stats final prep

In a certain board game a player's turn begins with three rolls of a pair of dice. If the player rolls doubles all three times there is a penalty. The probability of rolling doubles in a single roll of a pair of fair dice is 1/6. Find the probability of rolling doubles all three times.

0.0046

The distance from the seat back to the front of the knees of seated adult males is normally distributed with mean 23.8 inches and standard deviation 1.22 inches. The distance from the seat back to the back of the next seat forward in all seats on aircraft flown by a budget airline is 26 inches. Find the proportion of adult men flying with this airline whose knees will touch the back of the seat in front of them.

0.0359

The sensitivity of a drug test is the probability that the test will be positive when administered to a person who has actually taken the drug. Suppose that there are two independent tests to detect the presence of a certain type of banned drugs in athletes. One has sensitivity 0.75; the other has sensitivity 0.85. If both are applied to an athlete who has taken this type of drug, what is the chance that his usage will go undetected?

0.0375

A corporation has advertised heavily to try to insure that over half the adult population recognizes the brand name of its products. In a random sample of 20 adults, 14 recognized its brand name. What is the probability that 14 or more people in such a sample would recognize its brand name if the actual proportion p of all adults who recognize the brand name were only 0.50?

0.0577

A regulation hockey puck must weigh between 5.5 and 6 ounces. In an alternative manufacturing process the mean weight of pucks produced is 5.75 ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most 0.005 of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights.)

0.089

About 12% of all individuals write with their left hands. A class of 130 students meets in a classroom with 130 individual desks, exactly 14 of which are constructed for people who write with their left hands. Find the probability that exactly 14 of the students enrolled in the class write with their left hands.

0.1019

The systolic blood pressure X of adults in a region is normally distributed with mean 112 mm Hg and standard deviation 15 mm Hg. A person is considered "prehypertensive" if his systolic blood pressure is between 120 and 130 mm Hg. Find the probability that the blood pressure of a randomly selected person is prehypertensive.

0.1830

An outside financial auditor has observed that about 4% of all documents he examines contain an error of some sort. Assuming this proportion to be accurate, find the probability that a random sample of 700 documents will contain at least 30 with some sort of error. You may assume that the normal distribution applies.

0.3483

Dogberry's alarm clock is battery operated. The battery could fail with equal probability at any time of the day or night. Every day Dogberry sets his alarm for 6:30 a.m. and goes to bed at 10:00 p.m. Find the probability that when the clock battery finally dies, it will do so at the most inconvenient time, between 10:00 p.m. and 6:30 a.m.

0.3542

A tourist who speaks English and German but no other language visits a region of Slovenia. If 35% of the residents speak English, 15% speak German, and 3% speak both English and German, what is the probability that the tourist will be able to talk with a randomly encountered resident of the region?

0.47

Heights X of adult men are normally distributed with mean 69.1 inches and standard deviation 2.92 inches. Juliet, who is 63.25 inches tall, wishes to date only men who are taller than she but within 6 inches of her height. Find the probability that the next man she meets will have such a height.

0.4971

Suppose that in a certain region of the country the mean duration of first marriages that end in divorce is 7.8 years, standard deviation 1.2 years. Find the probability that in a sample of 75 divorces, the mean age of the marriages is at most 8 years.

0.9251

An accountant has observed that 5% of all copies of a particular two-part form have an error in Part I, and 2% have an error in Part II. If the errors occur independently, find the probability that a randomly selected form will be error-free.

0.931

Suppose the mean number of days to germination of a variety of seed is 22, with standard deviation 2.3 days. Find the probability that the mean germination time of a sample of 160 seeds will be within 0.5 day of the population mean.

0.9940

A high-speed packing machine can be set to deliver between 11 and 13 ounces of a liquid. For any delivery setting in this range the amount delivered is normally distributed with mean some amount μ and with standard deviation 0.08 ounce. To calibrate the machine it is set to deliver a particular amount, many containers are filled, and 25 containers are randomly selected and the amount they contain is measured. Find the probability that the sample mean will be within 0.05 ounce of the actual mean amount being delivered to all containers.

0.9982

Suppose the mean amount of cholesterol in eggs labeled "large" is 186 milligrams, with standard deviation 7 milligrams. Find the probability that the mean amount of cholesterol in a sample of 144 eggs will be within 2 milligrams of the population mean.

0.9994

Suppose the mean cost across the country of a 30-day supply of a generic drug is $46.58, with standard deviation $4.84. Find the probability that the mean of a sample of 100 prices of 30-day supplies of this drug will be between $45 and $50.

