chosen, find the probability distribution for X, the number chosen.b.
5.64 Garbage Collection A city administrator claims that 80% of all people in the city favour garbage collection by contract to a private concern (in contrast to collection by city employees). To check the theory that the proportion of people in the city favouring private collection is 0.8, you
5.63 Cancer Survivor Rates The 10-year survival rate for bladder cancer is approximately 50%. If 20 people who have bladder cancer are properly treated for the disease, what is the probability that:a. At least I will survive for 10 years?b. At least 10 will survive for 10 years?c. At least 15 will
5.62 Coins, continued Refer to Exercise 5.61. Suppose the coin is definitely unbalanced and the probability of a head is equal to p=0.1. Follow the instructions in partsa, b,c, andd. Note that the prob- ability distribution loses its symmetry and becomes skewed when p is not equal to 1/2.
5.61 Tossing a Coin A balanced coin is tossed three times. Let X equal the number of heads observed.a. Use the formula for the binomial probability distri- bution to calculate the probabilities associated with x= 0, 1, 2, and 3.b. Construct the probability distribution.c. Find the mean and standard
5.60 Under what conditions would you use the hypergeometric probability distribution to evaluate the probability of x successes in n trials?
5.59 Under what conditions can the Poisson ran- dom variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?
5.58 List the five identifying characteristics of the binomial experiment.
5.57 Seed Treatments Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large- scale experiment to determine what dilution of the fungicide to apply, five treated seeds and five untreated seeds were planted in clay soil and
5.56 Teaching Credentials In Southern Ontario, a growing number of persons pursuing a teaching credential are choosing paid internships over traditional student teaching programs. A group of eight candidates for three local teaching positions consisted of five candidates who had enrolled in paid
5.55 Gender Bias? A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no pref- erence is given for choosing either gender. Let x equal the number of women chosen to fill the two positions.a. Write the formula for
5.54 Defective Computer Chips A piece of electronic equipment contains six computer chips, two of which are defective. Three computer chips are randomly chosen for inspection, and the number of defective chips is recorded. Find the probability distribution for X, the number of defective computer
5.53 Candy Choices A candy dish contains five blue and three red candies. A child reaches up and selects three candies without looking.a. What is the probability that there are two blue and one red candies in the selection?b. What is the probability that the candies are all red?c. What is the
5.52 Let X be a hypergeometric random variable with N=15, n=3, and M = 4.a. Calculate p(0), p(1), p(2), and p(3).b. Construct the probability histogram for x.c. Use the formulas given in Section 5.4 to calculate and .d. What proportion of the population of measurements fall into the interval (u
5.51 Let X be the number of successes observed in a sample of n = 5 items selected from N = 10. Suppose that, of the N = 10 items, 6 are considered "successes."a. Find the probability of observing no successes.b. Find the probability of observing at least two successes.c. Find the probability of
5.50 Let X be the number of successes observed in a sample of n=4 items selected from a population of N=8. Suppose that of the N = 8 items, 5 are considered "successes."a. Find the probability of observing all successes.b. Find the probability of observing one success.c. Find the probability of
5.49 Evaluate these probabilities: Cic CC Cc a. b. C. C C
An eight-cylinder automobile engine has two misfiring spark plugs. The mechanic removes all four plugs from one side of the engine. What is the probability the two misfiring spark plugs are among those removed?
A rental truck agency services its vehicles on a regular basis, checking for mechanical problems. Suppose that the agency has six moving vans, two of which need to have new brakes. During a routine check, the vans are tested one at a time. 1. What is the probability that the last van with brake
A particular industrial product is shipped in lots of 20. Testing to determine whether each item is defective is costly; hence, the manufacturer samples production rather than using a 100% inspection plan. A sampling plan constructed to minimize the number of defectives shipped to customers calls
A case of wine has 12 bottles, 3 of which contain spoiled wine. A sample of 4 bottles is randomly selected from the case. 1. Find the probability distribution for X, the number of bottles of spoiled wine in the sample. 2. What are the mean and variance of X?
5.48 E. coli Outbreak Increased research and discussion have focused on the number of illnesses involving the organism Escherichia coli (01257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims. Suppose that sporadic outbreaks of E. coli have occurred in
5.47 Bacteria in Water Samples If a drop of water is placed on a slide and examined under a micro- scope, the number X of a particular type of bacteria present has been found to have a Poisson probability distribution. Suppose the maximum permissible count per water specimen for this type of
5.46 Cross-Border Drinking, continued Refer to Exercise 5.45.a. Calculate the mean and standard deviation for X, the number of fatalities per year.b. Within what limits would you expect the number of fatalities per year to fall?
