Let X denote the number of successes in n Bernoulli trials with a success probability of p. (a) Find the exact probabilities of each of the following: (i) X ~ 7 when n = 25, p = .4 (ii) 11 ≤ X ≤ 16whenn = 20, p = .7 (iii) X ≥ 9 when n = 16, p = .5 (b) Use a normal approximation for each situation in part (a).
> At one large midwest university, about 40% of the college seniors have a social science major. Fourteen seniors are selected at random. Let X denote the number that have a social science major. Determine (a) p [ 3 ≤ X ≤ g ] (b) p [ 3 < X ≤ g ] (c) P[3 <
> A school newspaper claims that 70% of the students support its view on a campus issue. A random sample of 20 students is taken, and 10 students agree with the newspaper. Find P[10 or less agree] if 70% support the view and comment on the plausibility of
> For each situation, state if a binomial distribution is reasonable for the random variable X. Justify your answer. (a) A multiple-choice examination consists of 10 problems, each of which has 5 suggested answers. A student marks answers by pure guesses
> In bidding for a contract to cater a large wedding, a caterer expects she will have a net profit of $3000 if she wins the contract and net loss of $60 if h er bid fails. If the probability of getting the contract is .3, calculate her expected return.
> National safety statistics suggest that about 33% of the persons treated in an emergency room because of moped accidents are under 16 years of age. Suppose you count the number of persons under 16 among the next 14 moped accident victims to come to the e
> The probability that a voter will believe a rumor about a politician is .3. If 20 voters are told individually: (a) Find the probability that none of the 20 believes the rumor. (b) Find the probability that seven or more believe the rumor. (c) Det erm
> A viral infection is spread by contact with an infected person. Let the probability that a healthy person gets the infection in one contact be p = .4. (a) An infected person has contact with six healthy persons. Specify the distribution of X = No. of pe
> The proportion of people having the blood type O-positive in a large southern city is .38. For two randomly selected donors: (a) Find the probability of at least one type 0 . (b) Find the expected number of type 0 . (c) Repeat parts (a) and (b) if the
> If in three Bernoulli trials the probability that the first two trials are both failures is 9/49 what is the probability that the first two are successes and the third is a failure?
> A basketball team scores 35% of the times it gets the ball. Find the probability that the first basket occurs on its third possession.
> If the probability of having a male child is . 5, find the probability that the third child is the first son.
> Give an example of repeated trials with two possible outcomes where: (a) The model of Bernoulli trials is reasonable. (b) The condition of independence is violated. (c) The condition of equal P ( S) is violated.
> Is the model of Bernoulli trials plausible in each of the following situations? Identify any serious violations of the conditions. (a) A beginning golfer tries 40 putts, 5 feet from the hole, while practicing on the putting green. (b) Persons applying f
> Let the random variable Y denote the proportion of times a head occurs in three tosses of a coin, that is, Y = ( No. of heads in 3 tosses )/ 3. (a) Obtain the probability distribution Y. (b) Draw the probability histogram. (c) Calculate the E ( Y) and
> A book club announces a sweepstakes in order to attract new subscribers. The prizes and the corresponding chances are listed here (typically, the prizes are listed in bold print in an advertisement flyer while the chances are entered in fine print or not
> A probability distribution can also be described by a function that gives the accumulated probability at or below each value of X. Specifically, Cumulative distribution function at c = Sum of probabilities of all values x ≤ c For the
> Suppose that X can take the values 0, 1, 2, 3, and 4, and the probability distribution of X is incompletely specified by the function Find(a) f(4) (b)P [X ≥ 2] (c) E(X)and (d) sd(X).