0.9994

A machine for making precision cuts in dimension lumber produces studs with lengths that vary with standard deviation 0.003 inch. Five trial cuts are made to check the machine's calibration. The mean length of the studs produced is 104.998 inches with sample standard deviation 0.004 inch. Construct a 99.5% confidence interval for the mean lengths of all studs cut by this machine. Assume lengths are normally distributed. Hint: Not all the numbers given in the problem are used.

104.998±0.004

A government agency was charged by the legislature with estimating the length of time it takes citizens to fill out various forms. Two hundred randomly selected adults were timed as they filled out a particular form. The times required had mean 12.8 minutes with standard deviation 1.7 minutes. Construct a 90% confidence interval for the mean time taken for all adults to fill out this form.

12.8±0.20

An insurance company will sell a $90,000 one-year term life insurance policy to an individual in a particular risk group for a premium of $478. Find the expected value to the company of a single policy if a person in this risk group has a 99.62% chance of surviving one year.

136

X is a normally distributed random variable X with mean 15 and standard deviation 0.25. Find the values xL and xR of X that are symmetrically located with respect to the mean of X and satisfy P(xL < X < xR) = 0.80.

15.32

In a study of dummy foal syndrome, the average time between birth and onset of noticeable symptoms in a sample of six foals was 18.6 hours, with standard deviation 1.7 hours. Assuming that the time to onset of symptoms in all foals is normally distributed, construct a 90% confidence interval for the mean time between birth and onset of noticeable symptoms.

18.6±1.4

A town council commissioned a random sample of 85 households to estimate the number of four-wheel vehicles per household in the town. The results are shown in the following frequency table. Construct a 98% confidence interval for the average number of four-wheel vehicles per household in the town.

2.54±0.30

In order to estimate the mean amount of damage sustained by vehicles when a deer is struck, an insurance company examined the records of 50 such occurrences, and obtained a sample mean of $2,785 with sample standard deviation $221. Construct a 95% confidence interval for the mean amount of damage in all such accidents.

2785±61

City planners wish to estimate the mean lifetime of the most commonly planted trees in urban settings. A sample of 16 recently felled trees yielded mean age 32.7 years with standard deviation 3.1 years. Assuming the lifetimes of all such trees are normally distributed, construct a 99.8% confidence interval for the mean lifetime of all such trees.

32.7±2.9

A sample of 250 workers aged 16 and older produced an average length of time with the current employer ("job tenure") of 4.4 years with standard deviation 3.8 years. Construct a 99.9% confidence interval for the mean job tenure of all workers aged 16 or older.

4.4±0.79

About 2% of alumni give money upon receiving a solicitation from the college or university from which they graduated. Find the average number monetary gifts a college can expect from every 2,000 solicitations it sends.

40

A college athletic program wishes to estimate the average increase in the total weight an athlete can lift in three different lifts after following a particular training program for six weeks. Twenty-five randomly selected athletes when placed on the program exhibited a mean gain of 47.3 lb with standard deviation 6.4 lb. Construct a 90% confidence interval for the mean increase in lifting capacity all athletes would experience if placed on the training program. Assume increases among all athletes are normally distributed.

47.3±2.19

For all settings a packing machine delivers a precise amount of liquid; the amount dispensed always has standard deviation 0.07 ounce. To calibrate the machine its setting is fixed and it is operated 50 times. The mean amount delivered is 6.02 ounces with sample standard deviation 0.04 ounce. Construct a 99.5% confidence interval for the mean amount delivered at this setting.

6.02±0.03

Tests of a new tire developed by a tire manufacturer led to an estimated mean tread life of 67,350 miles and standard deviation of 1,120 miles. The manufacturer will advertise the lifetime of the tire (for example, a "50,000 mile tire") using the largest value for which it is expected that 98% of the tires will last at least that long. Assuming tire life is normally distributed, find that advertised value.

65,054

All students in a large enrollment multiple section course take common in-class exams and a common final, and submit common homework assignments. Course grades are assigned based on students' final overall scores, which are approximately normally distributed. The department assigns a C to students whose scores constitute the middle 2/3 of all scores. If scores this semester had mean 72.5 and standard deviation 6.14, find the interval of scores that will be assigned a C.

66.5, 78.5

The monthly amount of water used per household in a small community is normally distributed with mean 7,069 gallons and standard deviation 58 gallons. Find the three quartiles for the amount of water used.

7030.14, 7069, 7107.86

A thread manufacturer tests a sample of eight lengths of a certain type of thread made of blended materials and obtains a mean tensile strength of 8.2 lb with standard deviation 0.06 lb. Assuming tensile strengths are normally distributed, construct a 90% confidence interval for the mean tensile strength of this thread.

8.2±0.04

Let X denote the number of boys in a randomly selected three-child family. Assuming that boys and girls are equally likely, construct the probability distribution of X.

