Statistics Chapter 5

Determine whether the given procedure results in a binomial distribution​ (or a distribution that can be treated as​ binomial). If the procedure is not​ binomial, identify at least one requirement that is not satisfied. Six different senators from the current U.S. Congress are randomly selected without replacement and whether or not​ they've served over 2 terms is recorded.

No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.

Determine whether the given procedure results in a binomial distribution. If it is not​ binomial, identify the requirements that are not satisfied. Randomly selecting 200 citizens of a state and recording their nationalities

No, because there are more than two possible outcomes.

Assume that a procedure yields a binomial distribution with n=5 trials and a probability of success of p=0.8. Use a binomial probability table to find the probability that the number of successes x is exactly 1.

P(1) = .006

Assume that random guesses are made for nine multiple choice questions on an SAT​ test, so that there are n=9 ​trials, each with probability of success​ (correct) given by p=0.35. Find the indicated probability for the number of correct answers. Find the probability that the number x of correct answers is fewer than 4.

P(X < 4) = .6089

The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls. x P(x) 0 0.001 1 0.019 2 0.041 3 0.116 4 0.204 5 0.237 6 0.201 7 0.117 8 0.041 9 0.015 10 0.008 Use the range rule of thumb to identify a range of values that are not significant. Based on the​ result, is 1 girl in 10 births a significantly low number of​ girls? Explain.

The maximum value in this range is 8.4 girls. The minimum value in this range is 1.6 girls. Yes, 1 girl is a significantly low number of​ girls, because 1 girl is below the range of values that are not significant.

Refer to the accompanying​ table, which describes the number of adults in groups of five who reported sleepwalking. Find the mean and standard deviation for the numbers of sleepwalkers in groups of five. x P(x) 0 0.193 1 0.351 2 0.295 3 0.132 4 0.025 5 0.004

The mean is 1.5 sleepwalker(s) The standard deviation is 1.0 sleepwalker(s).

Assume that when adults with smartphones are randomly​ selected, 51​% use them in meetings or classes. If 10 adult smartphone users are randomly​ selected, find the probability that at least 8 of them use their smartphones in meetings or classes.

The probability is .0621

Assume that when adults with smartphones are randomly​ selected, 47​% use them in meetings or classes. If 13 adult smartphone users are randomly​ selected, find the probability that fewer than 4 of them use their smartphones in meetings or classes.

The probability is .0712

Assume that when adults with smartphones are randomly​ selected, 59​% use them in meetings or classes. If 8 adult smartphone users are randomly​ selected, find the probability that exactly 3 of them use their smartphones in meetings or classes.

The probability is .1332

Assume that random guesses are made for 6 multiple-choice questions on a test with 5 choices for each​ question, so that there are n=6 trials, each with probability of success​ (correct) given by p=0.2. Find the probability of no correct answers.

The probability of no correct answers is .262

A survey showed that 84​% of adults need correction​ (eyeglasses, contacts,​ surgery, etc.) for their eyesight. If 19 adults are randomly​ selected, find the probability that no more than 11 of them need correction for their eyesight. Is 11 a significantly low number of adults requiring eyesight​ correction?

The probability that no more than 1 of the 19 adults require eyesight correction is .000. Yes, because the probability of this occurring is small.

The probability of a randomly selected adult in one country being infected with a certain virus is 0.003. In tests for the​ virus, blood samples from 17 people are combined. What is the probability that the combined sample tests positive for the​ virus? Is it unlikely for such a combined sample to test​ positive? Note that the combined sample tests positive if at least one person has the virus.

The probability that the combined sample will test positive is .049. It is not unlikely for such a combined sample to test​ positive, because the probability that the combined sample will test positive is greater than 0.05.

The probability of a randomly selected adult in one country being infected with a certain virus is 0.005. In tests for the​ virus, blood samples from 26 people are combined. What is the probability that the combined sample tests positive for the​ virus? Is it unlikely for such a combined sample to test​ positive? Note that the combined sample tests positive if at least one person has the virus.

The probability that the combined sample will test positive is .122. It is not unlikely for such a combined sample to test​ positive, because the probability that the combined sample will test positive is greater than 0.05.

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 57 tablets, then accept the whole batch if there is only one or none that​ doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a 44​% rate of​ defects, what is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

The probability that this whole shipment will be accepted is .3294 The company will accept 33​% of the shipments and will reject 67​% of the​ shipments, so many of the shipments will be rejected.