> A roulette wheel has 38 slots, of which 18 are red, 18 black, and 2 green. A gambler will play three times, each time betting $5 on red . The gambler gets $ 10 if red occurs and loses the bet otherwise. Let X denote the net gain of the gambler in 3 plays
> John and two of his friends have one-on-one basketball tournaments almost every week. Each person plays each other person once. Let X be the number of games that John wins next week X . Based on the results of the games over the past year, John construct
> A botany student is asked to match the popular names of three house plants with their obscure botanical names. Suppose the student never heard of these names and is trying to match by sheer guess. Let X denote the number of correct matches. (a) Obtain t
> Suppose the number of parking tickets X issued during a police officer's shift has the probability distribution (a) Find the mean and standard deviation of the number of parking tickets issued. (b) Let Find (c) Suppose the numbers of tickets issued on di
> The number of overnight emergency calls X to the answering service of a heating and air conditioning firm have the probabilities .05, .1, .15, .35, .20, and .15 for 0, 1, 2, 3, 4, and 5 calls, respectively. (a) Find the probability of fewer than 3 calls
> In the finals of a match play golf tournament, the winner will get $90,000 and the loser$ 15,000. Find the expected winnings of player B if (a) the two finalists are evenly matched and (b) player B has probability .8 of winning.
> A student buys a lottery ticket for $2. The prize is a bicycle worth $500. The winning ticket will be chosen at random from 2000 tickets that were sold. (a) What is the probability that the student will win a bicycle? (b) Determine the student's expect
> Refer to Exercise 5.91. (a) List the x values that lie in the interval µ - (, to µ + (, and calculate P [ µ - (≤ X ≤ µ + ( ] . (b) List the x values that lie in
> Referring to Exercise 5.25, find the mean and standard deviation of the number of customers. Data from Exercise 5.25: Based on recent records, the manager of a car painting center has determined the following probability distribution for the number of c
> Two of the integers { 1, 3, 5, 6, 7 } are chosen at random without replacement. Let X denote the difference = largest -smallest. (a) List all choices and the corresponding values of X. (b) List the distinct values of X and det ermine their probabilities
> The following distribution has been proposed for the number of times a student will eat at a gourmet restaurant dinner next week. (a) Calculate the mean and variance. (b) Plot the probability histogram and locate µ.
> Refer to the monthly intersection accident data. Considering an even longer record leads to a distribution for X = number of accidents in a month. (a) Calculate E(X). (b) Calculate sd ( X). (c) Draw the probability histogram and locate the mean.
> A list of the world's 10 largest companies, in terms of revenue, contains 4 from the United States, 3 from China, and 1 each from Germany, Japan, and the Netherlands. A potential investor randomly selects 3 of the companies to research further. (a) Find
> Refer to Exercise 5.88 but now suppose the sampling is done in two stages: First a list is selected at random and then, from that list, two persons are selected at random without replacement. Let Y denote the number of females in the sample. (a) List the
> A large science department at the university made a list of its top 20 juniors and 25 seniors. The first list contains 7 females and the second 10 females. One person is randomly selected from each list and the two selections are independent. Let X denot
> Let X denote the difference ( no. of heads - no. of tails) in three tosses of a coin. (a) List the possible values of X. (b) List the elementary outcomes associated with each value of X.
> One day, the visits of n = 48 students to the social network site produced the summary statistics x = 35.96 minutes and s = 29.11 minutes n = 48 (a) Calculate a 90% confidence interval for the mean daily time spent on the social network site. (b) Compa
> Refer to the data on the amount of reflected light from urban areas in the Data Bank. A computer calculation for a test of H0 : µ = 84 versus H1 : µ ≠84 has the output (a) What is the conclusion of the test wh
> In a given situation, suppose H0 was rejected at a = .05. Answer the following questions as "yes," "no," or "can't tell" as the case may be. (a) Would H0 also be rejected at a = .03? (b) Would H0 also be rejected at a = .10? (c) Is the P-value larger
> A test will be conducted to see how long a seven-ounce tube of toothpaste lasts. The researcher wants to establish that the mean time is greater than 30.5 days. From a random sample of size 75, an investigator obtains x = 32.3 and s = 6.2 days. (a) Form
> Calculate the mean and standard deviation for the probability distribution.