P (0) = 1/8 P (1) = 3/8 P (2) = 3/8 P (3) = 1/8

Make a statement in ordinary English that describes the complement of each event (do not simply insert the word "not"). a. In the roll of a die: "five or more." b. In a roll of a die: "an even number." c. In two tosses of a coin: "at least one heads." d. In the random selection of a college student: "Not a freshman."

a. "four or less" b. "an odd number" c. "no heads" or "all tails" d. "a freshman"

Information about a random sample is given. Verify that the sample is large enough to use it to construct a confidence interval for the population proportion. Then construct a 90% confidence interval for the population proportion. n = 25, pˆ=0.7p^=0.7 n = 50, pˆ=0.7

a. (0.5492, 0.8508) b. (0.5934, 0.8066)

X is a normally distributed random variable with mean 500 and standard deviation 25. Find the probability indicated. a. P(X < 400) b. P(466<X<625)

a. 0.0000 b. 0.9131

X is a normally distributed random variable with mean 15 and standard deviation 1. Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation. a. P(X < 12), P(X > 18) b. P(X < 14), P(X > 16) c. P(X < 11.25), P(X > 18.75) d. P(X < 12.67), P(X > 17.33)

a. 0.0013, 0.0013 b. 0.1587, 0.1587 c. 0.0001, 0.0001 d. 0.0099, 0.0099

*A manufacturer examines its records over the last year on a component part received from outside suppliers. The breakdown on source (supplier A, supplier B) and quality (H: high, U: usable, D: defective) is shown in the two-way contingency table. The record of a part is selected at random. Find the probability of each of the following events. a. The part was defective. b. The part was either of high quality or was at least usable, in two ways: (i) by adding numbers in the table, and (ii) using the answer to (a) and the Probability Rule for Complements. c. The part was defective and came from supplier B. d. The part was defective or came from supplier B, in two ways: by finding the cells in the table that correspond to this event and adding their probabilities, and (ii) using the Additive Rule of Probability.

a. 0.0023 b. 0.9977 c. 0.0009 d. 0.3014

In spite of the requirement that all dogs boarded in a kennel be inoculated, the chance that a healthy dog boarded in a clean, well-ventilated kennel will develop kennel cough from a carrier is 0.008. a. If a carrier (not known to be such, of course) is boarded with three other dogs, what is the probability that at least one of the three healthy dogs will develop kennel cough? b. If a carrier is boarded with four other dogs, what is the probability that at least one of the four healthy dogs will develop kennel cough? c. The pattern evident from parts (a) and (b) is that if K+1K+1 dogs are boarded together, one a carrier and K healthy dogs, then the probability that at least one of the healthy dogs will develop kennel cough is P(X≥1)=1−(0.992)KP(X≥1)=1−(0.992)K, where X is the binomial random variable that counts the number of healthy dogs that develop the condition. Experiment with different values of K in this formula to find the maximum number K+1K+1 of dogs that a kennel owner can board together so that if one of the dogs has the condition, the chance that another dog will be infected is less than 0.05.

a. 0.0238 b. 0.0316 c. 6

Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated. a. P(Z < −1.72) b. P(Z < 2.05) c. P(Z < 0) d. P(Z > −2.11) e. P(Z > 1.63) f. P(Z > 2.36)

a. 0.0427 b. 0.9798 c. 0.5 d. 0.9826 e. 0.0516 f. 0.0091

The probability that a 7-ounce skein of a discount worsted weight knitting yarn contains a knot is 0.25. Goneril buys ten skeins to crochet an afghan. a. Find the probability that (i) none of the ten skeins will contain a knot; (ii) at most one will. b. Find the expected number of skeins that contain knots. c. Find the most likely number of skeins that contain knots.

a. 0.0563 and 0.2440 b. 2.5 c. 2

A professional proofreader has a 98% chance of detecting an error in a piece of written work (other than misspellings, double words, and similar errors that are machine detected). A work contains four errors. a. Find the probability that the proofreader will miss at least one of them. b. Show that two such proofreaders working independently have a 99.96% chance of detecting an error in a piece of written work. c. Find the probability that two such proofreaders working independently will miss at least one error in a work that contains four errors.

a. 0.0776 b. 0.9996 c. 0.0016

A normally distributed population has mean 25.6 and standard deviation 3.3. a. Find the probability that a single randomly selected element X of the population exceeds 30. b. Find the mean and standard deviation of X⎯⎯X- for samples of size 9. c. Find the probability that the mean of a sample of size 9 drawn from this population exceeds 30.

