Chapter 5
For the binomial distribution, which formula finds the standard deviation?
-npq
The standard deviation of the Poisson distribution is calculated using _______.
-u
A binomial experiment with 100 trials was repeated 100 times, and the histogram was obtained for the number of successes. Which of the following is the most reasonable value for p, the probability of success on any one trial?
0.5 0.9
Consider the binomial probability distribution function P(x)=25!x!(25−x)!0.6x0.425−x. What is the probability of success?
0.6
The notation __________ is used for the probability of failure on any trial in a binomial experiment.
1-p
A national park service is using reported sightings of bears to estimate the population of bears in a particular national park. Suppose that the number x of reported sightings per week follows the Poisson distribution P(x)=6.2xe−6.2x!. What is the standard deviation of the number of bears spotted per week in this park?
2.49 bears per week
Consider the binomial probability distribution function P(x)=25!x!(25−x)!0.6x0.425−x. How many trials are in the experiment?
25
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
A national park service is using reported sightings of bears to estimate the population of bears in a particular national park. Suppose that the number x of reported sightings per week follows the Poisson distribution P(x)=6.2xe−6.2x!. What is the average number of bears spotted per week in this park?
6.2 bears per week
Suppose that cars pass a certain tollbooth at an average rate of 10 cars per minute between the hours of 7 a.m. and 9 a.m. The random variable X counts the number of cars that pass the tollbooth between 7 a.m. and 7:30 a.m. What is the value of lambda, λ, for the Poisson process?
=10 cars per minute
Which of the following statements is not true about binomial probability distributions?
As the probability of success increases, the probability distribution for a binomial variable becomes bell shaped.
Indicate a similarity between a Poisson distribution and a binomial distribution.
Both distributions have an independence requirement.
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. Nine 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.
Suppose a fair die is rolled ten times and the result is recorded each time. Does this constitute a binomial experiment? Why or why not?
No, because there are more than two outcomes for each trial.
Cards are drawn with replacement from a standard deck until a king is drawn. Does this constitute a binomial experiment? Why or why not?
No, because there is not a fixed number of trials.
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. Five hundred different voters in a region with two major political parties, A and B, are randomly selected from the population of 8500 registered voters. Each is asked if he or she is a member of political party A, recording Yes or No. (Similar question)
No, the trail are not independent and the sample is more than 5% of the population.
Twenty percent of adults in a particular community have at least a bachelor's degree. Suppose x is a binomial random variable that counts the number of adults with at least a bachelor's degree in a random sample of 100 adults from the community. Which of the following probability statements indicates the probability that at least 30 adults have at least a bachelor's degree?
P(x>-30)
A technology-support call center for a software company knows that they receive an average of 10 calls per hour between 8 a.m. and 5 p.m. on weekdays. If the random variable X counts the number of calls that are received between 9 a.m. and 10 a.m., then X follows a _______ probability distribution.
Poisson
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
If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. This represents the _______.
Rare Event Rule
A national park service is using reported sightings of bears to estimate the population of bears in a particular national park. Suppose that the number x of reported sightings per week follows a Poisson distribution. Based on the histogram shown to the right, what is the average number of bears spotted per week in this park?
The average number of bears spotted per week is 4.
In the Poisson probability distribution function P(x)=λxx!e−λ, what does λ represent?
The average number of occurrences of the event in an interval
Under what conditions will a binomial probability histogram be approximately bell-shaped?
The histogram will be approximately bell-shaped if n is large or if p is close to 0.5.
What is the mean of a probability distribution?
The mean is the expected value of the random variable.
In the Poisson probability distribution function P(x)=λxx!e−λ, what does x represent?
The number of occurrences in an interval of fixed length
Which of the following is NOT a requirement of the Poisson Distribution?
The occurrences must be dependent.
What does it mean to say that the trials in a binomial experiment are independent of each other?
The outcome of one trial does not affect the outcomes of the other trials.
Based on a survey, when 1009 consumers were asked if they are comfortable with drones delivering their purchases, 42% said yes. The probability of randomly selecting 30 of the 1009 consumers and getting exactly 24 who are comfortable with the drones is represented as 0+. What does 0+ indicate? Does 0+ indicate that it is impossible to get exactly 24 consumers who are comfortable with drones?
The probability 0+ indicates that the probability is a very small positive value. It indicates that the event is possible, but very unlikely.
In the Poisson probability distribution function P(x)=λxx!e−λ, what does P(x) represent?
The probability of obtaining x occurrences in the fixed interval
Which of the following is not a condition for a random variable to follow a Poisson process?
The probability of the occurrence of the event is the same for each trial.
Is the random variable given in the accompanying table discrete or continuous? Explain.