> A literary critic wants to establish that the mean number of words p er sentence, appearing in a newly discovered short story, is different from 9.1 words. A sample of 36 sentences provided the data x = 8.6 and s = 1.2 (a) Formulate the null and alterna
> In each case, identify the null hypothesis (H0) and the alternative hypothesis (H1) using the appropriate symbol for the parameter of interest. (a) A consumer group plans to test-drive several cars of a new model in order to document that its average hi
> The number of PCBs was measured in 40 samples of soil that were treated with contaminated sludge. The following summary statistics were obtained. x = 3.56, s = .5 ppm (a) Perform a test of hypotheses with the intent of establishing that the mean PCB co
> Refer to Exercise 6.53 and to the data on velocity for females. Make a normal-scores plot for the square root of the velocity. Comment on the agreement with the ideal straight-line pattern. Data from Exercise 6.53: The MINITAB computer language makes it
> Referring to Exercise 6.49, use MINITAB or another package program to make a normal-scores plot of the computer anxiety scores in the Data Bank. Data from Exercise 6.49: Use a computer program to make a normal-scores plot for the volume of timber data i
> In all of William Shakespeare's works, he used 884,64 74 different words. Of these, 14,376 appeared only once. In 1985 a 429-word poem was discovered that may have been written by Shakespeare. To keep the probability calculations simple, assume that the
> On a Saturday afternoon, 14 7 customers are observed during check-out and the number paying by card, credit or debit, is recorded. Records from the store suggest that 43% of customers pay by card. Approximate the probability that: (a) More than 60 will p
> Because 10% of the reservation holders are "no-shows," a U.S. airline sells 400 tickets for a flight that can accommodate 3 70 passengers. (a) Find the approximate probability that one or more reservation holders will not be accommodated on the flight.
> The number of successes X has a binomial distribution. State whether or not the normal approximation is appropriate in each of the following situations: (a) n = 400 , p = .23 (b) n = 20 , p = .03 (c) n = 90 , p = .98
> A particular program, say, program A, previously drew 30% of the television audience. To determine if a recent rescheduling of the programs on a competing channel has adversely affected the audience of program A, a random sample of 400 viewers are asked
> Probabilities, provided by the engaged couple, can help control the number of guests at their wedding. At the planning stage, decisions are necessarily based on the number of persons expected to respond yes to the RSVP. Concerning one group of four adult
> Suppose the random variable X is normally distributed with mean µ and standard deviation CJ. If Y is a linear function of X - that is, Y = a + bX, where a and bare constants-then Y is also normally distributed with Mean = a + bµ sd = |b| ( For instance,
> Suppose the amount of a popular sport drink in bottles leaving the filling machine has a normal distribution with mean 101.5 milliliters (ml) and standard deviation 1.6 ml. (a) If the bottles are labeled 100 ml, what proportion of the bottles contain le
> Suppose the duration of trouble-free operation of a new robotic vacuum cleaner is normally distributed with mean 750 days and standard deviation 100 days. (a) What is the probability that the vacuum cleaner will work for at least two years without troub
> The scores on an examination are normally distributed with mean µ = 70 and standard deviation ( = 8. Suppose that the instructor decides to assign letter grades according to the following scheme (left endpoint included). Find the percentage
> Th e bonding strength of a drop of plastic glue is normally distributed with mean 100 pounds and standard deviation 8 pounds. A broken plastic strip is repaired with a drop of this glue and then subjected to a test load of 90 pounds. What is the probabil
> The lifting capacities of a class of industrial workers are normally distributed with mean 65 pounds and standard deviation 8 pounds. What proportion of these workers can lift an 80-pound load?
> Suppose that a student's verbal score X from next year's Graduate Record Exam can be considered an observation from a normal population having mean 499 and standard deviation 120. Find (a) P ( X > 600 ] (b) 90th percentile of the distribution. (c) Pro
> If X has a normal distribution with µ = 90 and ( = 4 find b such that (a) P(X < b) .6700 (b) P(X > b) .0110 (c) P(IX - 90 1 < b) = .966 (d) Check all of your answers using software.
> The bell-shaped histogram for the heights of three-year-old red pine seedlings on page 1 79 is consistent with the assumption of a normal distribution having mean = 280 and sd = 58 millimeters. Let X denote the height, at three years of age, of the next
> Given the following probability distribution concerning the number of Web sites visited almost every day: (a) Construct the probability histogram. (b) Find E(X), (2, and (.
> A normal distribution with mean 11 5 and standard deviation 22 hundredths of an inch describes variation in female salmon growth in freshwater. (a) If a newly caught female salmon has growth 108, what is the corresponding standardized score? (b) If a s
> If Z is a standard normal random variable, what is the probability that (a) Z exceeds .62? (b) Z lies in the interval ( -1 .40, 1 .40)? (c)|Z| exceeds 3.0? (d) |Z| is less than 2.0?