5.45 Cross-Border Drinking Alcohol-related crashes increased in Windsor after bar hours extended, the Windsor Star reported in November 7, 2005. The number of injuries and fatalities from alcohol- related car accidents rose by 45% in Windsor since the Ontario government extended drinking hours to 2
5.44 Intensive Care The number X of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to five people per day.a. What is the probability that the number of people entering the intensive care unit on a particular day
5.43 Airport Safety The increased number of small commuter planes in major airports has heightened concern over air safety. An eastern airport has recorded a monthly average of five near-misses on landings and takeoffs in the past five years.a. Find the probability that during a given month there
5.42 Poisson vs. Binomial II To illustrate how well the Poisson probability distribution approxi- mates the binomial probability distribution, calculate the Poisson approximate values for p(0) and p(1) for a binomial probability distribution with n = 25 and p=0.05. Compare the answers with the
5.41 Poisson vs. Binomial Let X be a binomial random variable with n = 20 and p=0.1.a. Calculate P(X 2) using Table 1 in Appendix I to obtain the exact binomial probability.b. Use the Poisson approximation to calculate P(X 2).c. Compare the results of parts a andb. Is the approxi- mation accurate?
5.40 Let X be a Poisson random variable with mean =2.5. Use Table 2 in Appendix I to calculate these probabilities:a. P(X5)b. P(X>6)c. P(X=2)d. P(1x4)
5.39 Let X be a Poisson random variable with mean => 2. Calculate these probabilities:a. P(X=0)c. P(X>1)b. P(X-1)d. P(X=5)
5.38 Consider a Poisson random variable X with 3. Use Table 2 in Appendix I to find the following probabilities:a. P(X 3)c. P(X=3)b. P(X>3)d. P(3x5)
5.37 Consider a Poisson random variable X with =3. Use the Poisson formula to calculate the following probabilities:a. P(X=0)b. P(X=1)c. P(X>1)
5.36 Consider a Poisson random variable X with =2.5. Use the Poisson formula to calculate the following probabilities:a. P(X=0)c. P(X=2)b. P(X=1)d. P(X 2)
A manufacturer of power lawn mowers buys one-horsepower, two-cycle engines in lots of 1000 from a supplier. She then equips each of the mowers produced by her plant with one of the engines. History shows that the probability of any one engine from that supplier proving unsatisfactory is 0.001. In a
In 2001, an 82-year-old man in Ontario claimed that a convenience store clerk defrauded him of $250,000 by telling him that his ticket was not a winner and keeping it. The story received national attention in 2006 with a news story aired on the CBC. Analysis of Ontario Lottery Commission winning
Suppose a life insurance company insures the lives of 5000 men aged 42. If actuarial studies show the probability that any 42-year-old man will die in a given year to be 0.001, find the exact probability that the company will have to pay X=4 claims during a given year.
Refer to Example 5.11, where we calculated probabilities for a Poisson distribution with u=2 and =4. Use the cumulative Poisson table to find the probabilities of these events: 1. No accidents during a 1-week period 2. At most three accidents during a 2-week period
On February 7, 1976, Darryl Sittler scored 10 points in a hockey game while playing for the Toronto Maple Leafs. Wayne Gretzky has the most points of any hockey player in history, averaging 2.629 points per game in the 394 regular season games played in the five seasons 1981-82 to 1985-86. Goals in
The average number of traffic accidents on a certain section of highway is two per week. Assume that the number of accidents follows a Poisson distribution with =2. 1. Find the probability of no accidents on this section of highway during a 1-week period. 2. Find the probability of at most three
5.35 Less Vegetable Servings Many Canadians report consuming fewer servings of vegetables in the winter than in the summer, according to an Ipsos Reid/Campbell Company of Canada survey. Specifically, almost half of Canadians indicate they consume fewer vegetable servings on a typical winter day
5.34 Taste Test for PTC The taste test for PTC (phenylthiocarbamide) is a favourite exercise for every human genetics class. It has been established that a single gene determines the characteristic, and that 70% of Canadians are "tasters," while 30% are "non-tasters." Suppose that 20 Canadians are
5.33 Fast Food and Gas Stations Forty percent of all Canadians who travel by car look for gas sta- tions and food outlets that are close to or visible from the highway. Suppose a random sample of n = 25 Canadians who travel by car are asked how they determine where to stop for food and gas. Let x
