Question 1

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

Accepted Answer

The answer of If π remains constant in a binomial...

Question 2

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

Accepted Answer

The diameters of bolts would not have a binomial distribution, as the outcomes are not independent and each bolt could have a different diameter. The diameter of one bolt would also not affect the diameter of another bolt.

Question 3

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

Accepted Answer

Poisson distribution is commonly used to model the number of events occurring in a fixed time interval, such as customers arriving at a store. Therefore, the number of customers arriving at a department store in a 5-minute period can be modeled using a Poisson distribution.

Question 4

If π 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), where n is the number of trials and p is the probability of success on each trial. If π, which is another notation for p, remains constant and n increases, then the variance will increase as well because the number of possible outcomes and the spread of those outcomes will be larger. Therefore, the statement is true.

Question 5

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: Price: $10.00 $12.50 $15.00
Probability: 0.35 0.40 0.25
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

To find the budget for next year's rat orders, we need to find the expected value of the price of 100 rats. 
Expected value = (Price1 x Probability1) + (Price2 x Probability2) + (Price3 x Probability3)

Expected value = (10 x 0.35) + (12.5 x 0.4) + (15 x 0.25) = $6.5

Therefore, the lab should budget for $6.5 x 100 = $650 per week.
 
So the answer is (C) $650.

Question 6

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average 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 distribution is given by the formula P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate (3 in this case) and k is the number of events (6 in this case). Plugging in these values gives approximately 0.05041.

Question 7

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 hypergeometric distribution is used when a sample is taken from a finite population without replacement, and the probability of success changes from trial to trial. In this case, the analyst is selecting a sample of 3 companies from a population of 60 start-ups, where only 3 of them have become highly successful. This matches the criteria for a hypergeometric distribution, since the analyst is taking a sample without replacement and the probability of success (i.e. becoming highly successful) changes as companies are selected. The binomial distribution assumes a fixed probability of success, which is not the case here, and the Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, which is not relevant to this scenario.

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 a Poisson distribution represents the average or expected number of events (in this case, tickets written) per unit of time (in this case, one day). Therefore, the statement "If we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day" is the most accurate interpretation of the value of the mean.

Question 9

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 but does not have an upper limit. Therefore, X can be arbitrarily large, and there is no maximum value n that it can take.

Question 10

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

Accepted Answer

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

Question 11

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

The Poisson distribution is used to model the number of occurrences of rare events over a fixed interval of time or space. In this case, the arrival of customers can be considered a rare event, and the rate of arrival (1.8 customers per minute) is constant over time. Therefore, the Poisson distribution can be used to find the probability that there will be no customer arriving at a checkout counter in a given minute.

Question 12

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 average 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 mean cost of rats used in lab experiments being $13.00 per week indicates the expected or average cost for all weekly rat purchases.

Question 13

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

Accepted Answer

In a Poisson distribution, the mean and variance are both equal to λ, which is the average number of occurrences in a given time or space interval.

Question 14

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

Accepted Answer

This is true. The mean of a probability distribution is calculated by taking the sum of the product of each possible outcome and its probability, so it can also be expressed as the expected value of the distribution.

Question 15

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 the sample is taken without replacement from a finite population. In this case, the population is composed of 9 females and 18 males, so the number of males selected follows a hypergeometric distribution.

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

We can use the binomial distribution formula to calculate the probability of the judge making fewer than 2 incorrect verdicts:

P(X < 2) = P(X = 0) + P(X = 1)

where X represents the number of incorrect verdicts in 10 decisions.

Using the binomial distribution formula, we get:

P(X = k) = (10 choose k) * (0.1)^k * (0.9)^(10-k)

P(X = 0) = (10 choose 0) * (0.1)^0 * (0.9)^10 = 0.3486784401
P(X = 1) = (10 choose 1) * (0.1)^1 * (0.9)^9 = 0.387420489

Thus,

P(X < 2) = 0.3486784401 + 0.387420489 = 0.7360989291

This matches the given probability, so the answer is True (A).

Question 17

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

Accepted Answer

The Poisson distribution is used to model a discrete random variable, not a continuous random variable. It is typically used to model events that occur at a fixed rate, such as the number of cars passing through a toll booth in an hour or the number of arrivals at a hospital in a day.

Question 18

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 does not have a binomial distribution since the probability of selecting a male changes after each selection without replacement.

Question 19

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

Accepted Answer

A binomial distribution requires that each trial has only two possible outcomes (success or failure) and the probability of success remains constant from trial to trial. In the case of customers arriving at a department store, there are more than two possible outcomes (e.g. some may buy something while others may not), and the probability of a customer arriving may not be constant over time (e.g. it may be higher during certain times of day or on certain days of the week). Therefore, a binomial distribution would not be appropriate to model this situation.

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

A binomial distribution can be used to model the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (in this case, either getting the question right or wrong) and the probability of success is the same for each trial. The conditions for using a binomial distribution are met in this scenario, so it is the appropriate choice.

Quiz 5: Discrete Probability Distributions