> Find the 20th, 40th, 60th, and 80th percentiles of the standard normal distribution.
> For the standard normal distribution, find the value z such that (a) Area to its left is .0838. (b) Area to its left is .04 7. (c) Area to its right is .2611. (d) Area to its right is .12.
> For a standard normal random variable Z, find (a) P[Z < 1.56 ) (b) P [ Z > 1.245) (c) P[ .61 < Z < 1.92 ) (d) P [ -1.47 < Z < 1.055)
> In the context of the height of red pine seedlings presented at the front of the chapter, describe the reasoning that leads from a histogram to the concept of a probability density curve. (Think of successive histograms based on 100 heights, 500 heights,
> For X having the density in Exercise 6.55. Find (a) P(X > .8) (b) P(.4 ≤ X ≤ .8) and (c) P(.4 Data from Exercise 6.55: Determine (a) the median and (b) the quartiles for the distribution shown in the following
> Determine (a) the median and (b) the quartiles for the distribution shown in the following illustration.
> Refer to Exercise 9.51. Do these data provide strong evidence that the mean time to blossom is less than 42 days? Test with a = .01. (a) Formulate the null and alternative hypotheses. (b) Determine the test statistic. (c) Give the form of the rejectio
> The water quality is acceptable if the mean amount of suspended solids is less than 49 mg/I. Construct an a = .05 test to establish that the quality is acceptable. (a) Specify H0 and H1 . (b) State the test statistic. (c) What does the test conclude?
> Among cable TV customers, let X denote the number of television sets in a single-family residential dwelling. From an examination of the subscription records of 361 residences in a city, the frequency distribution is obtained. (a) Based on these data, ob
> Concerning the product volume for green gasoline. Obtain (a) A point estimate ofµ and its 95% error margin. (b) A 90% confidence interval for the mean. (c) Explain why you are 90% confident that the interval in part (b) covers the true unknown mean.
> The time to blossom of 21 plants has x = 38.4 days and s = 5.1 days. Give a 95% confidence interval for the mean time to blossom.
> Recorded here are the germination times (number of days) for seven seeds of a new strain of snap bean. Stating any assumptions that you make, determine a 95% confidence interval for the true mean germination time for this strain.
> The weights from a random sample of 20 golden retriever dogs have mean 76.1 pounds and standard deviation 5.9 pounds. Assume that the weights of the dogs have a normal distribution. (a) Construct a 98% confidence interval for the population mean. (b) Wh
> Use software to find the probability (a) T ≤ -1 .3 when d.f = 12. (b) T > 1.7 when d.f = 21. (c) |T| > 1.35 when d.f = 33. (d) Find c such that P [T > c] = .015 when d.f = 18. (e) Find c such that P [I T l > c] = .04 when d.f = 42.
> Record the t.05 values for d.f of 5, 10, 15, 20, and 29. Does this percentile increase or decrease with increasing degrees of freedom?
> Use software to find the number b so that (a) P[T < b] = .95 when d.f = 5. (b) P[ - b < T < b] = .95 when d.f = 37. (c) P[T > b] = .01 when d.f 2. (d) P[ T > b] = .99 when d.f = 13.
> Do the data in Exercise 9.22 substantiate the conjecture that the true standard deviation of the acidity measurements is larger than 0.4? Test at a = .05. Data from Exercise 9.22: Measurements of the acidity (pH) of rain samples were recorded at 13 site
> Referring to Exercise 9.21, construct a 95% confidence interval for the population standard deviation of the diameters of Indian mounds. Data from Exercise 9.21: The following measurements of the diameters (in feet) of Indian mounds in southern Wisconsi
> Refer to the data of lizard lengths in Exercise 9.10. (a) Determine a 90% confidence interval for the population standard deviation (. (b) Should H0 : ( = 9 be rejected in favor of H1 : ( ≠9 at a = .10? Data from Exercise 9.10: A zo
> Based on recent records, the manager of a car painting center has determined the following probability distribution for the number of customers per day. (a) If the center has the capacity to serve two customers per day, what is the probability that one o
> During manufacture, the thickness of laser printer paper is monitored. Data from several random samples each day during the year suggest that thickness follows a normal distribution. A sample of n = 10 thickness measurements (ten-thousandths inch) yields
> Refer to Exercise 9.41. Construct a 95% confidence interval for the true standard deviation of the thickness of sheets produced on this shift. Data from Exercise 9.41: Plastic sheets produced by a machine are periodically monitored for possible fluctuat
> Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. Uncontrollable heterogeneity in the viscosity of the liquid mold makes some variation in thickness measurements unavoidable. However, if the true stan
> Find a 90% confidence interval for (, based on the n = 40 measurements of heights of red pine seedlings. State any assumption you make about the population.