5.32 Pet Peeves Across the board, 22% of car leisure travellers rank "traffic and other drivers" as their pet peeve while travelling. Of car leisure travel- lers in the densely populated U.S. Northeast, 33% list this as their pet peeve. A random sample of n = 8 such travellers in the Northeast were
5.31 Colour Preferences in Mice In a psychol- ogy experiment, a researcher plans to test the colour preference of mice under certain experimental condi- tions. She designs a maze in which the mouse must choose one of two paths, coloured either red or blue, at each of 10 intersections. At the end of
5.30 Whitefly Infestation Suppose that 10% of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected and checked for whitefly.a. What is the average number of fields sampled that are infested with whitefly?b. Within
5.29 Medical Bills II Consider the medical payment problem in Exercise 5.28 in a more realistic setting. Of all patients admitted to the clinic, 30% fail to pay their bills and the debts are eventually forgiven. If the clinic treats 2000 different patients over a period of 1 year, what is the mean
5.28 Medical Bills Records show that 30% of all patients admitted to an alternative medicine clinic fail to pay their bills and that eventually the bills are for- given. Suppose n=4 new patients represent a random selection from the large set of prospective patients served by the clinic. Find these
5.27 O Canada! The National Hockey League (NHL) has 80% of its players born outside the United States, and of those born outside the United States, 50% are born in Canada. Suppose that n = 12 NHL players were selected at random. Let X be the number of players in the sample who were born outside of
5.26 Harry Potter Of all the Harry Potter books purchased in a recent year, about 60% were purchased for readers 14 or older. If 12 Harry Potter fans who bought books that year are surveyed, find the following probabilities.a. At least five of them are 14 or older.b. Exactly nine of them are 14 or
5.25 Car Colours Car colour preferences change over the years and according to the particular model that the customer selects. In a recent year, 10% of all luxury cars sold were black. If 25 cars of that year and type are randomly selected, find the following probabilities:a. At least five cars are
5.24 Blood Types In a certain population, 85% of the people have Rh-positive blood. Suppose that two people from this population get married. What is the probability that they are both Rh-negative, thus making it inevitable that their children will be Rh-negative?
5.23 Security Systems A home security system is designed to have a 99% reliability rate. Suppose that nine homes equipped with this system experience an attempted burglary. Find the probabilities of these events:a. At least one of the alarms is triggered.b. More than seven of the alarms are
5.22 MCAT Scores In 2006 the average com- bined MCAT score (physical science, verbal reason- ing, biological science) for students in Canada was 24.7. Suppose that approximately 45% of all students took this test, and that 100 students are randomly selected from throughout Canada.? Which of the
5.19 Let X be a binomial random variable with n = 20 and p=0.1.a. Calculate P(X4) using the binomial formula.b. Calculate P(X 4) using Table 1 in Appendix I.c. Use the Excel output below to calculate P(X 4). Compare the results of partsa, b, andc. d. Calculate the mean and standard deviation of the
5.18 In Exercise 5.17, the mean and standard devia- tion for a binomial random variable were calculated for a fixed sample size, n = 100, and for different values of p. Graph the values of the standard deviation for the five values of p given in Exercise 5.17. For what value of p does the standard
5.17 Find the mean and standard deviation for a binomial distribution with n = 100 and these values of p:a. p=0.01d. p=0.7b. p=0.9e. p=0.5c. p=0.3
5.16 Find the mean and standard deviation for a binomial distribution with these values:a. n = 1000, p = 0.3c. n=500, p=0.5b. n=400, p=0.01d. n 1600, p 0.8
5.15 Use Table 1 in Appendix I to find the following: a. P(X
5.14 Find P(X k) in each case:a. n=20, p=0.05, k=2b. n 15, p=0.7, k=8 -c. n 10, p=0.9, k=9
5.13 Use Table 1 in Appendix I to evaluate the following probabilities for n=6 and p=0.8:a. P(X4)c. P(X 1) Verify these answers using the values of p(x) calcu- lated in Exercise 5.9.
5.12 Use Table 1 in Appendix I to find the sum of the binomial probabilities from X=0 to X=k for these cases:a. n 10, p=0.1, k=3b. n 15, p=0.6, k=7c. n=25, p=0.5, k=14
5.11 Let X be a binomial random variable with n = 10 and p 0.4. Find these values:a. P(X=4)d. P(X 4)b. P(X4)c. P(X>4)e. = npf. = Vnpq
5.10 If X has a binomial distribution with p = 0.5, will the shape of the probability distribution be symmetric, skewed to the left, or skewed to the right?