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 45 ​tablets, then accept the whole batch if there is only one or none that​ doesn't meet the required specifications. If one shipment of 5000 aspirin tablets actually has a 66​% rate of​ defects, what is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

The probability that this whole shipment will be accepted is 0.2392. The company will accept 24​% of the shipments and will reject 76​% of the​ shipments, so many of the shipments will be rejected.

Is the random variable given in the accompanying table discrete or​ continuous? Explain. Number_of_girls P(x) 0 0.063 1 0.250 2 0.375 3 0.250 4 0.063

The random variable given in the accompanying table is discrete because there are a finite number of values.

The table to the right lists probabilities for the corresponding numbers of girls in three births. What is the random​ variable, what are its possible​ values, and are its values​ numerical? Number_of_girls P(x) 0 0.125 1 0.375 2 0.375 3 0.125

The random variable is​ x, which is the number of girls in three births. The possible values of x are​ 0, 1,​ 2, and 3. The values of the random value x are numerical.

For 100​ births, P(exactly 57 girls) = 0.0301 and ​P(57 or more girls) = 0.097. Is 57 girls in 100 births a significantly high number of​ girls? Which probability is relevant to answering that​ question? Consider a number of girls to be significantly high if the appropriate probability is 0.05 or less.

The relevant probability is P(57 or more girls), so 57 girls in 100 births is not a significantly high number of girls because the relevant probability is greater than 0.05.

Based on a​ survey, assume that 43​% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when four consumers are randomly​ selected, exactly two of them are comfortable with delivery by drones. Identify the values of​ n, x,​ p, and q.

The value of n is 4 The value of x is 2 The value of p is .43 The value of q is .57

Determine whether the given procedure results in a binomial distribution​ (or a distribution that can be treated as​ binomial). If the procedure is not​ binomial, identify at least one requirement that is not satisfied. The YSORT method of gender​ selection, developed by the Genetics​ & IVF​ Institute, was designed to increase the likelihood that a baby will be a boy. When 120 couples use the YSORT method and give birth to 120 ​babies, the genders of the babies are recorded.

Yes, because the procedure satisfies all the criteria for a binomial distribution.

Determine whether or not the procedure described below results in a binomial distribution. If it is not​ binomial, identify at least one requirement that is not satisfied. Six hundred different voters in a region with two major political​ parties, A and​ B, are randomly selected from the population of 4.1 million registered voters. Each is asked if he or she is a member of political party​ A, recording Yes or No.

Yes, the result is a binomial probability distribution.

When conducting research on color blindness in​ males, a researcher forms random groups with five males in each group. The random variable x is the number of males in the group who have a form of color blindness. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied. x P(x) 0 0.667 1 0.276 2 0.053 3 0.003 4 0.001 5 0.000 Does the table show a probability​ distribution? Select all that apply. A. Yes, the table shows a probability distribution. B. No, not every probability is between 0 and 1 inclusive. C. No, the sum of all the probabilities is not equal to 1. D. No, the random variable x is categorical instead of numerical. E. No, the random variable​ x's number values are not associated with probabilities. Find the mean of the random variable x. Find the standard deviation of the random variable x.

Yes, the table shows a probability distribution. μ = .4 male(s) σ = .6 male(s)

Groups of adults are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they would feel comfortable in a​ self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied Does the table show a probability​ distribution? Select all that apply. ​A. Yes, the table shows a probability distribution. B. No, the random variable x is categorical instead of numerical. C. No, the random variable​ x's number values are not associated with probabilities. D. No, the sum of all the probabilities is not equal to 1. Your answer is correct. E. No, not every probability is between 0 and 1 inclusive.

Yes, the table shows a probability distribution. μ = .9 adult(s) σ = .8 adult(s)

A sociologist randomly selects single adults for different groups of​ three, and the random variable x is the number in the group who say that the most fun way to flirt is in person. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied. x P(x) 0 0.087 1 0.348 2 0.400 3 0.165 Does the table show a probability​ distribution? Select all that apply. ​A. Yes, the table shows a probability distribution. B. No, the random variable x is categorical instead of numerical. C. No, the random variable​ x's number values are not associated with probabilities. D. No, the sum of all the probabilities is not equal to 1. Your answer is correct. E. No, not every probability is between 0 and 1 inclusive.

