Question 1

The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a binomial distribution.

Accepted Answer

The number of males selected from the sample is not fixed and can vary, so it does not have a binomial distribution.

Question 2

Another name for the mean of a probability distribution is its expected value.

Accepted Answer

The expected value of a probability distribution is also known as its mean.

Question 3

The Poisson distribution can be used to model a continuous random variable.

Accepted Answer

The Poisson distribution is a discrete probability distribution often used to model the number of events occurring in a fixed interval of time or space. It cannot be used to model continuous random variables. For continuous random variables, other distributions such as the normal distribution or exponential distribution may be more appropriate.

Question 4

The largest value that a Poisson random variable X can have is n.

Accepted Answer

A Poisson random variable can take any non-negative integer value, from 0 to infinity, so there is no upper limit on its value.

Question 5

In a Poisson distribution, the mean and variance are equal.

Accepted Answer

This is a property of the Poisson distribution. The mean and variance of a Poisson distribution with parameter λ are both equal to λ.

Question 6

The number of customers arriving at a department store in a 5-minute period has a binomial distribution.

Accepted Answer

The binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The arrival of customers at a department store is not a series of independent trials, and the probability of a customer arriving is likely to vary over time. A Poisson distribution may be more appropriate for modeling the arrival of customers at a department store.

Question 7

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts.Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week.Interpret this value.
A)Most of the weeks resulted in rat costs of $13.00.
B)The median cost for the distribution of rat costs is $13.00.
C)The expected or mean cost for all weekly rat purchases is $13.00.
D)The rat cost that occurs more often than any other is $13.00.

Accepted Answer

The given information states that the mean cost of rats used in lab experiments turned out to be $13.00 per week. This means that if we were to add up the costs of rats for all 52 weeks and divide it by 52, we would get an average cost of $13.00 per week. Hence, C is the correct interpretation. Option A and D are incorrect because the fact that the mean is $13.00 does not imply that most rats were purchased for $13.00 or that $13.00 is the mode. Option B is incorrect because we are given the mean and not the median.

Question 8

The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels.Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day.Interpret the value of the mean.
A)The number of tickets that is written most often is 6.5 tickets per day.
B)Half of the days have less than 6.5 tickets written, and half of the days have more than 6.5 tickets written.
C)If we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day.
D)The mean has no interpretation since 0.5 ticket can never be written.

Accepted Answer

The mean of 6.5 tickets per day represents the expected number of tickets written per day if we sampled all days. It is not the number written most often (mode) or a midpoint where half of the days have less or more tickets. The fact that 0.5 tickets cannot be written does not invalidate the interpretation of the mean as an average or expected value.

Question 9

In a Poisson distribution, the mean and standard deviation are equal.

Accepted Answer

The answer of In a Poisson distribution, the mean and...

Question 10

The diameters of 10 randomly selected bolts have a binomial distribution.

Accepted Answer

The diameter of bolts would more likely have a continuous distribution rather than a discrete binomial distribution.

Question 11

If  remains constant in a binomial distribution, an increase in n will increase the variance.

Accepted Answer

The formula for the variance of a binomial distribution is Var(X) = np(1-p). Since p is held constant, an increase in n will directly increase the variance as a function of n. The variance of a binomial distribution is equal to np(1-p). As n increases while p remains constant, the product np increases, which increases the variance.

Question 12

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process.The mean number of airplanes arriving per minute is 3.The probability that exactly 6 planes arrive in the next minute is 0.05041.

Accepted Answer

The probability of observing exactly $k$ events in a Poisson process is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate (mean) of occurrences in a fixed interval of time. Here, $\lambda = 3$ and $k = 6$. Plugging these values into the formula gives $P(6; 3) = \frac{e^{-3} 3^6}{6!} \approx 0.0504$, which matches the provided probability.

Question 13

The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a hypergeometric distribution.

Accepted Answer

A hypergeometric distribution is used when sampling without replacement, which is the case in this scenario. Therefore, the number of males selected in the sample does follow a hypergeometric distribution.

Question 14

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

Accepted Answer

The Poisson distribution is often used to model the number of arrivals over a fixed time period, so it is likely that the number of customers arriving in a 5-minute period could be modeled using a Poisson distribution.

Question 15

A financial analyst is presented with information on the past records of 60 start-up companies and told that in fact only 3 of them have managed to become highly successful. He selected 3 companies from this group as the candidates for success.To analyze his ability to spot the companies that will eventually become highly successful, he will use what type of probability distribution?
A)binomial distribution.
B)Poisson distribution.
C)hypergeometric distribution.
D)none of the above.

Accepted Answer

The financial analyst is selecting 3 companies from a finite population (the 60 start-up companies), and is interested in the probability of selecting the 3 successful companies from this population (without replacement). Therefore, the appropriate distribution to use is the hypergeometric distribution. The binomial distribution assumes a fixed number of trials and independent events, which is not the case here. The Poisson distribution is used to model rare events occurring over a large number of trials and is not appropriate for this scenario.

Question 16

Suppose that a judge's decisions follow a binomial distribution and that his verdict is incorrect 10% of the time.In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736.

Accepted Answer

Let's use the binomial distribution formula to calculate the probability of the judge making fewer than 2 incorrect verdicts in 10 decisions:
P(X<2) = P(X=0) + P(X=1)
P(X=0) = (0.9)^10 = 0.3487 (probability of making 0 incorrect verdicts in 10 decisions)
P(X=1) = 10C1 (0.9)^9 (0.1)^1 = 0.3874 (probability of making 1 incorrect verdict in 10 decisions)
P(X<2) = 0.3487 + 0.3874 = 0.7361
This matches the given probability of 0.736, so the answer is A)True.

Question 17

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts.Prices for 100 rats follow the following distribution:  How much should the lab budget for next year's rat orders be, assuming this distribution does not change?
A)$520
B)$637
C)$650
D)$780

Accepted Answer

The answer of A lab orders 100 rats a week...

Question 18

If  remains constant in a binomial distribution, an increase in n will not change the mean.

Accepted Answer

In a binomial distribution, the mean is given by μ = nπ, where n is the number of trials and π is the probability of success. Therefore, if n increases and π remains constant, the mean will also increase.

Question 19

On the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store.What type of probability distribution can be used to find out the probability that there will be no customer arriving at a checkout counter?
A)binomial distribution.
B)Poisson distribution.
C)hypergeometric distribution.
D)none of the above.

Accepted Answer

A Poisson distribution can be used to model the arrival of customers at a checkout counter. The Poisson distribution is used to model the number of occurrences of a certain event (in this case, the arrival of customers) in a fixed interval of time or space. The Poisson distribution is appropriate when the occurrences are random and independent, and the rate of occurrence is constant over the interval. In this case, we have a constant arrival rate of 1.8 customers per minute, and we want to find the probability of no customers arriving in a given time interval, so the Poisson distribution is the appropriate choice.

Question 20

A multiple-choice test has 30 questions.There are 4 choices for each question.A student who has not studied for the test decides to answer all questions randomly.What type of probability distribution can be used to figure out his chance of getting at least 20 questions right?
A)binomial distribution.
B)Poisson distribution.
C)hypergeometric distribution.
D)none of the above.

Accepted Answer

The scenario described fits the requirements of the binomial distribution, which has 2 outcomes (correct or incorrect) and a fixed number of trials (30 questions). The student either gets the question right or wrong with a probability of 0.25 (since there are 4 choices for each question and only 1 is correct). The question asks for the probability of getting at least 20 right, which can be calculated using the cumulative distribution function of the binomial distribution.

Quiz 5: Discrete Probability Distributions