a. 0.0918 b. μX⎯⎯⎯=25.6μX-=25.6, σX⎯⎯⎯=1.1σX-=1.1 c. 0.0000

Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated. a. P(−2.15 < Z < −1.09) b. P(−0.93 < Z < 0.55) c. P(0.68 < Z < 2.11)

a. 0.1221 b. 0.5326 c. 0.2309

*The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to employment type and level of life insurance. An adult is selected at random. Find each of the following probabilities. a. The person has a high level of life insurance. b. The person has a high level of life insurance, given that he does not have a professional position. c. The person has a high level of life insurance, given that he has a professional position. d. Determine whether or not the events "has a high level of life insurance" and "has a professional position" are independent.

a. 0.15 b. 0.14 c. 0.25 d. not independent

A state public health department wishes to investigate the effectiveness of a campaign against smoking. Historically 22% of all adults in the state regularly smoked cigars or cigarettes. In a survey commissioned by the public health department, 279 of 1,500 randomly selected adults stated that they smoke regularly. a. Find the sample proportion. b. Find the probability that, when a sample of size 1,500 is drawn from a population in which the true proportion is 0.22, the sample proportion will be no larger than the value you computed in part (a). You may assume that the normal distribution applies. c. Give an interpretation of the result in part (b). How strong is the evidence that the campaign to reduce smoking has been effective?

a. 0.186 b. 0.0007 c. In a population in which the true proportion is 22% the chance that a random sample of size 1500 would produce a sample proportion of 18.6% or less is only 7/100 of 1%. This is strong evidence that currently a smaller proportion than 22% smoke.

X is a binomial random variable with parameters n = 12 and p = 0.82. Compute the probability indicated. a. P(11) b. P(9) c. P(0) d. P(13)

a. 0.2434 b. 0.2151 c. 0.1812≈00.1812≈0 d. 0

An ordinary die is "fair" or "balanced" if each face has an equal chance of landing on top when the die is rolled. Thus the proportion of times a three is observed in a large number of tosses is expected to be close to 1/6 or 0.16⎯⎯.0.16-. Suppose a die is rolled 240 times and shows three on top 36 times, for a sample proportion of 0.15. a. Find the probability that a fair die would produce a proportion of 0.15 or less. You may assume that the normal distribution applies. b. Give an interpretation of the result in part (b). How strong is the evidence that the die is not fair? c. Suppose the sample proportion 0.15 came from rolling the die 2,400 times instead of only 240 times. Rework part (a) under these circumstances. d. Give an interpretation of the result in part (c). How strong is the evidence that the die is not fair?

a. 0.2451 b. We would expect a sample proportion of 0.15 or less in about 24.5% of all samples of size 240, so this is practically no evidence at all that the die is not fair. 0.0139 c. We would expect a sample proportion of 0.15 or less in only about 1.4% of all samples of size 2400, so this is strong evidence that the die is not fair.

*The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage. A person is selected at random. Find the probability of each of the following events. a. The person is a smoker. b. The person is under 30. c. The person is a smoker who is under 30.

a. 0.25 b. 0.25 c. 0.05

Scores on a common final exam in a large enrollment, multiple-section freshman course are normally distributed with mean 72.7 and standard deviation 13.1. a. Find the probability that the score X on a randomly selected exam paper is between 70 and 80. b. Find the probability that the mean score X⎯⎯X- of 38 randomly selected exam papers is between 70 and 80.

a. 0.2955 b. 0.8977

An English-speaking tourist visits a country in which 30% of the population speaks English. He needs to ask someone directions. a. Find the probability that the first person he encounters will be able to speak English. b. The tourist sees four local people standing at a bus stop. Find the probability that at least one of them will be able to speak English.

a. 0.3 b. 0.7599

In a random sample of size 1,100, 338 have the characteristic of interest. a. Compute the sample proportion pˆp^ with the characteristic of interest. b. Verify that the sample is large enough to use it to construct a confidence interval for the population proportion. c. Construct an 80% confidence interval for the population proportion p. d. Construct a 90% confidence interval for the population proportion p. e. Comment on why one interval is longer than the other.

a. 0.3073 b. pˆ±3pˆqˆn‾‾‾√=0.31±0.04p^±3p^q^n=0.31±0.04 and [0.27,0.35]⊂[0,1][0.27,0.35]⊂[0,1] c. (0.2895, 0.3251) d. (0.2844, 0.3302) e. Asking for greater confidence requires a longer interval.