Yes, the table shows a probability distribution. μ = 1.6 adult(s) σ = .9 adult(s)

Five males with an​ X-linked genetic disorder have one child each. The random variable x is the number of children among the five who inherit the​ X-linked genetic disorder. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied. x P(x) 0 0.029 1 0.152 2 0.319 3 0.319 4 0.152 5 0.029 Does the table show a probability​ distribution? Select all that apply. A. Yes, the table shows a probability distribution. B. No, the random variable​ x's number values are not associated with probabilities. C. No, the sum of all the probabilities is not equal to 1. D. No, the random variable x is categorical instead of numerical. E. No, not every probability is between 0 and 1 inclusive. Find the mean of the random variable x. Find the standard deviation of the random variable x.

Yes, the table shows a probability distribution. μ = 2.5 ​child(ren) σ = 1.1 child(ren)

A Gallup poll of 1236 adults showed that​ 12% of the respondents believe that it is bad luck to walk under a ladder. Consider the probability that among 30 randomly selected people from the 1236 who were​ polled, there are at least 2 who have that belief. Given that the subjects surveyed were selected without​ replacement, the events are not independent. Can the probability be found by using the binomial probability​ formula? Why or why​ not?

Yes. Although the selections are not​ independent, they can be treated as being independent by applying the​ 5% guideline.

The accompanying table describes the random variable​ x, the numbers of adults in groups of five who reported sleepwalking. Complete parts​ (a) through​ (d) below. x P(x) 0 0.167 1 0.367 2 0.301 3 0.139 4 0.021 5 0.005 a. Find the probability of getting exactly 4 sleepwalkers among 5 adults. b. Find the probability of getting 4 or more sleepwalkers among 5 adults. c. Which probability is relevant for determining whether 4 is a significantly high number of sleepwalkers among 5​ adults: the result from part​ (a) or part​ (b)? d. Is 4 a significantly high number of 4 sleepwalkers among 5​ adults? Why or why​ not? Use 0.05 as the threshold for a significant event.

a) .021 b) .026 c) Since the probability of getting 5 sleepwalkers includes getting 4​ sleepwalkers, the result from part​ (b) is the relevant probability. d) Yes, since the appropriate probability is less than​ 0.05, it is a significantly high number.

The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts​ (a) through​ (d) below. Number of Girls x P(x) 0 0.002 1 0.022 2 0.116 3 0.236 4 0.248 5 0.236 6 0.116 7 0.022 8 0.002 a. Find the probability of getting exactly 1 girl in 8 births. b. Find the probability of getting 1 or fewer girls in 8 births. c. Which probability is relevant for determining whether 1 is a significantly low number of girls in 8​ births: the result from part​ (a) or part​ (b)? d. Is 1 a significantly low number of girls in 8​ births? Why or why​ not? Use 0.05 as the threshold for a significant event.

a) .022 b) .024 c) Since getting 0 girls is an even lower number of girls than getting 1​ girl, the result from part​ (b) is the relevant probability. d) ​Yes, since the appropriate probability is less than​ 0.05, it is a significantly low number.

The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts​ (a) through​ (d) below. Number of Girls x P(x) 0 0.003 1 0.017 2 0.105 3 0.213 4 0.324 5 0.213 6 0.105 7 0.017 8 0.003 a. Find the probability of getting exactly 6 girls in 8 births. b. Find the probability of getting 6 or more girls in 8 births. c. Which probability is relevant for determining whether 6 is a significantly high number of girls in 8​ births: the result from part​ (a) or part​ (b)? d. Is 6 a significantly high number of girls in 8​ births? Why or why​ not? Use 0.05 as the threshold for a significant event.

a) .105 b) .125 c) The result from part​ b, since it is the probability of the given or more extreme result. d) No, since the appropriate probability is greater than​ 0.05, it is not a significantly high number.

