Stat 5.2 HW

If we sample from a small finite population without​ replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two​ types, we can use the hypergeometric distribution. If a population has A objects of one​ type, while the remaining B objects are of the other​ type, and if n objects are sampled without​ replacement, then the probability of getting x objects of type A and n−x objects of type B under the hypergeometric distribution is given by the following formula. In a lottery​ game, a bettor selects six numbers from 1 to 56 ​(without repetition), and a winning six​-number combination is later randomly selected. Find the probabilities of getting exactly four winning numbers with one ticket.​ (Hint: Use A=6​, B=50​, n=6​, and x=4​.)

P(4) = 0.0006 P(x)= [[A!(A−x)!x!] [B!(B−n+x)!(n−x)!]] ÷[(A+B)!(A+B−n)!n!] P(x)= [[6!(6−4)!4!] [50!(50−6+4)!(6−4)!]] / [(6+50)!(6+50−6)!6!]

Assume that a procedure yields a binomial distribution with n=8 trials and a probability of success of p=0.90. Use a binomial probability table to find the probability that the number of successes x is exactly 7.

P(7) = 0.383

Assume that random guesses are made for seven multiple choice questions on an SAT​ test, so that there are n=7 ​trials, each with probability of success​ (correct) given by p=0.5. 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) = 0.5

A main goal in statistics is to interpret and understand the meaning of statistical values. The​ _______ can be very helpful in understanding the meaning of the mean and standard deviation.

Range Rule of Thumb

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.

A survey showed that 76​% of adults need correction​ (eyeglasses, contacts,​ surgery, etc.) for their eyesight. Is 1 a significantly low number of adults requiring eyesight​ correction?

Yes​, because the probability of this occurring is small

Based on a​ poll, 40​% of adults believe in reincarnation. Assume that 8 adults are randomly​ selected, and find the indicated probability. If 8 adults are randomly​ selected, is 7 a significantly high number who believe in​ reincarnation?

Yes​, because the probability that 7 or more of the selected adults believe in reincarnation is less than 0.05.

Identify the expression for calculating the mean of a binomial distribution.

np

In the binomial probability​ formula, the variable x represents the​ _______.

number of successes

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 18 couples. Find the mean and the standard deviation for the numbers of girls in groups of 18 births.

μ=9

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 36. Find the mean for the numbers of peas with green pods in the groups of 36.

μ=9

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 18 couples. Find the standard deviation for the numbers of girls in groups of 18 births.

σ=2.1

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 36. Find the standard deviation for the numbers of peas with green pods in the groups of 36.

σ=2.6

Multiple-choice questions each have four possible answers (a, b, c, d)​, one of which is correct. Assume that you guess the answers to three such questions. Based on the preceding​ results, what is the probability of getting exactly two correct answers when three guesses are​ made?

​P(CCW​)+​P(CWC​)+​P(WCC​) = 0.046875 +0.046875 +0.046875 = 0.140625

Multiple-choice questions each have four possible answers (a, b, c, d)​, one of which is correct. Assume that you guess the answers to three such questions. Beginning with CWC​, make a complete list of the different possible arrangements of two correct answers and one wrong answer​, then find the probability for each entry in the list.

​P(CWC​) = P(C)•​P(W)•​P(C) (0.25)•(0.75)•(0.25) = 0.046875 ​ P(WCC​) = P(W)•​P(C)•​P(C) (0.75)•(0.25)•(0.25) =0.046875

For the binomial​ distribution, which formula finds the standard​ deviation?

√npq

A survey showed that 76​% of adults need correction​ (eyeglasses, contacts,​ surgery, etc.) for their eyesight. If 19 adults are randomly​ selected, find the probability that no more than 1 of them need correction for their eyesight.

0.000

Based on a​ poll, 40​% of adults believe in reincarnation. Assume that 8 adults are randomly​ selected, and find the indicated probability. What is the probability that all of the selected adults believe in​ reincarnation?

0.001

Based on a​ poll, 40​% of adults believe in reincarnation. Assume that 8 adults are randomly​ selected, and find the indicated probability. What is the probability that exactly 7 of the selected adults believe in​ reincarnation?

0.008

Based on a​ poll, 40​% of adults believe in reincarnation. Assume that 8 adults are randomly​ selected, and find the indicated probability. What is the probability that at least 7 of the selected adults believe in​ reincarnation?

0.009

If a procedure meets all of the conditions of a binomial distribution except the number of trials is not​ fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by P(x)=p(1−p)x−1​, where p is the probability of success on any one trial. Subjects are randomly selected for a health survey. The probability that someone is a universal donor​ (with group O and type Rh negative​ blood) is 0.07. Find the probability that the first subject to be a universal blood donor is the seventh person selected.

0.0453 P(7) = 0.07(1-0.07)^7-1

Multiple-choice questions each have four possible answers (a, b, c, d)​, one of which is correct. Assume that you guess the answers to three such questions. Use the multiplication rule to find P(CWC​), where C denotes a correct answer and W denotes a wrong answer.