5.8 Use the formula for the binomial probability distribution to calculate the values of p(x), and con- struct the probability histogram for X when n = 6 and p=0.2. [HINT: Calculate P(X=k) for seven different values of k.]
5.7 Let X be a binomial random variable with n = 7, p=0.3. Find these values:a. P(X=4)d. = npb. P(X 1)c. P(X>1)e. = Vnpq
5.6 Evaluate these binomial probabilities:a. C (0.2)(0.8)8c. C (0.2)(0.8)6b. C (0.2)(0.8)7d. P(X 1) when n = 8, p=0.2e. P(two or fewer successes)
5.5 Evaluate these binomial probabilities:a. C(0.3)(0.7)6b. C (0.05)(0.95)+c. C (0.5) (0.5)7d. C (0.2)(0.8)6
5.4 The Urn Problem, continued Refer to Exercise 5.3. Assume that the sampling was conducted with replacement. That is, assume that the first ball was selected from the jar, observed, and then replaced, and that the balls were then mixed before the second ball was selected. Explain why X, the
5.3 The Urn Problem A jar contains five balls: three red and two white. Two balls are randomly selected without replacement from the jar, and the number X of red balls is recorded. Explain why X is or is not a binomial random variable. (HINT: Compare the characteristics of this experiment with the
5.2 Consider a binomial random variable with n = 9 and p=0.3. Let X be the number of successes in the sample.a. Find the probability that X is exactly 2.b. Find the probability that X is less than 2.c. Find P(X>2).d. Find P(2x4).
5.1 Consider a binomial random variable with n = 8 and p=0.7. Let X be the number of successes in the sample.a. Find the probability that X is 3 or less.b. Find the probability that X is 3 or more.c. Find P(X
Would you rather take a multiple-choice or a full recall test? If you have absolutely no knowledge of the material, you will score zero on a full recall test. However, if you are given five choices for each question, you have at least one chance in five of guessing correctly! If a multiple-choice
- A regimen consisting of a daily dose of vitamin C was tested to determine its effective- ness in preventing the common cold. Ten people who were following the prescribed regimen were observed for a period of one year. Eight survived the winter without a cold. Suppose the probability of surviving
Refer to Example 5.7 and the binomial random variable X with n = 5 and p=0.6. Use the cumulative binomial table in Table 5.1 to find the remaining binomial probabilities: p(0), p(1), p(2), p(4), and p(5). Construct the probability histogram for the random vari- able X and describe its shape and
Use the cumulative binomial table for n=5 and p=0.6 to find the probabilities of these events: 1. Exactly three successes 2. Three or more successes
Over a long period of time it has been observed that a given marksman can hit a target on a single trial with probability equal to 0.8. Suppose he fires four shots at the target. 1. What is the probability that he will hit the target exactly two times? 2. What is the probability that he will hit
Find P(X=2) for a binomial random variable with n = 10 and p = 0.1.
A patient fills a prescription for a 10-day regimen of two pills daily. Unknown to the pharmacist and the patient, the 20 tablets consist of 18 pills of the prescribed medica- tion and 2 pills that are the generic equivalent of the prescribed medication. The patient selects two pills at random for
Suppose there are approximately 1,000,000 adults in a city and an unknown proportion, p. favour term limits for politicians. A sample of 1000 adults will be chosen in such a way that every one of the 1,000,000 adults has an equal chance of being selected, and each adult is asked whether he or she
The Islamic calendar is lunar. The beginning and ending of the calendar are determined by the sighting of the crescent moon (new moon). Muslims are supposed to sight the crescent everywhere they live. It is a purely lunar calendar, having 12 lunar months in a year of about 354 (12 x 29.53=354.36)
A poker hand consists of five cards from a deck of 52 ordinary cards. A player received five cards and did not look at them. Suppose that among these five cards, one card is the ace of spades, but this is not known to the player. Let X be the number of cards in the player's hand that are turned
How to Use Table 2 to Calculate Poisson Probabilities
How to Use Table 1 to Calculate Binomial Probabilities
A box of condiments has ten small packages, in which three are ketchup, three are mustard, and four are relish. A sample of three packages is randomly selected (without replacement) from the box.a. Find the probability distribution for X, the number of mustard packages in the sample.b. What are

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