A basketball player makes 60% of the free throws that he attempts, except that if he has just tried and missed a free throw then his chances of making a second one go down to only 30%. Suppose he has just been awarded two free throws. a. Find the probability that he makes both. b. Find the probability that he makes at least one. (A tree diagram could help.)

a. 0.36 b. 0.72

*The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (A, B, C, or None) and opinion on a bond issue. A person is selected at random. Find each of the following probabilities. a. The person is in favor of the bond issue. b. The person is in favor of the bond issue, given that he is affiliated with party A. c. The person is in favor of the bond issue, given that he is affiliated with party B.

a. 0.4 b. 0.43 c. 0.38

The useful life of a particular make and type of automotive tire is normally distributed with mean 57,500 miles and standard deviation 950 miles. a. Find the probability that such a tire will have a useful life of between 57,000 and 58,000 miles. b. Hamlet buys four such tires. Assuming that their lifetimes are independent, find the probability that all four will last between 57,000 and 58,000 miles. (If so, the best tire will have no more than 1,000 miles left on it when the first tire fails.) Hint: There is a binomial random variable here, whose value of p comes from part (a).

a. 0.4038 b. 0.0266

A random sample of size 121 is taken from a population in which the proportion with the characteristic of interest is p = 0.47. Find the indicated probabilities. a. P(0.45≤Pˆ≤0.50)P(0.45≤P^≤0.50) b. P(Pˆ≥0.50)

a. 0.4154 b. 0.2546

The sample space that describes all three-child families according to the genders of the children with respect to birth order is S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg}S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg} In the experiment of selecting a three-child family at random, compute each of the following probabilities, assuming all outcomes are equally likely. a. The probability that the family has at least two boys. b. The probability that the family has at least two boys, given that not all of the children are girls. c. The probability that at least one child is a boy. d. The probability that at least one child is a boy, given that the first born is a girl.

a. 0.5 b. 0.57 c. 0.875 d. 0.75

A machine for filling 2-liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation 0.002 liter and mean whatever amount the machine is set to deliver. a. If the machine is set to deliver 2 liters (so the mean amount delivered is 2 liters) what proportion of the bottles will contain at least 2 liters of soft drink? b. Find the minimum setting of the mean amount delivered by the machine so that at least 99% of all bottles will contain at least 2 liters.

a. 0.5 b. 2.005

The amount X of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean 64 ounces and standard deviation 0.25 ounce. a. What proportion of all containers contain less than a half gallon (64 ounces)? Explain. b. What is the median amount of orange juice in such containers? Explain.

a. 0.5 b. 64

*The following two-way contingency table gives the breakdown of the population of married or previously married women beyond child-bearing age in a particular locale according to age at first marriage and number of children. A woman is selected at random. Find the probability of each of the following events. a. The woman was in her twenties at her first marriage. b. The woman was 20 or older at her first marriage. c. The woman had no children. d. The woman was in her twenties at her first marriage and had at least three children.

a. 0.55 b. 0.76 c. 0.19 d. 0.11

A normally distributed population has mean 1,214 and standard deviation 122. a. Find the probability that a single randomly selected element X of the population is between 1,100 and 1,300. b. Find the mean and standard deviation of X⎯⎯X- for samples of size 25. c. Find the probability that the mean of a sample of size 25 drawn from this population is between 1,100 and 1,300.

a. 0.5818 b. μX⎯⎯⎯=1214μX-=1214, σX⎯⎯⎯=24.4σX-=24.4 c. 0.9998

An airline claims that 72% of all its flights to a certain region arrive on time. In a random sample of 30 recent arrivals, 19 were on time. You may assume that the normal distribution applies. a. Compute the sample proportion. b. Assuming the airline's claim is true, find the probability of a sample of size 30 producing a sample proportion so low as was observed in this sample.

a. 0.63 b. 0.1446

*If each die in a pair is "loaded" so that one comes up half as often as it should, six comes up half again as often as it should, and the probabilities of the other faces are unaltered, then the probability distribution for the sum X of the number of dots on the top faces when the two are rolled is SEE IMAGE. Compute each of the following. a. P(5≤X≤9).P(5≤X≤9). b. P(X ≥ 7). c. The mean μ of X. (For fair dice this number is 7.) d. The standard deviation σ of X. (For fair dice this number is about 2.415.)

a. 0.6528 b. 0.7153 c. μ = 7.8333 d. σ2=5.4866σ2=5.4866 e. σ = 2.3424

X is a normally distributed random variable with mean 57 and standard deviation 6. Find the probability indicated. a. P(X < 59.5) b. P(X < 46.2) c. P(X > 52.2) d. P(X > 70)

a. 0.6628 b. 0.7881 c. 0.0359 d. 0.0150

*In a hamster breeder's experience the number X of live pups in a litter of a female not over twelve months in age who has not borne a litter in the past six weeks has the probability distribution. a. Find the probability that the next litter will produce five to seven live pups. b. Find the probability that the next litter will produce at least six live pups. c. Compute the mean and standard deviation of X. Interpret the mean in the context of the problem.

a. 0.79 b. 0.60 c. μ = 5.8, σ = 1.2570

Suppose speeds of vehicles on a particular stretch of roadway are normally distributed with mean 36.6 mph and standard deviation 1.7 mph. a. Find the probability that the speed X of a randomly selected vehicle is between 35 and 40 mph. b. Find the probability that the mean speed X⎯⎯X- of 20 randomly selected vehicles is between 35 and 40 mph.