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?
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 girls) and P( or more girls). Is 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(55 or more girls), so girls in 100 births is not a significantly high number of girls because the relevant probability is greater than 0.05.
Identify the requirements for a discrete probability distribution.
The sum of the probabilities must equal one. Each probability must be between zero and one inclusive.
Which of the following is not a criterion for the binomial distribution?
The trials must be dependent.
Which of the following is not a requirement of the binomial probability distribution?
The trials must be dependent.
Based on a survey, assume that 58% of consumers are comfortable having drones deliver their purchases. Suppose we want to find the probability that when six consumers are randomly selected, exactly two of them are comfortable with the drones. What is wrong with using the multiplication rule to find the probability of getting two consumers comfortable with drones followed by four consumers not comfortable, as in this calculation: (0.58)(0.58)(0.42)(0.42)(0.42)(0.42)=0.0105? (Similar question)
There are other arrangements consisting of two consumers who are comfortable and four who are not. The probabilities corresponding to those other arrangements should also be included in the result
Explain how to find the mean of a discrete random variable.
To find the mean of a random variable, multiply each value of the random variable by its probability and then add those products.
Describe how the value of p affects the shape of the binomial probability histogram.
When n is small, the shape of the binomial distribution is determined by p. If p is close to zero, the distribution is skewed right. If p is close to 0.5, the distribution is approximately symmetric. If p is close to one, the distribution is skewed left.
It is assumed that approximately 15% of adults in the U.S. are left-handed. Consider the probability that among 100 adults selected in the U.S., there are at least 30 who are left-handed. Given that the adults surveyed were selected without replacement, can the probability be found by using the binomial probability formula with x counting the number who are left-handed? Why or why not?
Yes, because the 100 adults represent less than 5% of the U.S. adult population, the trials can be treated as independent.
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 140 couples use the YSORT method and give birth to 140 babies, the sex of the babies is recorded.
Yes, because the procedure satisfies all the criteria for a binomial distribution.
Three cards are drawn with replacement from a standard deck, and the number of kings is noted. Does this constitute a binomial experiment? Why or why not?
Yes, because there are three independent draws. For each draw there are two outcomes (king and not king) and a constant probability of getting a king.
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.
In a probability histogram, there is a correspondence between _______.
area and probability
A technology-support call center for a software company knows that 10% of its customers call tech support during installation of its software. If the random variable X counts the number of customers out of 1,000 who call, then X follows a _______ probability distribution.
binomial
A _______ random variable has infinitely many values associated with measurements.
continuous
A __________ random variable has infinitely many values which can be plotted on a number line in an uninterrupted fashion.
continuous
A _______ random variable has either a finite or a countable number of values.
discrete
A Poisson probability distribution is a _______ distribution in which the random variable x counts the number of occurrences of an event in a fixed _______.
discrete;intervals
The binomial probability distribution is a __________ probability distribution that describes probabilities for experiments in which there are two __________ outcomes.
discrete;mutually exclusive
The _______ of a discrete random variable represents the mean value of the outcomes.
excepted value
The notation __________ is used for the number of trials in a binomial experiment.
n
Identify the expression for calculating the mean of a binomial distribution.
np
If a random variable X follows a Poisson process modeled by the probability distribution function P(x)=λxx!e−λ, then what does its standard deviation, σx, equal?
ox=-u
The notation __________ is used for the probability of sucess on any trial in a binomial experiment.
p
A _______ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
random
A(n) __________ is a numerical measure of the outcome of a probability experiment.
random variable
The probability of obtaining x successes in n independent trials of a binomial experiment is given by P(x)=nCxpx(1−p)n−x, where p is the probability of success. What does the nCx represent in the formula?
the number of ways to get x successes in n trials
The probability of obtaining x successes in n independent trials of a binomial experiment is given by P(x)=nCxpx(1−p)n−x, where p is the probability of success. What does the (1−p)n−x represent in the formula?
the probability of failure raised to the number of failures
The probability of obtaining x successes in n independent trials of a binomial experiment is given by P(x)=nCxpx(1−p)n−x, where p is the probability of success. What does the px represent in the formula?
the probability of success raised to the number of successes
The notation __________ is used for the binomial random variable which counts the number of successes in n independent trials of an experiment.
x
Determine whether the distribution is a discrete probability distribution. If not, state why.
No, because some of the probabilities are negative and they must all be between 0 and 1 inclusive.
Determine whether the distribution is a discrete probability distribution. If not, state why.
No, because the probabilities do not sum to 1.
Three cards are drawn without replacement from a standard deck, and the number of kings is noted. Does this constitute a binomial experiment? Why or why not?
No, because the probability of getting a king is not the same for each of the three draws.