0.046875 ​P(CCW​) = P(C​)•​P(C​)•​P(W​) (0.25)•(0.25)•(0.75) =0.046875

Assume that when adults with smartphones are randomly​ selected, 45​% use them in meetings or classes. If 25 adult smartphone users are randomly​ selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.

0.0520 P(x) = [n! / (n-x)!x!] (p^x) (q^n-x) n = 25 x = 15 p = 0.45 q = 0.55 P(x) = [25! / (25-15)!15!] (0.45^15) (0.55^25-15) = 0.0520

Assume that when adults with smartphones are randomly​ selected, 58​% use them in meetings or classes. If 8 adult smartphone users are randomly​ selected, find the probability that at least 6 of them use their smartphones in meetings or classes.

0.2750 P(x) = [n! / (n-x)!x!] (p^x) (q^n-x) n = 8 x = 6 p = 0.58 q = 0.42 P(x) = [8! / (8-6)!6!] (0.58^6) (0.42^8-16) = 0.2750

Based on a​ poll, among adults who regret getting​ tattoos, 24​% say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Find the probability that none of the selected adults say that they were too young to get tattoos.

0.3336

Based on a​ poll, among adults who regret getting​ tattoos, 24​% say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.

0.4214

Based on a​ survey, assume that 44​% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when five 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 p is

0.44

Based on a​ survey, assume that 44​% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when five 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 q is

0.56 1-p = q

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 47 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 3​% rate of​ defects, what is the probability that this whole shipment will be​ accepted?

0.5862

Based on a​ poll, among adults who regret getting​ tattoos, 24​% say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Find the probability that the number of selected adults saying they were too young is 0 or 1.

0.7550

When purchasing bulk orders of​ batteries, a toy manufacturer uses this acceptance sampling​ plan: Randomly select and test 55 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 3 batteries do not meet specifications. A shipment contains 3000 ​batteries, and 1​% of them do not meet specifications. Will almost all such shipments be​ accepted, or will many be​ rejected?

0.9977

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 18 couples. 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 18 couples. Use the range rule of thumb to find the values separating results that are significantly high

13.2

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 36. Use the range rule of thumb to find the values separating results that are significantly high.

14.2

Based on a​ survey, assume that 44​% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when five 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 x is

2

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 36. Use the range rule of thumb to find the values separating results that are significantly low

3.8

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 18 couples. Use the range rule of thumb to find the values separating results that are significantly low

4.8

Based on a​ survey, assume that 44​% of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when five 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

5

If calculations are​ time-consuming and if a sample size is no more than​ 5% of the size of the​ population, the​ _______ states to treat the selections as being independent​ (even if the selections are technically​ dependent).

5% Guideline for Cumbersome Calculations

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 sex​ 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 sex of the babies is recorded. Does the procedure represent a binomial​ distribution?

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

Based on a​ poll, among adults who regret getting​ tattoos, 24​% say that they were too young when they got their tattoos. Assume that four adults who regret getting tattoos are randomly​ selected, and find the indicated probability. If we randomly select four ​adults, is 1 a significantly low number who say that they were too young to get​ tattoos?

No, because the probability that at most 1 of the selected adults say that they were too young is greater than 0.05.

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. Recording the number of televisions in 150 households

No, because there are more than two possible outcomes

Based on a​ survey, assume that 49​% of consumers are comfortable having drones deliver their purchases. Suppose we want to find the probability that when six consumers are randomly​ selected, exactly four of them are comfortable with the drones. What is wrong with using the multiplication rule to find the probability of getting four consumers comfortable with drones followed by two consumers not​ comfortable, as in this​ calculation: (0.49)(0.49)(0.49)(0.49)(0.51)(0.51)=0.0150​?

There are other arrangements consisting of four consumers who are comfortable and two who are not. The probabilities corresponding to those other arrangements should also be included in the result.

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 47 tablets, then accept the whole batch if there is only one or none that​ doesn't meet the required specifications. Will almost all such shipments be​ accepted, or will many be​ rejected?

The company will accept 58.62​% of the shipments and will reject 41.38​% of the​ shipments, so many of the shipments will be rejected.

When purchasing bulk orders of​ batteries, a toy manufacturer uses this acceptance sampling​ plan: Randomly select and test 55 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 3 batteries do not meet specifications. A shipment contains 3000 ​batteries, and 1​% of them do not meet specifications. What is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

The company will accept 99.77​% of the shipments and will reject 0.23​% of the​ shipments, so almost all of the shipments will be accepted.

Assume that hybridization experiments are conducted with peas having the property that for​ offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 36. Is a result of 1 peas with green pods a result that is significantly​ low? Why or why​ not?

The result is significantly​ low, because 1 peas with green pods is less than 3.8 peas.

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 18 couples. Is the result of 16 girls a result that is significantly​ high? What does it suggest about the effectiveness of the​ method?

The result is significantly​ high, because 16 girls is greater than 13.2 girls. A result of 16 girls would suggest that the method is effective.