a. 0.8036 b. 1.0000

The amount X of beverage in a can labeled 12 ounces is normally distributed with mean 12.1 ounces and standard deviation 0.05 ounce. A can is selected at random. a. Find the probability that the can contains at least 12 ounces. b. Find the probability that the can contains between 11.9 and 12.1 ounces.

a. 0.9772 b. 0.5000

A security feature on some web pages is graphic representations of words that are readable by human beings but not machines. When a certain design format was tested on 450 subjects, by having them attempt to read ten disguised words, 448 subjects could read all the words. a. Give a point estimate of the proportion p of all people who could read words disguised in this way. b. Show that the sample is not sufficiently large to construct a confidence interval for the proportion of all people who could read words disguised in this way.

a. 0.9956 b. (0.9862, 1.005)

X is a binomial random variable with the parameters shown. Use the tables in Chapter 12 "Appendix" to compute the probability indicated. a. n = 10, p = 0.25, P(X ≤ 6) b. n = 10, p = 0.75, P(X ≤ 6) c. n = 15, p = 0.75, P(X ≤ 6) d. n = 15, p = 0.75, P(12)P(12) e. n = 15, p=0.6⎯⎯p=0.6-, P(10≤X≤12)

a. 0.9965 b. 0.2241 c. 0.0042 d. 0.2252 e. 0.5390

Find the indicated value of Z. (It is easier to find −zc−zc and negate it.) a. z0.025 b. z0.20

a. 1.96 b. 0.84

A random sample is drawn from a population of known standard deviation 11.3. Construct a 90% confidence interval for the population mean based on the information given (not all of the information given need be used). n = 36, x⎯⎯=105.2x-=105.2, s = 11.2 n = 100, x⎯⎯=105.2x-=105.2, s = 11.2

a. 105.2±3.10105.2±3.10 b. 105.2±1.86

A random sample is drawn from a population of unknown standard deviation. Construct a 99% confidence interval for the population mean based on the information given. n = 49, x⎯⎯=17.1x-=17.1, s = 2.1 n = 169, x⎯⎯=17.1x-=17.1, s = 2.1

a. 17.1±0.7717.1±0.77 b. 17.1±0.42

*The owner of a proposed outdoor theater must decide whether to include a cover that will allow shows to be performed in all weather conditions. Based on projected audience sizes and weather conditions, the probability distribution for the revenue X per night if the cover is not installed is SEE IMAGE. The additional cost of the cover is $410,000. The owner will have it built if this cost can be recovered from the increased revenue the cover affords in the first ten 90-night seasons. a. Compute the mean revenue per night if the cover is not installed. b. Use the answer to (a) to compute the projected total revenue per 90-night season if the cover is not installed. c. Compute the projected total revenue per season when the cover is in place. To do so assume that if the cover were in place the revenue each night of the season would be the same as the revenue on a clear night. d. Using the answers to (b) and (c), decide whether or not the additional cost of the installation of the cover will be recovered from the increased revenue over the first ten years. Will the owner have the cover installed?

a. 2523.25 b. 227,092.5 c. 270,000 d. The owner will install the cover.

Scores on a national exam are normally distributed with mean 382 and standard deviation 26. a. Find the score that is the 50th percentile. b. Find the score that is the 90th percentile.

a. 382 b. 415

A random sample is drawn from a normally distributed population of unknown standard deviation. Construct a 99% confidence interval for the population mean based on the information given. a. n = 18, x⎯⎯=386x-=386, s = 24 b. n = 7, x⎯⎯=386x-=386, s = 24

a. 386±16.4386±16.4 b. 386±33.6

Wildlife researchers tranquilized and weighed three adult male polar bears. The data (in pounds) are: 926, 742, 1,109. Assume the weights of all bears are normally distributed. a. Construct an 80% confidence interval for the mean weight of all adult male polar bears using these data. b. Convert the three weights in pounds to weights in kilograms using the conversion 1 lb = 0.453 kg (so the first datum changes to (926)(0.453)=419(926)(0.453)=419). Use the converted data to construct an 80% confidence interval for the mean weight of all adult male polar bears expressed in kilograms. c. Convert your answer in part (a) into kilograms directly and compare it to your answer in (b). This illustrates that if you construct a confidence interval in one system of units you can convert it directly into another system of units without having to convert all the data to the new units.

a. 926±200926±200 b. 419±90419±90 c. 419±91

A random sample of size 14 is drawn from a normal population. The summary statistics are x⎯⎯=933x-=933 and s = 18. a. Construct an 80% confidence interval for the population mean μ. b. Construct a 90% confidence interval for the population mean μ. c. Comment on why one interval is longer than the other.

a. 933±6.5933±6.5 b. 933±8.5933±8.5 c. Asking for greater confidence requires a longer interval.