Determine whether the value is a discrete random​ variable, continuous random​ variable, or not a random variable. a. The distance a baseball travels in the air after being hit b. The number of people in a restaurant that has a capacity of 200 c. The political party affiliation of adults in the United States d. The exact time it takes to evaluate 27 + 72 e. The number of statistics students now reading a book f. The number of hits to a website in a day

a) It is a continuous random variable b) It is a discrete random variable c) It is not a random variable d) It is a continuous random variable e) It is a discrete random variable f) It is a discrete random variable

Determine whether the value is a discrete random​ variable, continuous random​ variable, or not a random variable. a. The time required to download a file from the Internet b. The number of light bulbs that burn out in the next week in a room with 19 bulbs c. The gender of college students d. The number of bald eagles in a country e. The distance a baseball travels in the air after being hit f. The amount of rain in City B during April

a) It is a continuous random variable b) It is a discrete random variable c) It is not a random variable d) It is a discrete random variable e) It is a continuous random variable f) It is a continuous random variable

Determine whether the following value is a continuous random​ variable, discrete random​ variable, or not a random variable. a. The weight of a hamburger b. The usual mode of transportation of people in City A c. The square footage of a pool d. The number of textbook authors now eating a meal e. The distance a football travels in the air after being thrown f. The number of hits to a website in a week

a) It is a continuous random variable b) It is not a random variable c) It is a continuous random variable d) It is a discrete random variable e) It is a continuous random variable f) It is a discrete random variable

Determine whether the following value is a continuous random​ variable, discrete random​ variable, or not a random variable. a. The square footage of a pool b. The hair color of adults in the United States c. The weight of a hamburger d. The number of runs scored during a baseball game e. The height of a randomly selected person f. The number of statistics students now doing their homework

a) continuous b) Not a random variable c) continuous d) discrete e) continuous f) discrete

Multiple-choice questions each have five possible answers (a, b, c, d, e)​, one of which is correct. Assume that you guess the answers to three such questions. a. Use the multiplication rule to find ​P(WWC), where C denotes a correct answer and W denotes a wrong answer. b. Beginning with WWC​, make a complete list of the different possible arrangements of one correct answer and two wrong answers​, then find the probability for each entry in the list. c. Based on the preceding​ results, what is the probability of getting exactly one correct answer when three guesses are​ made?

a. P(WWC) = .128 b) P(WWC) - see above P(WCW) = .128 P(CWW) = .128 c) .384

Based on a​ poll, 60% of adults believe in reincarnation. Assume that 7 adults are randomly​ selected, and find the indicated probability. Complete parts​ (a) through​ (d) below. a. What is the probability that exactly 6 of the selected adults believe in​ reincarnation? b. What is the probability that all of the selected adults believe in​ reincarnation? c. What is the probability that at least 6 of the selected adults believe in​ reincarnation? d. If 7 adults are randomly​ selected, is 6 a significantly high number who believe in​ reincarnation?

a. The probability that exactly 6 of the 7 adults believe in reincarnation is .131. b. The probability that all of the selected adults believe in reincarnation is .028. c. The probability that at least 6 of the selected adults believe in reincarnation is .159 d. ​No, because the probability that 6 or more of the selected adults believe in reincarnation is greater than 0.05.

Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a​ girl, but assume that the method has no​ effect, so the probability of a girl is 0.5. Assume that the groups consist of 44 couples. Complete parts​ (a) through​ (c) below. a. Find the mean and the standard deviation for the numbers of girls in groups of 44 births. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. c. Is the result of 42 girls a result that is significantly​ high? What does it suggest about the effectiveness of the​ method?

a. The value of the mean is μ = 22 b. Values of 15.4 girls or fewer are significantly low. c. The result is significantly​ high, because 42 girls is greater than 28.6 girls. A result of 42 girls would suggest that the method is effective.

Refer to the accompanying​ table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Find the mean and standard deviation for the number of girls in 8 births. Number of Girls x P(x) 0 0.005 1 0.031 2 0.106 3 0.216 4 0.277 5 0.228 6 0.106 7 0.029 8 0.002

μ = 4 girl(s) σ = 1.4 girl(s)

Ted is not particularly creative. He uses the pickup line​ "If I could rearrange the​ alphabet, I'd put U and I​ together." The random variable x is the number of women Ted approaches before encountering one who reacts positively. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied. x P(x) 0 0.001 1 0.007 2 0.034 3 0.058 Does the table show a probability​ distribution? Select all that apply. ​A. Yes, the table shows a probability distribution. B. No, the random variable x is categorical instead of numerical. C. No, the random variable​ x's number values are not associated with probabilities. D. No, the sum of all the probabilities is not equal to 1. Your answer is correct. E. No, not every probability is between 0 and 1 inclusive.

​D. No, the sum of all the probabilities is not equal to 1 The table does not show a probability distribution.