> Use software to find (a) T < -1.729 when d.f = 19. (b) |T| > 2.179 when d.f = 12. (c) -1.703 < T < 1. 703 when d.f 27. (d) -1.833 < T < 2.821 when d.f. 9.
> Use software to find the probability (a) x2 ≤ 17.2 when d .f = 9. (b) x2 > 13.7 when d.f. = 23. (c) Find c such that P(x2 > c) = .015 when d.f = 12. ( d) Find c such that P ( x2 < c) = .04 when d.f. = 33.
> Using MINITAB software to compute x2 percentiles. (a) Determine x2.05 for the x2 distribution with d.f = 19. (b) Determine P (x2 ≤ 30) for d.f. = 19. Repeat Part (a) but click on Cumulative probability. (c) Do both parts with d.f. = 11
> Name the x2 percentiles shown and find their values. (c) Find the percentile in part (a) if d.f = 37. (d) Find the percentile in part (b) if d.f = 11. (e) Verify your answers above using software.
> Using the t able for the x2 distributions, find: (a) The upper 5% point when d.f = 8. (b) The upper 1% point when d.f = 16. (c) The lower 2.5% point when d.f. = 9. (d) The lower 1% point when d.f = 27. (e) Verify your answers above using software.
> Establish the connection between the large sample Z test, which rejects H0 : µ = µ0 in favor of H1 : µ ≠µ0 , at a = .05, if and the 95% confidence interval
> Among six new apps available this month, only three will prove to be useful to you in the long run. Suppose you purchase three of the apps at random. Find the probability distribution of X, the number of useful apps you select.
> Refer to the data in Exercise 9.33. (a) Construct a 90% confidence interval forµ. (b) If you were to test H0 : µ = 17 versus H1 : µ ≠ 17 at a = . 10, what would you conclude from your result in part (a}? Why? (c) Perform the hypothesis test indicated
> Ten participants tried a new weight loss diet for two months. The resulting weight losses (pounds} are 20.2 11.0 15.4 9.9 7.9 15.6 15.5 23.1 11.0 11.4 (a) Construct a 95% confidence interval for the population mean amount µ of decrease in weight
> The petal width (mm} of one kind of iris has a normal distribution. Suppose that, from a random sample of widths, the t based 90% confidence interval for the population mean width is ( 16.8, 19.6 } mm. Answer each question "yes," "no," or "can't tell," a
> The 90% confidence interval for the mean weight of female wolves was found to be ( 67.13, 80.62} pounds. (a} What is the conclusion of testing H0 : µ = 81 versus H 1 : µ ≠ 81 at level a = .10? (b} What is the conclusion if H0 : µ = 69?
> Based on a random sample of tail lengths for 15 male kites, an investigator calculates the 95% confidence interval (183.0, 195.0} mm based on the t distribution. The normal assumption is reasonable. (a} What is the conclusion of the t test for H0 : µ = 1
> Using MINITAB software to compute t percentiles (a) Determine t.05 for the t-distribution with d.f = 27. Cale > Probability Distributions > t.... Click on inverse.... Type 27 in Degrees of Freedom. Click Input constant and Type .95. Click OK. (b) Determ
> Refer to the computer anxiety scores for female accounting students in Data Bank. A computer calculation for a test of H0 : µ = 2 versus H1 : µ ≠2 is given here. (a) What is the conclusion if you test with a =
> A few years ago, noon bicycle traffic past a busy section of campus had a mean of µ = 300. To see if any change in traffic has occurred, counts were taken for a sample of 15 weekdays. It was found that x = 340 and s = 30. (a) Construct an a = .05 test o