A random sample is drawn from a normally distributed population of known standard deviation 5. Construct a 99.8% confidence interval for the population mean based on the information given (not all of the information given need be used). a. n = 16, x⎯⎯=98x-=98, s = 5.6 b. n = 9, x⎯⎯=98x-=98, s = 5.6

a. 98±3.998±3.9 b. 98±5.2

Five thousand lottery tickets are sold for $1 each. One ticket will win $1,000, two tickets will win $500 each, and ten tickets will win $100 each. Let X denote the net gain from the purchase of a randomly selected ticket. a. Construct the probability distribution of X. b. Compute the expected value E(X)E(X) of X. Interpret its meaning. c. Compute the standard deviation σ of X.

a. P (-1) = 4987/5000 P (999) = 1/5000 P (499) = 2/5000 P (99) = 10/5000 b. -0.4 c. 17.8785

A manufacturer receives a certain component from a supplier in shipments of 100 units. Two units in each shipment are selected at random and tested. If either one of the units is defective the shipment is rejected. Suppose a shipment has 5 defective units. a. Construct the probability distribution for the number X of defective units in such a sample. (A tree diagram is helpful.) b. Find the probability that such a shipment will be accepted.

a. P (0) = 0.902 P (1) = 0.096 P (2) = 0.002 b. 0.902

A roulette wheel has 38 slots. Thirty-six slots are numbered from 1 to 36; half of them are red and half are black. The remaining two slots are numbered 0 and 00 and are green. In a $1 bet on red, the bettor pays $1 to play. If the ball lands in a red slot, he receives back the dollar he bet plus an additional dollar. If the ball does not land on red he loses his dollar. Let X denote the net gain to the bettor on one play of the game. a. Construct the probability distribution of X. b. Compute the expected value E(X)E(X) of X, and interpret its meaning in the context of the problem. c. Compute the standard deviation of X.

a. P(-1) = 20/38 P(1) = 18/38 b. E(X)=−0.0526E(X)=−0.0526 In many bets the bettor sustains an average loss of about 5.25 cents per bet. c. 0.9986

Classify each random variable as either discrete or continuous. a. The number of arrivals at an emergency room between midnight and 6:00 a.m. b. The weight of a box of cereal labeled "18 ounces." c. The duration of the next outgoing telephone call from a business office. d. The number of kernels of popcorn in a 1-pound container. e. The number of applicants for a job.

a. discrete b. continuous c. continuous d. discrete e. discrete

Classify each random variable as either discrete or continuous. a. The number of boys in a randomly selected three-child family. b. The temperature of a cup of coffee served at a restaurant. c. The number of no-shows for every 100 reservations made with a commercial airline. d. The number of vehicles owned by a randomly selected household. e. The average amount spent on electricity each July by a randomly selected household in a certain state.

a. discrete b. continuous c. discrete d. discrete e. continuous

An appliance store sells 20 refrigerators each week. Ten percent of all purchasers of a refrigerator buy an extended warranty. Let X denote the number of the next 20 purchasers who do so. a. Verify that X satisfies the conditions for a binomial random variable, and find n and p. b. Find the probability that X is zero. c. Find the probability that X is two, three, or four. d. Find the probability that X is at least five. 19

a. n = 20, p = 0.1 b. 0.1216 c. 0.5651 d. 0.0432

Determine whether or not the random variable X is a binomial random variable. If so, give the values of n and p. If not, explain why not. a. X is the number of dots on the top face of fair die that is rolled. b. X is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck. c. X is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which 0.02% of all parts are defective. d. X is the number of times the number of dots on the top face of a fair die is even in six rolls of the die. e. X is the number of dice that show an even number of dots on the top face when six dice are rolled at once.

a. not binomial; not success/failure. b. not binomial; trials are not independent. c. binomial; n = 10, p = 0.0002 d. binomial; n = 6, p = 0.5 e. binomial; n = 6, p = 0.5

Random samples of size n produced sample proportions pˆp^ as shown. In each case decide whether or not the sample size is large enough to assume that the sample proportion PˆP^ is normally distributed. a. n = 50, pˆ=0.48p^=0.48 b. n = 50, pˆ=0.12p^=0.12 c. n = 100, pˆ=0.12

a. pˆ±3pˆqˆn‾‾‾√=0.48±0.21p^±3p^q^n=0.48±0.21, yes b. pˆ±3pˆqˆn‾‾‾√=0.12±0.14p^±3p^q^n=0.12±0.14, no c. pˆ±3pˆqˆn‾‾‾√=0.12±0.10p^±3p^q^n=0.12±0.10, yes

