Question 1

The Poisson probability distribution is an example of a continuous probability distribution.

Accepted Answer

The Poisson probability distribution is an example of a discrete probability distribution.

Question 2

A jug contains five black marbles and five white marbles well mixed. A marble is removed and its colour is noted. A second marble is removed, without replacing the first marble, and its colour is also noted. If x is the total number of black marbles in the two draws, then x has a binomial distribution.

Accepted Answer

The number of black marbles in the second draw is dependent on the colour of the first marble removed, so the trials are not independent, violating an important requirement for the binomial distribution. Therefore, x does not have a binomial distribution.

Question 3

The probability distribution of a Poisson random variable provides a good model for data that represent the number of occurrences of a specified event in a given unit of time or space.

Accepted Answer

The statement is true. The Poisson distribution is commonly used to model data related to the occurrence of events such as accidents, arrivals at a service counter, or defects in a manufacturing process, where the number of events that occur in a fixed time or space interval follows a Poisson distribution.

Question 4

If x is a binomial random variable with n = 20, and p = 0.5, then P(x = 20) = 1.0.

Accepted Answer

P(x = 20) represents the probability of getting 20 successes out of 20 trials with a success probability of 0.5 for each trial. This is calculated using the binomial probability formula, which in this case would yield a value much less than 1.0, specifically $ (0.5)^{20} $.

Question 5

The binomial probability distribution could be used to describe the speed of tennis balls when the players are serving.

Accepted Answer

The binomial probability distribution is used to describe the probabilities of a certain number of successes or failures in a fixed number of independent trials. It is not appropriate for describing continuous variables such as the speed of tennis balls. A continuous probability distribution such as the normal distribution would be more appropriate for this type of variable.

Question 6

A binomial experiment is a sequence of n identical trials such that both of the following properties hold:
a.) each trial produces one of two outcomes that are conventionally called "success" and "failure," and
b.) each trial is independent of any other trial so that the probability of success or failure is constant from trial to trial.

Accepted Answer

This is the definition of a binomial experiment.

Question 7

A binomial probability distribution shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.

Accepted Answer

This statement is true. A binomial probability distribution is based on the outcomes of a binomial experiment, which is a type of experiment where there are only two possible outcomes for each trial (such as success or failure), the trials are independent, the probability of success (or failure) is constant for each trial, and the total number of trials is fixed. The probabilities associated with each possible value of the discrete random variable (such as the number of successes in a certain number of trials) can be calculated using the binomial probability formula.

Question 8

Where n = 100 is the number of trials and p = 0.04 is the probability of success in each trial, the mean of a binomial experiment is  .

Accepted Answer

The answer of Where n = 100 is the number...

Question 9

Where n = 150 is the number of trials and p = 0.6 is the probability of success in each trial, the standard deviation of a binomial experiment is 36.

Accepted Answer

The formula for the standard deviation of a binomial experiment is √(npq), where q = 1 - p. Using n = 150 and p = 0.6, we have q = 0.4. Thus, the standard deviation should be √(150 x 0.6 x 0.4) ≈ 6.12, which is not equal to 36. Therefore, the statement is false.

Question 10

A binomial experiment requires that the success and failure probabilities be constant from one trial to the next, and also that these two probabilities be equal to each other.

Accepted Answer

A binomial experiment requires that the success and failure probabilities be constant from one trial to the next, but they do not need to be equal to each other.

Question 11

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = 0.03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

Accepted Answer

The standard deviation of a binomial distribution is calculated using the formula σ = √(np(1-p)), where n is the sample size and p is the probability of success. Plugging in the given values, we get σ = √(200*0.03*(1-0.03)) = 3.18, not 5.82. Therefore, the statement is false.

Question 12

The distribution of the number of phone calls to a doctor's office in a one-hour time period is likely to be described by a binomial distribution.

Accepted Answer

The binomial distribution models the number of successes in a fixed number of independent trials, with a fixed probability of success. The number of phone calls to a doctor's office in a one-hour time period is not a fixed number of independent trials, and the probability of a call may depend on many factors, such as the time of day, the day of the week, and the weather. Therefore, it is unlikely that the number of phone calls would be described by a binomial distribution.

Question 13

As a rule of thumb, if the sample size n is large relative to the population size N, in particular, if n/N > 0.05, then the resulting experiment will NOT be binomial.

Accepted Answer

This is because if n is large relative to N, the individual trials may no longer be independent and identically distributed, which are necessary assumptions for a binomial experiment. In such cases, a different statistical model may be more appropriate.

Question 14

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = 0.03. Based on this information, the expected number of defects in the sample is six.

Accepted Answer

The expected value of a binomial distribution is given by E(X) = np, where n is the sample size and p is the probability of success (in this case, the probability of a defective part). Therefore, E(X) = 200 * 0.03 = 6, so the expected number of defects in the sample is six.

Question 15

The number of successes observed during the n trials of a binomial experiment is called the binomial random variable.

Accepted Answer

This is the definition of a binomial random variable.

Question 16

A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most two sales is less than 0.10.

Accepted Answer

We can use the binomial distribution formula to find the probability of making at most two sales:
P(X ≤ 2) = C(15, 0)(0.40)^0(0.60)^15 + C(15, 1)(0.40)^1(0.60)^14 + C(15, 2)(0.40)^2(0.60)^13
Using a calculator, we get P(X ≤ 2) ≈ 0.088, which is less than 0.10. Therefore, the statement is true.

Question 17

The Poisson probability distribution provides a good approximation to binomial probabilities when n is large and  = np is small, preferably with np < 7.

Accepted Answer

This statement is true. The Poisson distribution can be used as an approximation of the binomial distribution when the sample size is large and the probability of success is small. If np (the expected number of successes) is less than 7, the Poisson distribution provides a good approximation to the binomial distribution.

Question 18

The binomial distribution is used to describe continuous random variables.

Accepted Answer

The binomial distribution is used to describe discrete random variables, not continuous random variables.

Question 19

There are only two possible outcomes in a binomial experiment where n is the number of trials and p is the probability of success in each trial.

Accepted Answer

In a binomial experiment, each trial can only have two possible outcomes (success or failure) and the probability of success remains constant throughout all trials. Therefore, there are only two possible outcomes in a binomial experiment.

Question 20

The binomial random variable is the number of successes that occur in a certain period of time or space.

Accepted Answer

This is a correct definition of the binomial random variable.

Quiz 5: Several Useful Discrete Distributions