Identify the set of possible values for each random variable. (Make a reasonable estimate based on experience, where necessary.) a. The number of heads in two tosses of a coin. b. The average weight of newborn babies born in a particular county one month. c. The amount of liquid in a 12-ounce can of soft drink. d. The number of games in the next World Series (best of up to seven games). e. The number of coins that match when three coins are tossed at once.

a. {0.1.2}{0.1.2} b. an interval (a,b)(a,b) (answers vary) c. an interval (a,b)(a,b) (answers vary) d. {4,5,6,7} e. {2,3}

The sample space for tossing three coins is S={hhh,hht,hth,htt,thh,tht,tth,ttt}S={hhh,hht,hth,htt,thh,tht,tth,ttt} a. List the outcomes that correspond to the statement "All the coins are heads." b. List the outcomes that correspond to the statement "Not all the coins are heads." c. List the outcomes that correspond to the statement "All the coins are not heads."

a. {hhh}{hhh} b. {hht,hth,htt,thh,tht,tth,ttt}{hht,hth,htt,thh,tht,tth,ttt} c. {ttt}{ttt}

A population has mean 128 and standard deviation 22. a. Find the mean and standard deviation of X⎯⎯X- for samples of size 36. b. Find the probability that the mean of a sample of size 36 will be within 10 units of the population mean, that is, between 118 and 138.

a. μX⎯⎯⎯=128μX-=128, σX⎯⎯⎯=3.67σX-=3.67 b. 0.9936

A population has mean 72 and standard deviation 6. a. Find the mean and standard deviation of X⎯⎯X- for samples of size 45. b. Find the probability that the mean of a sample of size 45 will differ from the population mean 72 by at least 2 units, that is, is either less than 70 or more than 74.

a. μX⎯⎯⎯=72μX-=72, σX⎯⎯⎯=0.8944σX-=0.8944 b. 0.0250

A population has mean 75 and standard deviation 12. Random samples of size 121 are taken. Find the mean and standard deviation of the sample mean. How would the answers to part (a) change if the size of the samples were 400 instead of 121?

a. μX⎯⎯⎯=75μX-=75, σX⎯⎯⎯=1.09σX-=1.09 b. μX⎯⎯⎯μX- stays the same but σX⎯⎯⎯σX- decreases to 0.6

Find the value of z*z* that yields the probability shown. a. P(Z<z*)=0.1500P(Z<z*)=0.1500 b. P(Z<z*)=0.7500P(Z<z*)=0.7500 c. P(Z>z*)=0.3333P(Z>z*)=0.3333 d. P(Z>z*)=0.8000

a. −1.04 b. 0.67 c. 0.43 d. −0.84

Find the value of z*z* that yields the probability shown. P(Z<z*)=0.0075P(Z<z*)=0.0075 P(Z<z*)=0.9850P(Z<z*)=0.9850 P(Z>z*)=0.8997P(Z>z*)=0.8997 P(Z>z*)=0.0110

a. −2.43 b. 2.17 c. −1.28 d. 2.29

Suppose that 2% of all cell phone connections by a certain provider are dropped. Find the probability that in a random sample of 1,500 calls at most 40 will be dropped. First verify that the sample is sufficiently large to use the normal distribution.

p±3pqn‾‾‾√=0.02±0.01p±3pqn=0.02±0.01 and [0.01,0.03]⊂[0,1],0.9671

Suppose that 8% of all males suffer some form of color blindness. Find the probability that in a random sample of 250 men at least 10% will suffer some form of color blindness. First verify that the sample is sufficiently large to use the normal distribution.

p±3pqn‾‾‾√=0.08±0.05p±3pqn=0.08±0.05 and [0.03,0.13]⊂[0,1],0.1210

The proportion of a population with a characteristic of interest is p = 0.37. Find the mean and standard deviation of the sample proportion PˆP^ obtained from random samples of size 1,600.

μPˆ=0.37μP^=0.37, σPˆ=0.012

The proportion of a population with a characteristic of interest is p = 0.76. Find the mean and standard deviation of the sample proportion PˆP^ obtained from random samples of size 1,200.

μPˆ=0.76μP^=0.76, σPˆ=0.012

Random samples of size 225 are drawn from a population with mean 100 and standard deviation 20. Find the mean and standard deviation of the sample mean.

μX⎯⎯⎯=100μX-=100, σX⎯⎯⎯=1.33

On every passenger vehicle that it tests an automotive magazine measures, at true speed 55 mph, the difference between the true speed of the vehicle and the speed indicated by the speedometer. For 36 vehicles tested the mean difference was −1.2 mph with standard deviation 0.2 mph. Construct a 90% confidence interval for the mean difference between true speed and indicated speed for all vehicles.

−1.2±0.05